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Wikibilig - Хэрэглэгчийн оруулсан хувь нэмэр [mn]
2026-06-14T02:50:17Z
Хэрэглэгчийн оруулсан хувь нэмэр
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-10-11T04:59:50Z
<p>Angka: Category</p>
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'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ<sup>complex</sup> ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠭᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠥᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃ <br />
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|style = "height: 225px;"|ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠢᠢ ᠲᠥᠯᠦᠭᠡᠯᠡᠵᠦ ᠪᠤᠶ ᠠᠷᠭᠠᠨ᠋ᠳ᠋ ᠤᠨ<sup>Argand</sup> ᠳᠢᠶᠠᠭ᠋ᠷᠠᠮ ᠳᠡᠭᠡᠷ ᠡ᠋ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠸᠧᠺᠲ᠋ᠣᠷ<sup>vector</sup> ᠡᠭᠦᠰᠬᠡᠵᠦ ᠪᠤᠶ ᠬᠣᠣᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ <span style="writing-mode: horizontal-tb;"><i>(a, b)</i></span> ᠳᠦᠷᠰᠦᠯᠡᠨ ᠦᠵᠡᠭᠦᠯᠵᠦ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
ᠡᠭᠦᠨ ᠳ᠋ᠦ: <br><span style="writing-mode: horizontal-tb;"><i>Re</i></span> ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ᠂ <br><span style="writing-mode: horizontal-tb;"><i>Im</i></span> ᠢᠨᠦ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠊ᠠ᠋|ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ]] ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠣᠯᠬᠤ ᠠᠭᠠᠳ <br><span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠢᠨᠦ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠳ᠋ᠦ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠬᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠮᠥᠨ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
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<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
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|style = "height: 220px;"|ᠬᠣᠶᠠᠷ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃ <br />
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= ᠬᠠᠰᠤᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠭᠡ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
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<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math><br />
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= ᠦᠷᠡᠵᠢᠬᠦ =<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃<ref>''Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И.'' Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — <abbr>М.</abbr>: Просвещение, 1998. — 288 с. — <nowiki>ISBN 5-09-008036-4</nowiki>.</ref> <br />
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<math>(a + bi) (c + di) = </math> <math>= ac + bci - bd + adi = </math> <math>= (ac - bd) + (bc + ad)i </math> <br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|<span style="writing-mode: horizontal-tb;"><i>2 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭ ᠠ᠋ ᠢᠢ (ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) <span style="writing-mode: horizontal-tb;"><i>3 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭᠠᠨ ᠳᠤ (ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃<br />
ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠢ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠣᠷᠤᠶ ᠲᠠᠢ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠵᠤ᠂ <span style="writing-mode: horizontal-tb;"><i>√5</i></span> ᠢᠢᠠᠷ ᠰᠤᠩᠭᠠᠨ ᠠ᠋᠃ ᠡᠨᠡ ᠢᠨᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠾᠢᠫᠣᠲ᠋ᠧᠨᠦᠽ ᠤᠨ<sup>hypotenuse</sup> ᠤᠷᠲᠤ ᠪᠣᠯᠠᠢ᠃ <br />
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= ᠬᠤᠪᠢᠶᠠᠬᠤ =<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠬᠤᠪᠢᠶᠠᠬᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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<math>\frac{a+bi}{c+di}=</math> <br />
<math>=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=</math> <br />
<math>=\frac{ac+bd}{c^2+d^2}+\left(\frac{bc-ad}{c^2+d^2}\right)i.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ<sup>quadrate</sup> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨᠡ᠋᠃<ref>Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01</ref><br />
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<math>(x+yi)^2=x^2-y^2 + 2xyi.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> <span style="writing-mode: horizontal-tb;"><i>(b ≠ 0)</i></span> ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ <br />
<math> \pm (\gamma + \delta i)</math> ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄<br />
<math>\gamma = \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠠ <math>\delta = (\sgn b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠣᠯᠤᠮᠤᠢ᠃<br />
<span style="writing-mode: horizontal-tb;"><i>sgn b</i></span> ᠭᠡᠭᠴᠢ ᠢᠨᠦ <math>\sgn b = \begin{cases} \ \ 1, & b > 0 \\ \ \ 0, & b = 0 \\ -1, & b < 0 \end{cases}</math> ᠭᠡᠵᠦ ᠲᠠᠢᠢᠯᠠᠭᠳᠠᠨᠠ᠃ <br />
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ᠡᠭᠦᠨ ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ ᠢᠢᠨ ᠲᠤᠬᠠᠢᠢᠲᠠ ᠳ᠋ᠤ <math> \pm (\gamma + \delta i)</math> ᠲᠣᠭᠠ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠦ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠲᠣᠭᠠ ᠢᠢ ᠭᠠᠷᠭᠠᠵᠤ ᠢᠷᠡᠬᠦ ᠶᠣᠰᠤᠲᠠᠢ᠃ <br />
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=ᠬᠡᠷᠡᠭᠯᠡᠭᠡ=<br />
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== ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ==<br />
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|style = "height: 225px;"|ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠳᠠᠪᠲᠠᠯᠲᠠ ᠪᠠᠷ ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠠᠢᠢᠭᠤᠯᠬᠤ᠃<br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠯ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠳ᠋ᠤ ᠡᠭᠦᠰᠬᠡᠳᠡᠭ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ<sup>fractal</sup> ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠵᠡᠰᠱᠢᠶ ᠡ᠋ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡ ᠬᠦ ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ <span style="writing-mode: horizontal-tb;"><i>z = 0</i></span> ᠡᠴᠡ ᠡᠬᠢᠯᠡᠭᠡᠳ ᠢᠲ᠋ᠧᠷᠠᠼ<sup>iteration</sup> ᠬᠢᠬᠦ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;"><i>f<sub>c</sub>(z)=z<sup>2</sup>+c</i></span> ᠹᠦᠨ᠍ᠻᠼ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠢᠢᠬᠤ ᠨᠥᠬᠦᠴᠡᠯ ᠪᠦᠬᠦᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠤ ᠣᠯᠠᠨᠯᠢᠬ ᠪᠠᠢᠢᠳᠠᠭ᠃ <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ ᠪᠠᠢᠢᠭᠠᠯᠢ ᠢᠢᠨ ᠭᠠᠢᠢᠬᠠᠮᠰᠢᠭᠲᠤ ᠳᠦᠷᠰᠦ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠦᠨ ᠰᠠᠢᠢᠬᠠᠨ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠶᠡᠷᠦ ᠪᠤᠰᠤ ᠢᠢᠨ ᠭᠣᠶᠤᠮᠰᠠᠭ ᠬᠣᠣᠰᠯᠠᠯ ᠢ ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠳᠡᠭ᠃<ref>ᠷᠡᠨᠴᠡᠨ ᠦ ᠡᠩᠬᠡᠪᠠᠲᠤ᠃ '''ᠫᠢᠲ᠋ᠾᠠᠭᠣᠷ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌'''᠃ ᠮᠣᠩᠭᠤᠯ ᠤᠯᠤᠯᠰ ᠊ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠪᠠ ᠲᠣᠭᠠᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠢᠢᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ https://imdt.ac.mn/c/1013874?content=1150891&fbclid=IwAR1HVqeXwT-h7dijj9EeIUzDvFustH99GAr9LHYQIU61XIByuBHCkSm10So 2020</ref> <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠵᠠᠭᠠᠭ ᠢ ᠳᠠᠪᠠᠬᠤ ᠳᠤ ᠪᠠᠨ ᠵᠦᠯᠢᠶ ᠠ᠋ ᠢᠢᠨ<sup>Julia</sup> ᠹᠷᠠᠻᠲᠠᠯ ᠤᠳ ᠦᠷᠭᠡᠯᠵᠢ ᠬᠣᠯᠪᠤᠯᠲᠠ ᠪᠠᠨ ᠠᠯᠳᠠᠵᠤ᠂ ᠹᠠᠲᠣᠤ ᠢᠢᠨ<sup>Fatou</sup> ᠲᠣᠭᠤᠰᠤ ᠪᠣᠯᠤᠨ ᠬᠦᠪᠢᠷᠠᠳᠠᠭ᠃<ref>ᠨᠡᠭᠡᠭᠡᠯᠲᠡᠲᠡᠢ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠳ᠃ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠳᠠᠯ᠃ ᠪᠢᠳᠡᠨ ᠦ ᠡᠷᠭᠢᠨ ᠲᠣᠭᠣᠷᠢᠨ ᠳ᠋ᠠᠬᠢ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠬᠡᠷᠬᠢᠨ ᠠᠵᠢᠯᠯᠠᠳᠠᠭ ᠪᠤᠢ᠃ ᠹᠷᠠᠻᠲᠯᠠ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠶᠢᠨ ᠹᠷᠠᠻᠲᠯᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢ᠌ᠭ᠌᠃ https://ultrait.ru/mn/smartphones/otkrytie-fraktalov-beskonechnost-fraktalov-kak-ustroen-mir.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/10/02</ref><br />
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== ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ==<br />
[[Квант механик|ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌]]<sup>quantum mechanics</sup> ᠠᠲ᠋ᠣᠮ ᠠᠴᠠ ᠵᠢᠵᠢᠭ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ᠂ ᠶᠠᠭᠤᠨ ᠤ ᠡᠮᠦᠨᠡ ᠪᠣᠽᠣᠨ<sup>boson</sup> (ᠵᠢᠱᠢᠶᠡ ᠨᠢ ᠹᠣᠲ᠋ᠣᠨ<sup>photon</sup> ᠪᠠ ᠹᠧᠷᠮᠢᠣᠨ ᠤ<sup>fermion</sup> (ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠨᠧᠦᠲ᠋ᠷᠣᠨ) ᠬᠥᠳᠡᠯᠭᠡᠭᠡᠨ᠂ ᠬᠠᠷᠢᠯᠴᠠᠨ ᠦᠢᠯᠡᠴᠢᠯᠡᠯ ᠢ ᠰᠤᠳᠤᠯᠳᠠᠭ᠃ <br />
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ᠴᠢᠩᠭᠢᠬᠦ ᠳᠦ ᠪᠡᠨ ᠲᠡᠳᠡᠭᠡᠷ ᠦᠨ ᠦᠢᠯᠡ ᠬᠥᠳᠡᠯᠦᠯ ᠦᠨ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭ᠌ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠯᠲᠠ ᠶᠢ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠤᠳᠬᠠ ᠪᠠᠷ ᠭᠠᠷᠭᠠᠳᠠᠭ᠃ <br />
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ᠴᠤᠬᠤᠮ ᠳᠠᠭᠠᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠦᠨᠳᠦᠰᠦ ᠰᠠᠭᠤᠷᠢ ᠪᠣᠯᠳᠠᠭ᠃ <br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠲᠤ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ <sup>Schrödinger</sup> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠠᠴᠢ ᠬᠣᠯᠪᠣᠭᠳᠠᠯ ᠨᠢ ᠰᠣᠩᠭᠣᠳᠠᠭ ᠹᠢᠽᠢᠻ ᠳ᠋ᠡᠬᠢ ᠨᠧᠧᠲ᠋ᠣᠨ᠍ ᠤ ᠬᠣᠶᠠᠳᠤᠭᠠᠷ ᠬᠠᠤᠯᠢ ᠶᠢᠨ ᠠᠳᠠᠯᠢ ᠶᠤᠮ᠃ <br />
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ᠠᠯᠢ ᠠᠯᠢ ᠨᠢ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠷᠢᠯᠠᠯ᠂ ᠢᠮᠫᠦ᠋ᠯᠰ ᠢ <sup>impuls</sup> ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭᠴᠢᠯᠠᠨ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠳᠠᠭ᠃ <br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠨᠢ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ<sup>function</sup> ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠬᠦ ᠳᠦ ᠲᠣᠬᠢᠷᠠᠮᠵᠢ ᠲᠠᠢ ᠤᠴᠢᠷ ᠠᠴᠠ ᠹᠢᠽᠢᠻ ᠦᠨ ᠡᠨᠡ ᠰᠠᠯᠪᠤᠷᠢ ᠳᠤ ᠵᠠᠶ᠋ᠢᠯᠠᠰᠢ ᠦᠭᠡᠢ ᠱᠠᠭᠠᠷᠳᠠᠯᠭᠠ ᠲᠠᠢ ᠶᠤᠮ᠃ <br />
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ᠴᠠᠭᠠᠰᠢᠯᠠᠪᠠᠯ᠂ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠢᠭᠤᠳ ᠨᠥᠯᠥᠭᠡ ᠪᠡᠷ ᠬᠢᠮᠢ ᠶᠢᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠬᠥᠭᠵᠢᠯ ᠡᠷᠴᠢᠮᠵᠢᠭᠰᠡᠨ᠃ <br />
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1927 ᠣᠨ ᠳᠤ ᠸᠠᠯᠲ᠋ᠧᠷ ᠾᠠᠢᠢᠲ᠋ᠯᠧᠷ<sup>Walter Heitler</sup> ᠂ ᠹᠷᠢᠼ ᠯᠣᠨ᠍ᠳᠣᠨ<sup>Fritz London</sup> ᠨᠠᠷ ᠸᠠᠯᠧᠨ᠍ᠲ ᠤᠨ<sup>valency</sup> ᠬᠣᠯᠪᠣᠭᠠᠰᠤ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠲᠣᠮᠢᠶᠠᠯᠠᠭᠰᠠᠨ᠃ <br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠨᠢᠭᠡ ᠭᠣᠣᠯ ᠠᠰᠠᠭᠤᠳᠠᠯ ᠪᠣᠯ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ ᠦᠨ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ ᠤᠯᠬᠤ ᠶᠠᠪᠤᠳᠠᠯ ᠶᠤᠮ᠃ <br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠭᠡᠳᠡᠭ ᠨᠢ ᠲᠣᠳᠣᠷᠬᠠᠢ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠵᠤᠷᠪᠤᠰ ᠲᠤ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠵᠤ ᠪᠣᠯᠬᠤ ᠪᠠᠶ᠋ᠢᠷᠢᠰᠢᠯ ᠤᠳ ᠤᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠲᠠᠷᠬᠠᠯᠲᠠ ᠶᠤᠮ᠄ <br />
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<small><math>\frac{-\hbar}{2m}\frac{\partial^2\Psi(x,t)}{\partial x^2} + V(x,t)\Psi(x,t) = i\hbar\frac{\partial \Psi(x,t)}{\partial t}</math></small><br />
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ᠻᠢᠨᠧᠲᠢᠭ<sup>kinetic</sup> ᠧᠨᠧᠷᠭᠢ<sup>energy</sup><br />
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ᠫᠣᠲ᠋ᠧᠨ᠍ᠼᠢᠶᠠᠯ<sup>potencial</sup> ᠧᠨᠧᠷᠭᠢ<br />
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ᠨᠡᠢᠢᠲᠡ ᠧᠨᠧᠷᠭᠢ<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠲᠦ ᠰᠢᠲᠦᠭᠰᠡᠨ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠠᠭᠤᠷᠢ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠨᠢᠭᠡ ᠪᠣᠯ ᠳᠡᠭᠡᠷᠡ ᠳᠤᠷᠠᠳᠤᠭᠰᠠᠨ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡᠬᠦ ᠲᠣᠮᠢᠶᠠ ᠶᠢ ᠠᠰᠢᠭᠯᠠᠭᠰᠠᠨ ᠢᠶ᠋ᠠᠷ ᠤᠰᠤ ᠲᠥᠷᠦᠭᠴᠢ ᠶᠢᠨ ᠮᠣᠯᠧᠻᠤᠯ<sup>molecule</sup> ᠳᠠᠬᠢ ᠬᠣᠶᠠᠷ ᠠᠲ᠋ᠣᠮ ᠨᠢ ᠻᠣᠸᠠᠯᠧᠨ᠍ᠲ<sup>covalent</sup> ᠬᠣᠯᠪᠣᠭᠠ ᠭᠡᠳᠡᠭ ᠢ ᠡᠭᠦᠰᠬᠡᠨ ᠧᠯᠧᠻᠲ᠋ᠷᠣᠨ ᠨᠤᠭᠤᠳ ᠢᠢᠠᠨ ᠬᠤᠪᠢᠶᠠᠯᠴᠠᠵᠤ ᠪᠠᠶ᠋ᠢᠳᠠᠭ ᠢ ᠪᠠᠲᠤᠯᠠᠭᠰᠠᠨ᠃<ref>Josiah Wu. Real Life Applications of Complex Numbers. 2020 https://issuu.com/harrowhongkong/docs/final_scientific_harrovian_issue_vi-i/s/11488755</ref><br />
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== ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠢᠢᠨ ᠣᠨᠤᠯ ==<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠶᠢ ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ᠂ ᠢᠯᠠᠩᠭᠤᠶᠠ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠨ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠶ᠋ᠢᠳᠠᠯ ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠭᠡᠨ ᠳᠤ ᠬᠡᠷᠡᠭᠯᠡᠳᠡᠭ᠃ <br />
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ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠳᠤ 《ᠰᠢᠰᠲ᠋ᠧᠮ》 ᠭᠡᠳᠡᠭ ᠦᠭᠡ ᠶᠢ ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ ᠪᠥᠭᠡᠳ ᠡᠨᠡ ᠨᠢ ᠵᠠᠪᠠᠯ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠢ ᠬᠡᠯᠡᠳᠡᠭ ᠦᠭᠡᠢ᠃ <br />
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ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠡᠭᠦᠨ ᠢ ᠬᠥᠷᠥᠩᠭᠡ ᠶᠢᠨ ᠵᠠᠬᠠ ᠵᠡᠭᠡᠯᠢ ᠳ᠋ᠡᠬᠢ ᠥᠭᠡᠷᠡᠴᠢᠯᠡᠯᠲᠡ᠂ ᠬᠢᠮᠢ ᠶᠢᠨ ᠤᠷᠪᠠᠯ᠂ ᠦᠢᠯᠡ ᠶᠠᠪᠤᠴᠠ ᠶᠢ ᠰᠤᠳᠤᠯᠬᠤ ᠳᠤ ᠬᠡᠷᠡᠭᠯᠡᠵᠦ ᠪᠣᠯᠤᠨᠠ᠃ <br />
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ᠲᠡᠭᠦᠨᠴᠢᠯᠡᠨ ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠷᠣᠪᠣᠲ ᠤᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠳᠤ<sup>technology</sup> ᠥᠷᠭᠡᠨ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ ᠲᠤᠯᠠ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠠᠨᠠᠯᠢᠽ ᠴᠤ ᠪᠠᠰᠠ ᠲᠡᠭᠦᠨ ᠳᠦ ᠬᠡᠷᠡᠭᠯᠡᠭᠳᠡᠨᠡ ᠭᠡᠰᠡᠨ ᠦᠭᠡ᠃<ref>Ujjvala Y. Gawarguru, Mitali K. Tibdewal, Rajashri A. Naphade, Rahul M. Jethwani. The Review of Introduction & Application of Complex Number in Engineering. 2nd National Conference Recent Innovations in Science and Engineering (NC-RISE 17). Volume: 5 Issue: 9. pp55 – 57. ISSN: 2321-8169. https://ijritcc.org/download/conferences/NC-RISE_17/Track_6_(ASH)/1506931102_02-10-2017.pdf</ref><br />
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ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠳᠤ ᠵᠠᠷᠢᠮ ᠳᠤ ᠪᠠᠨ ᠯᠠᠫ᠊ᠯᠠᠰ ᠤᠨ<sup>Laplace</sup> ᠬᠤᠪᠢᠷᠠᠭᠠᠯᠲᠠ ᠶᠢ ᠠᠰᠢᠭᠯᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠳ ᠢ<sup>system</sup> ᠴᠠᠭ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠮᠤᠵᠢ ᠠᠴᠠ ᠳᠠᠪᠲᠠᠮᠵᠢ ᠶᠢᠨ ᠮᠤᠵᠢ ᠳᠤ ᠰᠢᠯᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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ᠳᠠᠷᠠᠭᠠ ᠨᠢ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠨ ᠲᠡᠭ ᠪᠠ ᠲᠤᠶ᠋ᠢᠯ ᠤᠳ ᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠰᠢᠨᠵᠢᠯᠡᠨᠡ᠃<br />
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ᠢᠵᠠᠭᠤᠷ ᠤᠨ ᠮᠥᠷ ᠦᠨ (root locus) ᠠᠷᠭᠠ ᠪᠠᠷ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠳᠦ ᠲᠡᠭ ᠪᠠ ᠲᠤᠶ᠋ᠢᠯ ᠤᠳ ᠵᠡᠭᠦᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠠᠶ᠋ᠢᠨᠠ ᠤᠤ᠂ ᠡᠰᠡᠪᠡᠯ ᠪᠠᠷᠠᠭᠤᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠠᠶ᠋ᠢᠨᠠ ᠤᠤ ᠭᠡᠳᠡᠭ ᠨᠢ ᠴᠢᠬᠤᠯᠠ᠃ <br />
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ᠥᠭᠡᠷᠡ ᠪᠡᠷ ᠬᠡᠯᠡᠪᠡᠯ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠨᠢ ᠲᠡᠭ ᠡᠴᠡ ᠶᠡᠬᠡ ᠪᠠᠢᠢᠨᠠ ᠤᠤ ᠡᠰᠡᠪᠡᠯ ᠪᠠᠭᠠ ᠪᠠᠶ᠋ᠢᠨᠠ ᠤᠤ ᠭᠡᠳᠡᠭ ᠨᠢ ᠴᠢᠬᠤᠯᠠ ᠶᠤᠮ᠃ <br />
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ᠴᠠᠭ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠬᠤᠪᠢᠷᠠᠯᠲᠠ ᠦᠭᠡᠢ ᠱᠤᠭᠤᠮᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠨᠢ᠄<br />
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- ᠪᠠᠷᠠᠭᠤᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠲᠤᠶ᠋ᠢᠯ ᠲᠠᠢ ᠪᠣᠯ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠂<br />
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- ᠵᠡᠭᠦᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠦᠬᠦ ᠲᠤᠶ᠋ᠢᠯ ᠨᠢ ᠪᠠᠶ᠋ᠢᠪᠠᠯ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠂<br />
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- ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠲᠤᠶ᠋ᠢᠯ ᠲᠠᠢ ᠪᠣᠯ ᠶᠠᠯᠢ ᠦᠭᠡᠶ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠃<br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ - complex number - комплексное число<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ - imaginary unit - мнимая единица <br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭᠠ - imaginary number - чисто мнимое число<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ - imaginaty part - мнимая часть<br />
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ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠ - real number - действительное число<br />
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ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ - real part - действительная часть<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ - complex plane - комлексная плоскость<br />
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ᠲᠡᠩᠬᠡᠯᠢᠭ - axis - ось<br />
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ᠣᠯᠠᠨᠯᠢᠭ - set - множество<br />
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ᠡᠶᠡᠷᠭᠦ - positive - положительный<br />
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ᠰᠥᠷᠭᠦ - negative - отрицательный<br />
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ᠲᠡᠭᠰᠢᠳᠬᠡᠯ - equation - уравнение<br />
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ᠲᠣᠮᠢᠶ ᠠ᠋ - formula - формула<br />
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ᠡᠷᠭᠢᠯᠲᠡ - rotation - поворот<br />
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ᠮᠠᠭᠠᠳᠯᠠᠯ - probability - вероятность<br />
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ᠬᠡᠮ - degree - градус<br />
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ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠢᠢᠨ ᠣᠨᠤᠯ - control theory - теория управления<br />
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ᠠᠲ᠋ᠣᠮ - atom - атом<br />
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ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮ - elementary particle - элементарная частица<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ - wave function - волновая функция<br />
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ᠳᠠᠪᠲᠠᠮᠵᠢ - frequency - частота<br />
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ᠹᠷᠠᠻᠲ᠋ᠠᠯ - fractal - фрактал<br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ - quantum mechanics - квантовая механика<br />
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ᠰᠣᠩᠭᠣᠳᠠᠭ ᠹᠢᠽᠢᠻ - classical physics - классическая физика<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==<br />
<div style="writing-mode: horizontal-tb; "><References/><br />
[[Ангилал:ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ]]</div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%AC%E1%A0%A0%E1%A0%AD%E1%A0%A4%E1%A0%B7%E1%A0%AE%E1%A0%A0%E1%A0%AD_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%8A%E1%A0%A0%E1%A0%8B&diff=2759
ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠊ᠠ᠋
2021-10-11T04:59:14Z
<p>Angka: Category</p>
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ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠢᠨᠦ ᠲᠡᠭ <span style="writing-mode: horizontal-tb;"><i>(0)</i></span> ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ [[ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ]] ᠢᠢ '''ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋''' ᠬᠠᠮᠠᠮᠦᠢ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠺᠸᠠᠲᠷᠠᠲ<sup>quadrate</sup> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠬᠠᠰᠤᠬᠤ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠬᠠᠷᠢᠭᠤ ᠭᠠᠷᠤᠨ ᠠ᠋᠃<ref name=":0">https://www.livescience.com/42748-imaginary-numbers.html Live Science. Elaine J. Hom. What Are Imaginary Numbers? 2014/01/22. </ref> <br />
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ᠲᠡᠭᠡᠬᠦ ᠪᠡᠷ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠠᠨᠤ ᠬᠠᠰᠤᠬᠤ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠶ ᠲᠣᠭᠠᠨ ᠤ ᠬᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠮᠥᠨ᠃ <br />
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ᠲᠡᠭ <span style="writing-mode: horizontal-tb;"><i>(0)</i></span> ᠲᠣᠭ ᠠ᠋ ᠪᠣᠳᠠᠲᠤ ᠪᠤᠶᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠠᠯᠢ ᠨᠢ ᠴᠤ ᠪᠠᠢᠢᠵᠤ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
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|[[File:Rotations_on_the_complex_plane.svg|thumb]]<br />
|style = "height: 150px;"|ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭᠠ ᠪᠦᠬᠦᠢ ᠲᠣᠭᠠᠴᠠᠭᠠᠯᠠᠯ ᠤᠨ ᠳᠦᠷᠰᠦᠯᠡᠯ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃<br />
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ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠳᠦᠷᠰᠦᠯᠡᠯ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;"><i>–1</i></span> ᠢᠢᠡᠷ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ᠢ ᠲᠣᠭᠤᠷᠢᠭᠰᠠᠨ <span style="writing-mode: horizontal-tb;"><i>180</i></span> ᠬᠡᠮ ᠦᠨ ᠡᠷᠭᠢᠯᠲᠡ ᠪᠣᠯᠤᠨ ᠠ᠋᠃ <br />
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<span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠲᠣᠭ ᠠ᠋ ᠪᠡᠷ ᠦᠷᠡᠵᠢᠯᠭᠦᠯᠬᠦ ᠢᠨᠦ ᠨᠠᠷᠠ ᠪᠤᠷᠤᠭᠤ “ᠡᠶᠡᠷᠭᠦ” ᠴᠢᠭ᠌ ᠲᠦ <span style="writing-mode: horizontal-tb;"><i>90</i></span> ᠬᠡᠮ ᠦᠨ ᠡᠷᠭᠢᠯᠲᠡ ᠪᠣᠯᠬᠤ ᠪᠠ ᠬᠡᠷᠪᠡ <span style="writing-mode: horizontal-tb;"><i>i2 = -1</i></span> ᠭᠡᠰᠡᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢᠨᠦ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ᠢ ᠲᠣᠭᠤᠷᠢᠭᠰᠠᠨ <span style="writing-mode: horizontal-tb;"><i>90</i></span> ᠬᠡᠮ ᠦᠨ ᠬᠣᠶᠠᠷ ᠡᠷᠭᠢᠯᠲᠡ ᠭᠡᠵᠦ ᠦᠵᠡᠪᠡᠯ ᠦᠷ ᠡ᠋ ᠳ᠋ᠦ᠋ᠩ ᠢᠨᠦ ᠭᠠᠭᠴᠠ <span style="writing-mode: horizontal-tb;"><i>180</i></span> ᠬᠡᠮ ᠦᠨ ᠡᠷᠭᠢᠯᠲᠡ ᠪᠣᠯᠤᠨ ᠠ᠋᠃ <br />
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“ᠰᠥᠷᠭᠦ” (ᠨᠠᠷᠠ ᠵᠥᠪ) ᠴᠢᠭ᠌ ᠲᠦ <span style="writing-mode: horizontal-tb;"><i>90</i></span> ᠬᠡᠮ ᠡᠷᠭᠢᠬᠦ ᠢᠨᠦ ᠮᠥᠨ ᠡᠨᠡ ᠳᠦᠷᠰᠦᠯᠡᠯ ᠳ᠋ᠦ ᠨᠡᠶ᠋ᠢᠴᠡᠬᠦ ᠪᠠ <span style="writing-mode: horizontal-tb;"><i>x2 = -1</i></span> ᠭᠡᠰᠡᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠶ᠋ᠢᠳ <span style="writing-mode: horizontal-tb;"><i>−i</i></span> ᠪᠣᠯᠬᠤ ᠢ ᠬᠠᠷᠠᠭᠤᠯᠵᠤ ᠪᠠᠢᠢᠨ ᠠ᠋᠃ <br />
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ᠶᠡᠷᠦᠩᠬᠡᠢᠢ ᠳ᠋ᠦ ᠪᠡᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ ᠦᠷᠡᠵᠢᠯᠭᠦᠯᠬᠦ ᠢᠨᠦ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ᠢ ᠲᠣᠭᠤᠷᠢᠭᠤᠯᠤᠨ ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭᠠᠨ ᠤ ᠠᠷᠭᠤᠮᠧᠨ᠍ᠲ<sup>argument</sup> ᠢᠢᠠᠷ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ᠂ ᠴᠠᠷ ᠬᠡᠮᠵᠢᠶ ᠡ᠋ ᠪᠡᠷ ᠢᠨᠦ ᠤᠷᠲᠤ ᠢᠢ ᠴᠢᠴᠠᠤᠯᠠᠬᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠪᠣᠯᠤᠨᠠ᠃ <br />
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ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭᠠᠨ ᠤ ᠠᠷᠭᠤᠮᠧᠨ᠍ᠲ ᠭᠡᠳᠡᠭ ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠡᠬᠢᠯᠠᠯ ᠴᠡᠭ ᠢ ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭᠠ z ᠲᠡᠢ ᠬᠣᠯᠪᠤᠵᠤ ᠪᠠᠢᠢᠭᠠ ᠱᠤᠭᠤᠮ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠢᠢᠨ ᠥᠨᠴᠦᠭ <span style="writing-mode: horizontal-tb;"><i>φ</i></span> ᠶᠤᠮ᠃<br />
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{| class="wikitable"<br />
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|style = "height: 150px;"|ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭᠠ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ<sup>complex</sup> ᠲᠣᠭᠠ ᠢᠢᠨ ᠠᠷᠭᠤᠮᠧᠨ᠍ᠲ<sup>argument</sup> ᠨᠢ ᠥᠨᠴᠦᠭ <span style="writing-mode: horizontal-tb;"><i>φ</i></span>᠃<br />
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==ᠬᠡᠷᠡᠭᠯᠡᠭᠡ==<br />
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[[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠬᠦ ᠠᠭᠠᠳ ᠧᠯᠧᠺᠲ᠋ᠷᠣᠨᠢᠭ᠌ ᠲᠦ<sup>electronic</sup> <span style="writing-mode: horizontal-tb;"><i>j</i></span> ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠨ ᠡ᠋᠃ <br />
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ᠡᠨᠡ ᠢᠨᠦ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠳᠡᠭ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠭᠦᠶᠦᠳᠡᠯ ᠡᠴᠡ ᠢᠯᠭᠠᠬᠤ ᠱᠠᠭᠠᠷᠳᠠᠯᠭ ᠠ᠋ ᠡᠴᠡ ᠤᠷᠭᠤᠨ ᠭᠠᠷᠤᠭᠰᠠᠨ ᠤᠯᠠᠮᠵᠢᠯᠠᠯ ᠶᠤᠮ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠧᠯᠧᠺᠲ᠋ᠷᠣᠨᠢᠭ᠌ ᠲᠦ᠂ ᠢᠯᠠᠩᠭᠤᠶ ᠠ᠋ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠡᠯ ᠳ᠋ᠦ ᠣᠨᠴᠤᠭᠤᠢ ᠠᠴᠢ ᠬᠣᠯᠪᠤᠭᠳᠠᠯ ᠲᠠᠢ᠃ <br />
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ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠡᠯ ᠰᠢᠨᠦ᠋ᠰ<sup>sinus</sup> ᠳᠣᠯᠭᠢᠶᠠᠨ ᠳᠤ ᠨᠡᠮᠡᠬᠦ ᠪᠠ ᠬᠠᠰᠤᠬᠤ ᠢᠢᠨ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠰᠣᠯᠢᠭᠳᠠᠳᠠᠭ᠃<ref>https://www.electricaltechnology.org/2013/07/active-reactive-apparent-and-complex.html Electrical Technology. Active, Reactive, Apparent and Complex Power. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/07/31</ref><br />
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|style = "height: 150px;"|ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠡᠯ ᠪᠦᠬᠦᠢ ᠡᠷᠴᠢᠮ ᠬᠦᠴᠦ᠃ <span style="writing-mode: horizontal-tb;"><i>P</i></span> ᠢᠨᠦ ᠢᠳᠡᠪᠬᠢᠲᠦ ᠴᠢᠳᠠᠯ᠂ <span style="writing-mode: horizontal-tb;"><i>Q</i></span> ᠢᠨᠦ ᠬᠠᠷᠢᠭᠤ ᠢᠳᠡᠪᠬᠢᠲᠦ ᠴᠢᠳᠠᠯ᠂ <span style="writing-mode: horizontal-tb;"><i>S</i></span> ᠨᠡᠶ᠋ᠢᠯᠡᠪᠦᠷᠢ ᠴᠢᠳᠠᠯ᠃<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠢ ᠶᠠᠪ ᠴᠠᠪ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠬᠤ ᠬᠡᠴᠡᠭᠦᠦ ᠲᠤᠯᠠ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠡᠯ ᠦᠳ ᠢ ᠳᠠᠪᠬᠤᠷᠯᠠᠬᠤ ᠨᠡᠨ ᠲᠦᠪᠡᠭ ᠲᠡᠢ᠃<br />
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ᠬᠠᠷᠢᠨ ᠲᠥᠰᠦᠭᠡᠯᠡᠯ ᠭᠦᠶᠦᠳᠡᠯ ᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠨᠤᠭᠤᠳ ᠲᠠᠶ ᠬᠣᠣᠰᠮᠠᠭᠯᠠᠭᠤᠯᠤᠭᠰᠠᠨ ᠢᠢᠠᠷ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠡᠯ ᠦᠨ ᠲᠣᠭᠠᠴᠠᠭ ᠠ᠋ ᠬᠢᠵᠦ᠂ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠤ ᠣᠰᠤᠯ ᠡᠴᠡ ᠤᠷᠢᠳᠤᠴᠢᠯᠠᠨ ᠰᠡᠷᠭᠡᠢᠢᠯᠡᠬᠦ ᠲᠤᠰᠠ ᠲᠠᠢ ᠪᠠᠢᠢᠳᠠᠭ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠮᠥᠨ ᠰᠢᠭ᠌ᠨᠠᠯ<sup>signal</sup> ᠪᠣᠯᠪᠠᠰᠤᠷᠠᠭᠤᠯᠬᠤ ᠲᠧᠺᠨᠣᠯᠣᠭᠢ ᠳ᠋ᠤ<sup>technology</sup> ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ <br />
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ᠡᠨᠡ ᠨᠢ ᠡᠭᠦᠷ ᠦᠨ ᠬᠣᠯᠪᠤᠭ ᠠ᠋᠂ ᠤᠲᠠᠰᠤ ᠦᠭᠡᠢ ᠳᠠᠮᠵᠢᠭᠤᠯᠤᠯᠲᠠ᠂ ᠷᠠᠳᠠᠷ<sup>radar</sup>᠂ ᠪᠢᠣᠯᠣᠭᠢ<sup>biology</sup> (ᠲᠠᠷᠢᠬᠢᠨ ᠤ ᠳᠣᠯᠭᠢᠶᠠᠨ) ᠰᠤᠳᠤᠯᠬᠤ ᠳ᠋ᠤ ᠬᠠᠮᠢᠶᠠᠷᠤᠨ ᠠ᠋᠃<br />
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ᠶᠡᠷᠦ᠂ ᠰᠢᠨᠦ᠋ᠰ ᠪᠠ ᠺᠣᠰᠢᠨᠦ᠋ᠰ<sup>cosinus</sup> ᠳᠣᠯᠭᠢᠶᠠᠨ ᠪᠦᠬᠦᠢ ᠵᠦᠢᠯ ᠢ ᠬᠡᠮᠵᠢᠬᠦ᠂ ᠲᠣᠭᠠᠴᠠᠭᠠᠯᠬᠤ ᠳ᠋ᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠴᠢᠬᠤᠯᠠ ᠬᠡᠷᠡᠭ ᠲᠡᠢ᠃<ref name=":0" /><br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠ - real number - действительное число<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ - imaginary unit - мнимая единица <br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭᠠ - imaginary number - чисто мнимое число<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ - complex number - комплексное число<br />
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ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ - real part - действительная часть<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ - imaginaty part - мнимая часть<br />
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ᠲᠡᠩᠬᠡᠯᠢᠭ - axis - ось<br />
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ᠣᠯᠠᠨᠯᠢᠭ - set - множество<br />
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ᠡᠶᠡᠷᠭᠦ - positive - положительный<br />
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ᠰᠥᠷᠭᠦ - negative - отрицательный<br />
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ᠲᠡᠭᠰᠢᠳᠬᠡᠯ - equation - уравнение<br />
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ᠡᠷᠭᠢᠯᠲᠡ - rotation - поворот<br />
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ᠬᠡᠮ - degree - градус<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==<br />
<div style="writing-mode: horizontal-tb; "><References/><br />
[[Ангилал:ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ]]</div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%BA%E1%A0%A3%E1%A0%AE%E1%A0%AB%E1%A0%8A%E1%A0%AF%E1%A0%A7%E1%A0%BA%E1%A0%B0_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%A0&diff=2758
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-10-11T03:18:07Z
<p>Angka: Link</p>
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'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ<sup>complex</sup> ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠭᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠥᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃ <br />
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|style = "height: 225px;"|ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠢᠢ ᠲᠥᠯᠦᠭᠡᠯᠡᠵᠦ ᠪᠤᠶ ᠠᠷᠭᠠᠨ᠋ᠳ᠋ ᠤᠨ<sup>Argand</sup> ᠳᠢᠶᠠᠭ᠋ᠷᠠᠮ ᠳᠡᠭᠡᠷ ᠡ᠋ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠸᠧᠺᠲ᠋ᠣᠷ<sup>vector</sup> ᠡᠭᠦᠰᠬᠡᠵᠦ ᠪᠤᠶ ᠬᠣᠣᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ <span style="writing-mode: horizontal-tb;"><i>(a, b)</i></span> ᠳᠦᠷᠰᠦᠯᠡᠨ ᠦᠵᠡᠭᠦᠯᠵᠦ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
ᠡᠭᠦᠨ ᠳ᠋ᠦ: <br><span style="writing-mode: horizontal-tb;"><i>Re</i></span> ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ᠂ <br><span style="writing-mode: horizontal-tb;"><i>Im</i></span> ᠢᠨᠦ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠊ᠠ᠋|ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ]] ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠣᠯᠬᠤ ᠠᠭᠠᠳ <br><span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠢᠨᠦ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠳ᠋ᠦ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠬᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠮᠥᠨ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
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<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
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{| class="wikitable"<br />
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|style = "height: 220px;"|ᠬᠣᠶᠠᠷ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃ <br />
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= ᠬᠠᠰᠤᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠭᠡ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
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<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math><br />
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= ᠦᠷᠡᠵᠢᠬᠦ =<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃<ref>''Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И.'' Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — <abbr>М.</abbr>: Просвещение, 1998. — 288 с. — <nowiki>ISBN 5-09-008036-4</nowiki>.</ref> <br />
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<math>(a + bi) (c + di) = </math> <math>= ac + bci - bd + adi = </math> <math>= (ac - bd) + (bc + ad)i </math> <br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|<span style="writing-mode: horizontal-tb;"><i>2 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭ ᠠ᠋ ᠢᠢ (ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) <span style="writing-mode: horizontal-tb;"><i>3 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭᠠᠨ ᠳᠤ (ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃<br />
ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠢ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠣᠷᠤᠶ ᠲᠠᠢ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠵᠤ᠂ <span style="writing-mode: horizontal-tb;"><i>√5</i></span> ᠢᠢᠠᠷ ᠰᠤᠩᠭᠠᠨ ᠠ᠋᠃ ᠡᠨᠡ ᠢᠨᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠾᠢᠫᠣᠲ᠋ᠧᠨᠦᠽ ᠤᠨ<sup>hypotenuse</sup> ᠤᠷᠲᠤ ᠪᠣᠯᠠᠢ᠃ <br />
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= ᠬᠤᠪᠢᠶᠠᠬᠤ =<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠬᠤᠪᠢᠶᠠᠬᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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<math>\frac{a+bi}{c+di}=</math> <br />
<math>=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=</math> <br />
<math>=\frac{ac+bd}{c^2+d^2}+\left(\frac{bc-ad}{c^2+d^2}\right)i.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ<sup>quadrate</sup> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨᠡ᠋᠃<ref>Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01</ref><br />
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<math>(x+yi)^2=x^2-y^2 + 2xyi.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> <span style="writing-mode: horizontal-tb;"><i>(b ≠ 0)</i></span> ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ <br />
<math> \pm (\gamma + \delta i)</math> ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄<br />
<math>\gamma = \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠠ <math>\delta = (\sgn b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠣᠯᠤᠮᠤᠢ᠃<br />
<span style="writing-mode: horizontal-tb;"><i>sgn b</i></span> ᠭᠡᠭᠴᠢ ᠢᠨᠦ <math>\sgn b = \begin{cases} \ \ 1, & b > 0 \\ \ \ 0, & b = 0 \\ -1, & b < 0 \end{cases}</math> ᠭᠡᠵᠦ ᠲᠠᠢᠢᠯᠠᠭᠳᠠᠨᠠ᠃ <br />
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ᠡᠭᠦᠨ ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ ᠢᠢᠨ ᠲᠤᠬᠠᠢᠢᠲᠠ ᠳ᠋ᠤ <math> \pm (\gamma + \delta i)</math> ᠲᠣᠭᠠ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠦ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠲᠣᠭᠠ ᠢᠢ ᠭᠠᠷᠭᠠᠵᠤ ᠢᠷᠡᠬᠦ ᠶᠣᠰᠤᠲᠠᠢ᠃ <br />
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=ᠬᠡᠷᠡᠭᠯᠡᠭᠡ=<br />
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== ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ==<br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠳᠠᠪᠲᠠᠯᠲᠠ ᠪᠠᠷ ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠠᠢᠢᠭᠤᠯᠬᠤ᠃<br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠯ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠳ᠋ᠤ ᠡᠭᠦᠰᠬᠡᠳᠡᠭ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ<sup>fractal</sup> ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠵᠡᠰᠱᠢᠶ ᠡ᠋ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡ ᠬᠦ ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ <span style="writing-mode: horizontal-tb;"><i>z = 0</i></span> ᠡᠴᠡ ᠡᠬᠢᠯᠡᠭᠡᠳ ᠢᠲ᠋ᠧᠷᠠᠼ<sup>iteration</sup> ᠬᠢᠬᠦ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;"><i>f<sub>c</sub>(z)=z<sup>2</sup>+c</i></span> ᠹᠦᠨ᠍ᠻᠼ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠢᠢᠬᠤ ᠨᠥᠬᠦᠴᠡᠯ ᠪᠦᠬᠦᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠤ ᠣᠯᠠᠨᠯᠢᠬ ᠪᠠᠢᠢᠳᠠᠭ᠃ <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ ᠪᠠᠢᠢᠭᠠᠯᠢ ᠢᠢᠨ ᠭᠠᠢᠢᠬᠠᠮᠰᠢᠭᠲᠤ ᠳᠦᠷᠰᠦ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠦᠨ ᠰᠠᠢᠢᠬᠠᠨ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠶᠡᠷᠦ ᠪᠤᠰᠤ ᠢᠢᠨ ᠭᠣᠶᠤᠮᠰᠠᠭ ᠬᠣᠣᠰᠯᠠᠯ ᠢ ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠳᠡᠭ᠃<ref>ᠷᠡᠨᠴᠡᠨ ᠦ ᠡᠩᠬᠡᠪᠠᠲᠤ᠃ '''ᠫᠢᠲ᠋ᠾᠠᠭᠣᠷ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌'''᠃ ᠮᠣᠩᠭᠤᠯ ᠤᠯᠤᠯᠰ ᠊ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠪᠠ ᠲᠣᠭᠠᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠢᠢᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ https://imdt.ac.mn/c/1013874?content=1150891&fbclid=IwAR1HVqeXwT-h7dijj9EeIUzDvFustH99GAr9LHYQIU61XIByuBHCkSm10So 2020</ref> <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠵᠠᠭᠠᠭ ᠢ ᠳᠠᠪᠠᠬᠤ ᠳᠤ ᠪᠠᠨ ᠵᠦᠯᠢᠶ ᠠ᠋ ᠢᠢᠨ<sup>Julia</sup> ᠹᠷᠠᠻᠲᠠᠯ ᠤᠳ ᠦᠷᠭᠡᠯᠵᠢ ᠬᠣᠯᠪᠤᠯᠲᠠ ᠪᠠᠨ ᠠᠯᠳᠠᠵᠤ᠂ ᠹᠠᠲᠣᠤ ᠢᠢᠨ<sup>Fatou</sup> ᠲᠣᠭᠤᠰᠤ ᠪᠣᠯᠤᠨ ᠬᠦᠪᠢᠷᠠᠳᠠᠭ᠃<ref>ᠨᠡᠭᠡᠭᠡᠯᠲᠡᠲᠡᠢ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠳ᠃ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠳᠠᠯ᠃ ᠪᠢᠳᠡᠨ ᠦ ᠡᠷᠭᠢᠨ ᠲᠣᠭᠣᠷᠢᠨ ᠳ᠋ᠠᠬᠢ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠬᠡᠷᠬᠢᠨ ᠠᠵᠢᠯᠯᠠᠳᠠᠭ ᠪᠤᠢ᠃ ᠹᠷᠠᠻᠲᠯᠠ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠶᠢᠨ ᠹᠷᠠᠻᠲᠯᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢ᠌ᠭ᠌᠃ https://ultrait.ru/mn/smartphones/otkrytie-fraktalov-beskonechnost-fraktalov-kak-ustroen-mir.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/10/02</ref><br />
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== ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ==<br />
[[Квант механик|ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌]]<sup>quantum mechanics</sup> ᠠᠲ᠋ᠣᠮ ᠠᠴᠠ ᠵᠢᠵᠢᠭ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ᠂ ᠶᠠᠭᠤᠨ ᠤ ᠡᠮᠦᠨᠡ ᠪᠣᠽᠣᠨ<sup>boson</sup> (ᠵᠢᠱᠢᠶᠡ ᠨᠢ ᠹᠣᠲ᠋ᠣᠨ<sup>photon</sup> ᠪᠠ ᠹᠧᠷᠮᠢᠣᠨ ᠤ<sup>fermion</sup> (ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠨᠧᠦᠲ᠋ᠷᠣᠨ) ᠬᠥᠳᠡᠯᠭᠡᠭᠡᠨ᠂ ᠬᠠᠷᠢᠯᠴᠠᠨ ᠦᠢᠯᠡᠴᠢᠯᠡᠯ ᠢ ᠰᠤᠳᠤᠯᠳᠠᠭ᠃ <br />
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ᠴᠢᠩᠭᠢᠬᠦ ᠳᠦ ᠪᠡᠨ ᠲᠡᠳᠡᠭᠡᠷ ᠦᠨ ᠦᠢᠯᠡ ᠬᠥᠳᠡᠯᠦᠯ ᠦᠨ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭ᠌ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠯᠲᠠ ᠶᠢ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠤᠳᠬᠠ ᠪᠠᠷ ᠭᠠᠷᠭᠠᠳᠠᠭ᠃ <br />
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ᠴᠤᠬᠤᠮ ᠳᠠᠭᠠᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠦᠨᠳᠦᠰᠦ ᠰᠠᠭᠤᠷᠢ ᠪᠣᠯᠳᠠᠭ᠃ <br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠲᠤ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ <sup>Schrödinger</sup> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠠᠴᠢ ᠬᠣᠯᠪᠣᠭᠳᠠᠯ ᠨᠢ ᠰᠣᠩᠭᠣᠳᠠᠭ ᠹᠢᠽᠢᠻ ᠳ᠋ᠡᠬᠢ ᠨᠧᠧᠲ᠋ᠣᠨ᠍ ᠤ ᠬᠣᠶᠠᠳᠤᠭᠠᠷ ᠬᠠᠤᠯᠢ ᠶᠢᠨ ᠠᠳᠠᠯᠢ ᠶᠤᠮ᠃ <br />
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ᠠᠯᠢ ᠠᠯᠢ ᠨᠢ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠷᠢᠯᠠᠯ᠂ ᠢᠮᠫᠦ᠋ᠯᠰ ᠢ <sup>impuls</sup> ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭᠴᠢᠯᠠᠨ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠳᠠᠭ᠃ <br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠨᠢ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ<sup>function</sup> ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠬᠦ ᠳᠦ ᠲᠣᠬᠢᠷᠠᠮᠵᠢ ᠲᠠᠢ ᠤᠴᠢᠷ ᠠᠴᠠ ᠹᠢᠽᠢᠻ ᠦᠨ ᠡᠨᠡ ᠰᠠᠯᠪᠤᠷᠢ ᠳᠤ ᠵᠠᠶ᠋ᠢᠯᠠᠰᠢ ᠦᠭᠡᠢ ᠱᠠᠭᠠᠷᠳᠠᠯᠭᠠ ᠲᠠᠢ ᠶᠤᠮ᠃ <br />
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ᠴᠠᠭᠠᠰᠢᠯᠠᠪᠠᠯ᠂ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠢᠭᠤᠳ ᠨᠥᠯᠥᠭᠡ ᠪᠡᠷ ᠬᠢᠮᠢ ᠶᠢᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠬᠥᠭᠵᠢᠯ ᠡᠷᠴᠢᠮᠵᠢᠭᠰᠡᠨ᠃ <br />
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1927 ᠣᠨ ᠳᠤ ᠸᠠᠯᠲ᠋ᠧᠷ ᠾᠠᠢᠢᠲ᠋ᠯᠧᠷ<sup>Walter Heitler</sup> ᠂ ᠹᠷᠢᠼ ᠯᠣᠨ᠍ᠳᠣᠨ<sup>Fritz London</sup> ᠨᠠᠷ ᠸᠠᠯᠧᠨ᠍ᠲ ᠤᠨ<sup>valency</sup> ᠬᠣᠯᠪᠣᠭᠠᠰᠤ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠲᠣᠮᠢᠶᠠᠯᠠᠭᠰᠠᠨ᠃ <br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠨᠢᠭᠡ ᠭᠣᠣᠯ ᠠᠰᠠᠭᠤᠳᠠᠯ ᠪᠣᠯ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ ᠦᠨ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ ᠤᠯᠬᠤ ᠶᠠᠪᠤᠳᠠᠯ ᠶᠤᠮ᠃ <br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠭᠡᠳᠡᠭ ᠨᠢ ᠲᠣᠳᠣᠷᠬᠠᠢ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠵᠤᠷᠪᠤᠰ ᠲᠤ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠵᠤ ᠪᠣᠯᠬᠤ ᠪᠠᠶ᠋ᠢᠷᠢᠰᠢᠯ ᠤᠳ ᠤᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠲᠠᠷᠬᠠᠯᠲᠠ ᠶᠤᠮ᠄ <br />
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<small><math>\frac{-\hbar}{2m}\frac{\partial^2\Psi(x,t)}{\partial x^2} + V(x,t)\Psi(x,t) = i\hbar\frac{\partial \Psi(x,t)}{\partial t}</math></small><br />
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ᠻᠢᠨᠧᠲᠢᠭ<sup>kinetic</sup> ᠧᠨᠧᠷᠭᠢ<sup>energy</sup><br />
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ᠫᠣᠲ᠋ᠧᠨ᠍ᠼᠢᠶᠠᠯ<sup>potencial</sup> ᠧᠨᠧᠷᠭᠢ<br />
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ᠨᠡᠢᠢᠲᠡ ᠧᠨᠧᠷᠭᠢ<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠲᠦ ᠰᠢᠲᠦᠭᠰᠡᠨ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠠᠭᠤᠷᠢ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠨᠢᠭᠡ ᠪᠣᠯ ᠳᠡᠭᠡᠷᠡ ᠳᠤᠷᠠᠳᠤᠭᠰᠠᠨ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡᠬᠦ ᠲᠣᠮᠢᠶᠠ ᠶᠢ ᠠᠰᠢᠭᠯᠠᠭᠰᠠᠨ ᠢᠶ᠋ᠠᠷ ᠤᠰᠤ ᠲᠥᠷᠦᠭᠴᠢ ᠶᠢᠨ ᠮᠣᠯᠧᠻᠤᠯ<sup>molecule</sup> ᠳᠠᠬᠢ ᠬᠣᠶᠠᠷ ᠠᠲ᠋ᠣᠮ ᠨᠢ ᠻᠣᠸᠠᠯᠧᠨ᠍ᠲ<sup>covalent</sup> ᠬᠣᠯᠪᠣᠭᠠ ᠭᠡᠳᠡᠭ ᠢ ᠡᠭᠦᠰᠬᠡᠨ ᠧᠯᠧᠻᠲ᠋ᠷᠣᠨ ᠨᠤᠭᠤᠳ ᠢᠢᠠᠨ ᠬᠤᠪᠢᠶᠠᠯᠴᠠᠵᠤ ᠪᠠᠶ᠋ᠢᠳᠠᠭ ᠢ ᠪᠠᠲᠤᠯᠠᠭᠰᠠᠨ᠃<ref>Josiah Wu. Real Life Applications of Complex Numbers. 2020 https://issuu.com/harrowhongkong/docs/final_scientific_harrovian_issue_vi-i/s/11488755</ref><br />
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== ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠢᠢᠨ ᠣᠨᠤᠯ ==<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠶᠢ ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ᠂ ᠢᠯᠠᠩᠭᠤᠶᠠ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠨ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠶ᠋ᠢᠳᠠᠯ ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠭᠡᠨ ᠳᠤ ᠬᠡᠷᠡᠭᠯᠡᠳᠡᠭ᠃ <br />
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ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠳᠤ 《ᠰᠢᠰᠲ᠋ᠧᠮ》 ᠭᠡᠳᠡᠭ ᠦᠭᠡ ᠶᠢ ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ ᠪᠥᠭᠡᠳ ᠡᠨᠡ ᠨᠢ ᠵᠠᠪᠠᠯ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠢ ᠬᠡᠯᠡᠳᠡᠭ ᠦᠭᠡᠢ᠃ <br />
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ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠡᠭᠦᠨ ᠢ ᠬᠥᠷᠥᠩᠭᠡ ᠶᠢᠨ ᠵᠠᠬᠠ ᠵᠡᠭᠡᠯᠢ ᠳ᠋ᠡᠬᠢ ᠥᠭᠡᠷᠡᠴᠢᠯᠡᠯᠲᠡ᠂ ᠬᠢᠮᠢ ᠶᠢᠨ ᠤᠷᠪᠠᠯ᠂ ᠦᠢᠯᠡ ᠶᠠᠪᠤᠴᠠ ᠶᠢ ᠰᠤᠳᠤᠯᠬᠤ ᠳᠤ ᠬᠡᠷᠡᠭᠯᠡᠵᠦ ᠪᠣᠯᠤᠨᠠ᠃ <br />
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ᠲᠡᠭᠦᠨᠴᠢᠯᠡᠨ ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠷᠣᠪᠣᠲ ᠤᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠳᠤ<sup>technology</sup> ᠥᠷᠭᠡᠨ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ ᠲᠤᠯᠠ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠠᠨᠠᠯᠢᠽ ᠴᠤ ᠪᠠᠰᠠ ᠲᠡᠭᠦᠨ ᠳᠦ ᠬᠡᠷᠡᠭᠯᠡᠭᠳᠡᠨᠡ ᠭᠡᠰᠡᠨ ᠦᠭᠡ᠃<ref>Ujjvala Y. Gawarguru, Mitali K. Tibdewal, Rajashri A. Naphade, Rahul M. Jethwani. The Review of Introduction & Application of Complex Number in Engineering. 2nd National Conference Recent Innovations in Science and Engineering (NC-RISE 17). Volume: 5 Issue: 9. pp55 – 57. ISSN: 2321-8169. https://ijritcc.org/download/conferences/NC-RISE_17/Track_6_(ASH)/1506931102_02-10-2017.pdf</ref><br />
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ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠳᠤ ᠵᠠᠷᠢᠮ ᠳᠤ ᠪᠠᠨ ᠯᠠᠫ᠊ᠯᠠᠰ ᠤᠨ<sup>Laplace</sup> ᠬᠤᠪᠢᠷᠠᠭᠠᠯᠲᠠ ᠶᠢ ᠠᠰᠢᠭᠯᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠳ ᠢ<sup>system</sup> ᠴᠠᠭ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠮᠤᠵᠢ ᠠᠴᠠ ᠳᠠᠪᠲᠠᠮᠵᠢ ᠶᠢᠨ ᠮᠤᠵᠢ ᠳᠤ ᠰᠢᠯᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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ᠳᠠᠷᠠᠭᠠ ᠨᠢ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠨ ᠲᠡᠭ ᠪᠠ ᠲᠤᠶ᠋ᠢᠯ ᠤᠳ ᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠰᠢᠨᠵᠢᠯᠡᠨᠡ᠃<br />
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ᠢᠵᠠᠭᠤᠷ ᠤᠨ ᠮᠥᠷ ᠦᠨ (root locus) ᠠᠷᠭᠠ ᠪᠠᠷ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠳᠦ ᠲᠡᠭ ᠪᠠ ᠲᠤᠶ᠋ᠢᠯ ᠤᠳ ᠵᠡᠭᠦᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠠᠶ᠋ᠢᠨᠠ ᠤᠤ᠂ ᠡᠰᠡᠪᠡᠯ ᠪᠠᠷᠠᠭᠤᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠠᠶ᠋ᠢᠨᠠ ᠤᠤ ᠭᠡᠳᠡᠭ ᠨᠢ ᠴᠢᠬᠤᠯᠠ᠃ <br />
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ᠥᠭᠡᠷᠡ ᠪᠡᠷ ᠬᠡᠯᠡᠪᠡᠯ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠨᠢ ᠲᠡᠭ ᠡᠴᠡ ᠶᠡᠬᠡ ᠪᠠᠢᠢᠨᠠ ᠤᠤ ᠡᠰᠡᠪᠡᠯ ᠪᠠᠭᠠ ᠪᠠᠶ᠋ᠢᠨᠠ ᠤᠤ ᠭᠡᠳᠡᠭ ᠨᠢ ᠴᠢᠬᠤᠯᠠ ᠶᠤᠮ᠃ <br />
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ᠴᠠᠭ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠬᠤᠪᠢᠷᠠᠯᠲᠠ ᠦᠭᠡᠢ ᠱᠤᠭᠤᠮᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠨᠢ᠄<br />
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- ᠪᠠᠷᠠᠭᠤᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠲᠤᠶ᠋ᠢᠯ ᠲᠠᠢ ᠪᠣᠯ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠂<br />
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- ᠵᠡᠭᠦᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠦᠬᠦ ᠲᠤᠶ᠋ᠢᠯ ᠨᠢ ᠪᠠᠶ᠋ᠢᠪᠠᠯ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠂<br />
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- ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠲᠤᠶ᠋ᠢᠯ ᠲᠠᠢ ᠪᠣᠯ ᠶᠠᠯᠢ ᠦᠭᠡᠶ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠃<br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ - complex number - комплексное число<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ - imaginary unit - мнимая единица <br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭᠠ - imaginary number - чисто мнимое число<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ - imaginaty part - мнимая часть<br />
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ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠ - real number - действительное число<br />
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ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ - real part - действительная часть<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ - complex plane - комлексная плоскость<br />
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ᠲᠡᠩᠬᠡᠯᠢᠭ - axis - ось<br />
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ᠣᠯᠠᠨᠯᠢᠭ - set - множество<br />
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ᠡᠶᠡᠷᠭᠦ - positive - положительный<br />
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ᠰᠥᠷᠭᠦ - negative - отрицательный<br />
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ᠲᠡᠭᠰᠢᠳᠬᠡᠯ - equation - уравнение<br />
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ᠲᠣᠮᠢᠶ ᠠ᠋ - formula - формула<br />
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ᠡᠷᠭᠢᠯᠲᠡ - rotation - поворот<br />
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ᠮᠠᠭᠠᠳᠯᠠᠯ - probability - вероятность<br />
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ᠬᠡᠮ - degree - градус<br />
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ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠢᠢᠨ ᠣᠨᠤᠯ - control theory - теория управления<br />
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ᠠᠲ᠋ᠣᠮ - atom - атом<br />
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ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮ - elementary particle - элементарная частица<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ - wave function - волновая функция<br />
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ᠳᠠᠪᠲᠠᠮᠵᠢ - frequency - частота<br />
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ᠹᠷᠠᠻᠲ᠋ᠠᠯ - fractal - фрактал<br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ - quantum mechanics - квантовая механика<br />
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ᠰᠣᠩᠭᠣᠳᠠᠭ ᠹᠢᠽᠢᠻ - classical physics - классическая физика<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==<br />
<div style="writing-mode: horizontal-tb; "><References/></div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%BA%E1%A0%A3%E1%A0%AE%E1%A0%AB%E1%A0%8A%E1%A0%AF%E1%A0%A7%E1%A0%BA%E1%A0%B0_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%A0&diff=2757
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-10-11T02:41:30Z
<p>Angka: Layout</p>
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<div><div style="max-height: 80vh; writing-mode: vertical-lr; font-family: 'Noto Sans Mongolian'"><br />
'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ<sup>complex</sup> ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠭᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠥᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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__TOC__<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃ <br />
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{| class="wikitable"<br />
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|[[File:Komplexe_zahlenebene.svg|thumb]]<br />
|style = "height: 225px;"|ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠢᠢ ᠲᠥᠯᠦᠭᠡᠯᠡᠵᠦ ᠪᠤᠶ ᠠᠷᠭᠠᠨ᠋ᠳ᠋ ᠤᠨ<sup>Argand</sup> ᠳᠢᠶᠠᠭ᠋ᠷᠠᠮ ᠳᠡᠭᠡᠷ ᠡ᠋ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠸᠧᠺᠲ᠋ᠣᠷ<sup>vector</sup> ᠡᠭᠦᠰᠬᠡᠵᠦ ᠪᠤᠶ ᠬᠣᠣᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ <span style="writing-mode: horizontal-tb;"><i>(a, b)</i></span> ᠳᠦᠷᠰᠦᠯᠡᠨ ᠦᠵᠡᠭᠦᠯᠵᠦ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
ᠡᠭᠦᠨ ᠳ᠋ᠦ: <br><span style="writing-mode: horizontal-tb;"><i>Re</i></span> ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ᠂ <br><span style="writing-mode: horizontal-tb;"><i>Im</i></span> ᠢᠨᠦ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠣᠯᠬᠤ ᠠᠭᠠᠳ <br><span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠢᠨᠦ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠳ᠋ᠦ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠬᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠮᠥᠨ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
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<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
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{| class="wikitable"<br />
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|[[File:Komplexe_addition.svg|thumb]]<br />
|style = "height: 220px;"|ᠬᠣᠶᠠᠷ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃ <br />
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= ᠬᠠᠰᠤᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠭᠡ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
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<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math><br />
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= ᠦᠷᠡᠵᠢᠬᠦ =<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃<ref>''Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И.'' Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — <abbr>М.</abbr>: Просвещение, 1998. — 288 с. — <nowiki>ISBN 5-09-008036-4</nowiki>.</ref> <br />
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<math>(a + bi) (c + di) = </math> <math>= ac + bci - bd + adi = </math> <math>= (ac - bd) + (bc + ad)i </math> <br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|<span style="writing-mode: horizontal-tb;"><i>2 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭ ᠠ᠋ ᠢᠢ (ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) <span style="writing-mode: horizontal-tb;"><i>3 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭᠠᠨ ᠳᠤ (ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃<br />
ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠢ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠣᠷᠤᠶ ᠲᠠᠢ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠵᠤ᠂ <span style="writing-mode: horizontal-tb;"><i>√5</i></span> ᠢᠢᠠᠷ ᠰᠤᠩᠭᠠᠨ ᠠ᠋᠃ ᠡᠨᠡ ᠢᠨᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠾᠢᠫᠣᠲ᠋ᠧᠨᠦᠽ ᠤᠨ<sup>hypotenuse</sup> ᠤᠷᠲᠤ ᠪᠣᠯᠠᠢ᠃ <br />
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= ᠬᠤᠪᠢᠶᠠᠬᠤ =<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠬᠤᠪᠢᠶᠠᠬᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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<math>\frac{a+bi}{c+di}=</math> <br />
<math>=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=</math> <br />
<math>=\frac{ac+bd}{c^2+d^2}+\left(\frac{bc-ad}{c^2+d^2}\right)i.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ<sup>quadrate</sup> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨᠡ᠋᠃<ref>Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01</ref><br />
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<math>(x+yi)^2=x^2-y^2 + 2xyi.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> <span style="writing-mode: horizontal-tb;"><i>(b ≠ 0)</i></span> ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ <br />
<math> \pm (\gamma + \delta i)</math> ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄<br />
<math>\gamma = \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠠ <math>\delta = (\sgn b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠣᠯᠤᠮᠤᠢ᠃<br />
<span style="writing-mode: horizontal-tb;"><i>sgn b</i></span> ᠭᠡᠭᠴᠢ ᠢᠨᠦ <math>\sgn b = \begin{cases} \ \ 1, & b > 0 \\ \ \ 0, & b = 0 \\ -1, & b < 0 \end{cases}</math> ᠭᠡᠵᠦ ᠲᠠᠢᠢᠯᠠᠭᠳᠠᠨᠠ᠃ <br />
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ᠡᠭᠦᠨ ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ ᠢᠢᠨ ᠲᠤᠬᠠᠢᠢᠲᠠ ᠳ᠋ᠤ <math> \pm (\gamma + \delta i)</math> ᠲᠣᠭᠠ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠦ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠲᠣᠭᠠ ᠢᠢ ᠭᠠᠷᠭᠠᠵᠤ ᠢᠷᠡᠬᠦ ᠶᠣᠰᠤᠲᠠᠢ᠃ <br />
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=ᠬᠡᠷᠡᠭᠯᠡᠭᠡ=<br />
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== ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ==<br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠳᠠᠪᠲᠠᠯᠲᠠ ᠪᠠᠷ ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠠᠢᠢᠭᠤᠯᠬᠤ᠃<br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠯ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠳ᠋ᠤ ᠡᠭᠦᠰᠬᠡᠳᠡᠭ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ<sup>fractal</sup> ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠵᠡᠰᠱᠢᠶ ᠡ᠋ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡ ᠬᠦ ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ <span style="writing-mode: horizontal-tb;"><i>z = 0</i></span> ᠡᠴᠡ ᠡᠬᠢᠯᠡᠭᠡᠳ ᠢᠲ᠋ᠧᠷᠠᠼ<sup>iteration</sup> ᠬᠢᠬᠦ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;"><i>f<sub>c</sub>(z)=z<sup>2</sup>+c</i></span> ᠹᠦᠨ᠍ᠻᠼ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠢᠢᠬᠤ ᠨᠥᠬᠦᠴᠡᠯ ᠪᠦᠬᠦᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠤ ᠣᠯᠠᠨᠯᠢᠬ ᠪᠠᠢᠢᠳᠠᠭ᠃ <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ ᠪᠠᠢᠢᠭᠠᠯᠢ ᠢᠢᠨ ᠭᠠᠢᠢᠬᠠᠮᠰᠢᠭᠲᠤ ᠳᠦᠷᠰᠦ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠦᠨ ᠰᠠᠢᠢᠬᠠᠨ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠶᠡᠷᠦ ᠪᠤᠰᠤ ᠢᠢᠨ ᠭᠣᠶᠤᠮᠰᠠᠭ ᠬᠣᠣᠰᠯᠠᠯ ᠢ ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠳᠡᠭ᠃<ref>ᠷᠡᠨᠴᠡᠨ ᠦ ᠡᠩᠬᠡᠪᠠᠲᠤ᠃ '''ᠫᠢᠲ᠋ᠾᠠᠭᠣᠷ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌'''᠃ ᠮᠣᠩᠭᠤᠯ ᠤᠯᠤᠯᠰ ᠊ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠪᠠ ᠲᠣᠭᠠᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠢᠢᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ https://imdt.ac.mn/c/1013874?content=1150891&fbclid=IwAR1HVqeXwT-h7dijj9EeIUzDvFustH99GAr9LHYQIU61XIByuBHCkSm10So 2020</ref> <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠵᠠᠭᠠᠭ ᠢ ᠳᠠᠪᠠᠬᠤ ᠳᠤ ᠪᠠᠨ ᠵᠦᠯᠢᠶ ᠠ᠋ ᠢᠢᠨ<sup>Julia</sup> ᠹᠷᠠᠻᠲᠠᠯ ᠤᠳ ᠦᠷᠭᠡᠯᠵᠢ ᠬᠣᠯᠪᠤᠯᠲᠠ ᠪᠠᠨ ᠠᠯᠳᠠᠵᠤ᠂ ᠹᠠᠲᠣᠤ ᠢᠢᠨ<sup>Fatou</sup> ᠲᠣᠭᠤᠰᠤ ᠪᠣᠯᠤᠨ ᠬᠦᠪᠢᠷᠠᠳᠠᠭ᠃<ref>ᠨᠡᠭᠡᠭᠡᠯᠲᠡᠲᠡᠢ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠳ᠃ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠳᠠᠯ᠃ ᠪᠢᠳᠡᠨ ᠦ ᠡᠷᠭᠢᠨ ᠲᠣᠭᠣᠷᠢᠨ ᠳ᠋ᠠᠬᠢ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠬᠡᠷᠬᠢᠨ ᠠᠵᠢᠯᠯᠠᠳᠠᠭ ᠪᠤᠢ᠃ ᠹᠷᠠᠻᠲᠯᠠ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠶᠢᠨ ᠹᠷᠠᠻᠲᠯᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢ᠌ᠭ᠌᠃ https://ultrait.ru/mn/smartphones/otkrytie-fraktalov-beskonechnost-fraktalov-kak-ustroen-mir.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/10/02</ref><br />
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|style = "height: 225px;"|ᠳᠦᠷᠰᠦ ᠠᠮᠢᠯᠠᠭᠤᠯᠤᠨ ᠵᠦᠯᠢᠶ ᠠ᠋ ᠢᠢᠨ <sup>Julia</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠦᠵᠡᠭᠦᠯᠦᠭᠰᠡᠨ ᠢᠨᠦ᠃<br />
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== ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ==<br />
ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌<sup>quantum mechanics</sup> ᠠᠲ᠋ᠣᠮ ᠠᠴᠠ ᠵᠢᠵᠢᠭ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ᠂ ᠶᠠᠭᠤᠨ ᠤ ᠡᠮᠦᠨᠡ ᠪᠣᠽᠣᠨ<sup>boson</sup> (ᠵᠢᠱᠢᠶᠡ ᠨᠢ ᠹᠣᠲ᠋ᠣᠨ<sup>photon</sup> ᠪᠠ ᠹᠧᠷᠮᠢᠣᠨ ᠤ<sup>fermion</sup> (ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠨᠧᠦᠲ᠋ᠷᠣᠨ) ᠬᠥᠳᠡᠯᠭᠡᠭᠡᠨ᠂ ᠬᠠᠷᠢᠯᠴᠠᠨ ᠦᠢᠯᠡᠴᠢᠯᠡᠯ ᠢ ᠰᠤᠳᠤᠯᠳᠠᠭ᠃ <br />
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ᠴᠢᠩᠭᠢᠬᠦ ᠳᠦ ᠪᠡᠨ ᠲᠡᠳᠡᠭᠡᠷ ᠦᠨ ᠦᠢᠯᠡ ᠬᠥᠳᠡᠯᠦᠯ ᠦᠨ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭ᠌ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠯᠲᠠ ᠶᠢ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠤᠳᠬᠠ ᠪᠠᠷ ᠭᠠᠷᠭᠠᠳᠠᠭ᠃ <br />
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ᠴᠤᠬᠤᠮ ᠳᠠᠭᠠᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠦᠨᠳᠦᠰᠦ ᠰᠠᠭᠤᠷᠢ ᠪᠣᠯᠳᠠᠭ᠃ <br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠲᠤ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ <sup>Schrödinger</sup> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠠᠴᠢ ᠬᠣᠯᠪᠣᠭᠳᠠᠯ ᠨᠢ ᠰᠣᠩᠭᠣᠳᠠᠭ ᠹᠢᠽᠢᠻ ᠳ᠋ᠡᠬᠢ ᠨᠧᠧᠲ᠋ᠣᠨ᠍ ᠤ ᠬᠣᠶᠠᠳᠤᠭᠠᠷ ᠬᠠᠤᠯᠢ ᠶᠢᠨ ᠠᠳᠠᠯᠢ ᠶᠤᠮ᠃ <br />
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ᠠᠯᠢ ᠠᠯᠢ ᠨᠢ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠷᠢᠯᠠᠯ᠂ ᠢᠮᠫᠦ᠋ᠯᠰ ᠢ <sup>impuls</sup> ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭᠴᠢᠯᠠᠨ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠳᠠᠭ᠃ <br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠨᠢ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ<sup>function</sup> ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠬᠦ ᠳᠦ ᠲᠣᠬᠢᠷᠠᠮᠵᠢ ᠲᠠᠢ ᠤᠴᠢᠷ ᠠᠴᠠ ᠹᠢᠽᠢᠻ ᠦᠨ ᠡᠨᠡ ᠰᠠᠯᠪᠤᠷᠢ ᠳᠤ ᠵᠠᠶ᠋ᠢᠯᠠᠰᠢ ᠦᠭᠡᠢ ᠱᠠᠭᠠᠷᠳᠠᠯᠭᠠ ᠲᠠᠢ ᠶᠤᠮ᠃ <br />
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ᠴᠠᠭᠠᠰᠢᠯᠠᠪᠠᠯ᠂ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠢᠭᠤᠳ ᠨᠥᠯᠥᠭᠡ ᠪᠡᠷ ᠬᠢᠮᠢ ᠶᠢᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠬᠥᠭᠵᠢᠯ ᠡᠷᠴᠢᠮᠵᠢᠭᠰᠡᠨ᠃ <br />
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1927 ᠣᠨ ᠳᠤ ᠸᠠᠯᠲ᠋ᠧᠷ ᠾᠠᠢᠢᠲ᠋ᠯᠧᠷ<sup>Walter Heitler</sup> ᠂ ᠹᠷᠢᠼ ᠯᠣᠨ᠍ᠳᠣᠨ<sup>Fritz London</sup> ᠨᠠᠷ ᠸᠠᠯᠧᠨ᠍ᠲ ᠤᠨ<sup>valency</sup> ᠬᠣᠯᠪᠣᠭᠠᠰᠤ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠲᠣᠮᠢᠶᠠᠯᠠᠭᠰᠠᠨ᠃ <br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠨᠢᠭᠡ ᠭᠣᠣᠯ ᠠᠰᠠᠭᠤᠳᠠᠯ ᠪᠣᠯ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ ᠦᠨ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ ᠤᠯᠬᠤ ᠶᠠᠪᠤᠳᠠᠯ ᠶᠤᠮ᠃ <br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠭᠡᠳᠡᠭ ᠨᠢ ᠲᠣᠳᠣᠷᠬᠠᠢ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠵᠤᠷᠪᠤᠰ ᠲᠤ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠵᠤ ᠪᠣᠯᠬᠤ ᠪᠠᠶ᠋ᠢᠷᠢᠰᠢᠯ ᠤᠳ ᠤᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠲᠠᠷᠬᠠᠯᠲᠠ ᠶᠤᠮ᠄ <br />
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<small><math>\frac{-\hbar}{2m}\frac{\partial^2\Psi(x,t)}{\partial x^2} + V(x,t)\Psi(x,t) = i\hbar\frac{\partial \Psi(x,t)}{\partial t}</math></small><br />
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ᠻᠢᠨᠧᠲᠢᠭ<sup>kinetic</sup> ᠧᠨᠧᠷᠭᠢ<sup>energy</sup><br />
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ᠫᠣᠲ᠋ᠧᠨ᠍ᠼᠢᠶᠠᠯ<sup>potencial</sup> ᠧᠨᠧᠷᠭᠢ<br />
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ᠨᠡᠢᠢᠲᠡ ᠧᠨᠧᠷᠭᠢ<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠲᠦ ᠰᠢᠲᠦᠭᠰᠡᠨ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠠᠭᠤᠷᠢ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠨᠢᠭᠡ ᠪᠣᠯ ᠳᠡᠭᠡᠷᠡ ᠳᠤᠷᠠᠳᠤᠭᠰᠠᠨ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡᠬᠦ ᠲᠣᠮᠢᠶᠠ ᠶᠢ ᠠᠰᠢᠭᠯᠠᠭᠰᠠᠨ ᠢᠶ᠋ᠠᠷ ᠤᠰᠤ ᠲᠥᠷᠦᠭᠴᠢ ᠶᠢᠨ ᠮᠣᠯᠧᠻᠤᠯ<sup>molecule</sup> ᠳᠠᠬᠢ ᠬᠣᠶᠠᠷ ᠠᠲ᠋ᠣᠮ ᠨᠢ ᠻᠣᠸᠠᠯᠧᠨ᠍ᠲ<sup>covalent</sup> ᠬᠣᠯᠪᠣᠭᠠ ᠭᠡᠳᠡᠭ ᠢ ᠡᠭᠦᠰᠬᠡᠨ ᠧᠯᠧᠻᠲ᠋ᠷᠣᠨ ᠨᠤᠭᠤᠳ ᠢᠢᠠᠨ ᠬᠤᠪᠢᠶᠠᠯᠴᠠᠵᠤ ᠪᠠᠶ᠋ᠢᠳᠠᠭ ᠢ ᠪᠠᠲᠤᠯᠠᠭᠰᠠᠨ᠃<ref>Josiah Wu. Real Life Applications of Complex Numbers. 2020 https://issuu.com/harrowhongkong/docs/final_scientific_harrovian_issue_vi-i/s/11488755</ref><br />
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== ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠢᠢᠨ ᠣᠨᠤᠯ ==<br />
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{| class="wikitable"<br />
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|[[File:Simple_feedback_system.svg|thumb]]<br />
|style = "height: 340px;"|ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠢᠢᠨ ᠲᠣᠭᠲᠠᠯᠴᠠᠭᠠᠨ ᠤ ᠡᠩ ᠦᠨ ᠪᠦᠳᠦᠭᠦᠪᠴᠢ᠃ <br><br><span style="writing-mode: horizontal-tb;"><i>X(s)</i></span> ᠣᠷᠤᠯᠲᠠ ᠢᠢᠨ ᠮᠡᠳᠡᠭᠡᠯᠡᠯ᠂ <br><br><span style="writing-mode: horizontal-tb;"><i>Y(s)</i></span> ᠭᠠᠷᠤᠯᠲᠠ ᠢᠢᠨ ᠮᠡᠳᠡᠭᠡᠯᠡᠯ᠂ <br><br><span style="writing-mode: horizontal-tb;"><i>G(s)</i></span> ᠤᠷᠤᠭᠰᠢ ᠳᠠᠮᠵᠢᠭᠤᠯᠬᠤ ᠹᠦᠨ᠍ᠻᠼ<sup>function</sup>᠂ <br><br><span style="writing-mode: horizontal-tb;"><i>H(s)</i></span> ᠡᠷᠭᠢᠬᠦ ᠬᠣᠯᠪᠤᠭᠠ<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠶᠢ ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ᠂ ᠢᠯᠠᠩᠭᠤᠶᠠ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠨ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠶ᠋ᠢᠳᠠᠯ ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠭᠡᠨ ᠳᠤ ᠬᠡᠷᠡᠭᠯᠡᠳᠡᠭ᠃ <br />
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ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠳᠤ 《ᠰᠢᠰᠲ᠋ᠧᠮ》 ᠭᠡᠳᠡᠭ ᠦᠭᠡ ᠶᠢ ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ ᠪᠥᠭᠡᠳ ᠡᠨᠡ ᠨᠢ ᠵᠠᠪᠠᠯ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠢ ᠬᠡᠯᠡᠳᠡᠭ ᠦᠭᠡᠢ᠃ <br />
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ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠡᠭᠦᠨ ᠢ ᠬᠥᠷᠥᠩᠭᠡ ᠶᠢᠨ ᠵᠠᠬᠠ ᠵᠡᠭᠡᠯᠢ ᠳ᠋ᠡᠬᠢ ᠥᠭᠡᠷᠡᠴᠢᠯᠡᠯᠲᠡ᠂ ᠬᠢᠮᠢ ᠶᠢᠨ ᠤᠷᠪᠠᠯ᠂ ᠦᠢᠯᠡ ᠶᠠᠪᠤᠴᠠ ᠶᠢ ᠰᠤᠳᠤᠯᠬᠤ ᠳᠤ ᠬᠡᠷᠡᠭᠯᠡᠵᠦ ᠪᠣᠯᠤᠨᠠ᠃ <br />
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ᠲᠡᠭᠦᠨᠴᠢᠯᠡᠨ ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠷᠣᠪᠣᠲ ᠤᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠳᠤ<sup>technology</sup> ᠥᠷᠭᠡᠨ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ ᠲᠤᠯᠠ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠠᠨᠠᠯᠢᠽ ᠴᠤ ᠪᠠᠰᠠ ᠲᠡᠭᠦᠨ ᠳᠦ ᠬᠡᠷᠡᠭᠯᠡᠭᠳᠡᠨᠡ ᠭᠡᠰᠡᠨ ᠦᠭᠡ᠃<ref>Ujjvala Y. Gawarguru, Mitali K. Tibdewal, Rajashri A. Naphade, Rahul M. Jethwani. The Review of Introduction & Application of Complex Number in Engineering. 2nd National Conference Recent Innovations in Science and Engineering (NC-RISE 17). Volume: 5 Issue: 9. pp55 – 57. ISSN: 2321-8169. https://ijritcc.org/download/conferences/NC-RISE_17/Track_6_(ASH)/1506931102_02-10-2017.pdf</ref><br />
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ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠳᠤ ᠵᠠᠷᠢᠮ ᠳᠤ ᠪᠠᠨ ᠯᠠᠫ᠊ᠯᠠᠰ ᠤᠨ<sup>Laplace</sup> ᠬᠤᠪᠢᠷᠠᠭᠠᠯᠲᠠ ᠶᠢ ᠠᠰᠢᠭᠯᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠳ ᠢ<sup>system</sup> ᠴᠠᠭ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠮᠤᠵᠢ ᠠᠴᠠ ᠳᠠᠪᠲᠠᠮᠵᠢ ᠶᠢᠨ ᠮᠤᠵᠢ ᠳᠤ ᠰᠢᠯᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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ᠳᠠᠷᠠᠭᠠ ᠨᠢ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠨ ᠲᠡᠭ ᠪᠠ ᠲᠤᠶ᠋ᠢᠯ ᠤᠳ ᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠰᠢᠨᠵᠢᠯᠡᠨᠡ᠃<br />
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ᠢᠵᠠᠭᠤᠷ ᠤᠨ ᠮᠥᠷ ᠦᠨ (root locus) ᠠᠷᠭᠠ ᠪᠠᠷ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠳᠦ ᠲᠡᠭ ᠪᠠ ᠲᠤᠶ᠋ᠢᠯ ᠤᠳ ᠵᠡᠭᠦᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠠᠶ᠋ᠢᠨᠠ ᠤᠤ᠂ ᠡᠰᠡᠪᠡᠯ ᠪᠠᠷᠠᠭᠤᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠠᠶ᠋ᠢᠨᠠ ᠤᠤ ᠭᠡᠳᠡᠭ ᠨᠢ ᠴᠢᠬᠤᠯᠠ᠃ <br />
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ᠥᠭᠡᠷᠡ ᠪᠡᠷ ᠬᠡᠯᠡᠪᠡᠯ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠨᠢ ᠲᠡᠭ ᠡᠴᠡ ᠶᠡᠬᠡ ᠪᠠᠢᠢᠨᠠ ᠤᠤ ᠡᠰᠡᠪᠡᠯ ᠪᠠᠭᠠ ᠪᠠᠶ᠋ᠢᠨᠠ ᠤᠤ ᠭᠡᠳᠡᠭ ᠨᠢ ᠴᠢᠬᠤᠯᠠ ᠶᠤᠮ᠃ <br />
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ᠴᠠᠭ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠬᠤᠪᠢᠷᠠᠯᠲᠠ ᠦᠭᠡᠢ ᠱᠤᠭᠤᠮᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠨᠢ᠄<br />
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- ᠪᠠᠷᠠᠭᠤᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠲᠤᠶ᠋ᠢᠯ ᠲᠠᠢ ᠪᠣᠯ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠂<br />
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- ᠵᠡᠭᠦᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠦᠬᠦ ᠲᠤᠶ᠋ᠢᠯ ᠨᠢ ᠪᠠᠶ᠋ᠢᠪᠠᠯ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠂<br />
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- ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠲᠤᠶ᠋ᠢᠯ ᠲᠠᠢ ᠪᠣᠯ ᠶᠠᠯᠢ ᠦᠭᠡᠶ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠃<br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ - complex number - комплексное число<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ - imaginary unit - мнимая единица <br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭᠠ - imaginary number - чисто мнимое число<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ - imaginaty part - мнимая часть<br />
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ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠ - real number - действительное число<br />
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ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ - real part - действительная часть<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ - complex plane - комлексная плоскость<br />
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ᠲᠡᠩᠬᠡᠯᠢᠭ - axis - ось<br />
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ᠣᠯᠠᠨᠯᠢᠭ - set - множество<br />
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ᠡᠶᠡᠷᠭᠦ - positive - положительный<br />
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ᠰᠥᠷᠭᠦ - negative - отрицательный<br />
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ᠲᠡᠭᠰᠢᠳᠬᠡᠯ - equation - уравнение<br />
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ᠲᠣᠮᠢᠶ ᠠ᠋ - formula - формула<br />
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ᠡᠷᠭᠢᠯᠲᠡ - rotation - поворот<br />
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ᠮᠠᠭᠠᠳᠯᠠᠯ - probability - вероятность<br />
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ᠬᠡᠮ - degree - градус<br />
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ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠢᠢᠨ ᠣᠨᠤᠯ - control theory - теория управления<br />
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ᠠᠲ᠋ᠣᠮ - atom - атом<br />
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ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮ - elementary particle - элементарная частица<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ - wave function - волновая функция<br />
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ᠳᠠᠪᠲᠠᠮᠵᠢ - frequency - частота<br />
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ᠹᠷᠠᠻᠲ᠋ᠠᠯ - fractal - фрактал<br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ - quantum mechanics - квантовая механика<br />
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ᠰᠣᠩᠭᠣᠳᠠᠭ ᠹᠢᠽᠢᠻ - classical physics - классическая физика<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==<br />
<div style="writing-mode: horizontal-tb; "><References/></div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%BA%E1%A0%A3%E1%A0%AE%E1%A0%AB%E1%A0%8A%E1%A0%AF%E1%A0%A7%E1%A0%BA%E1%A0%B0_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%A0&diff=2756
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-10-11T02:39:07Z
<p>Angka: Spelling</p>
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'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ<sup>complex</sup> ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠭᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠥᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃ <br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠢᠢ ᠲᠥᠯᠦᠭᠡᠯᠡᠵᠦ ᠪᠤᠶ ᠠᠷᠭᠠᠨ᠋ᠳ᠋ ᠤᠨ<sup>Argand</sup> ᠳᠢᠶᠠᠭ᠋ᠷᠠᠮ ᠳᠡᠭᠡᠷ ᠡ᠋ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠸᠧᠺᠲ᠋ᠣᠷ<sup>vector</sup> ᠡᠭᠦᠰᠬᠡᠵᠦ ᠪᠤᠶ ᠬᠣᠣᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ <span style="writing-mode: horizontal-tb;"><i>(a, b)</i></span> ᠳᠦᠷᠰᠦᠯᠡᠨ ᠦᠵᠡᠭᠦᠯᠵᠦ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
ᠡᠭᠦᠨ ᠳ᠋ᠦ: <br><span style="writing-mode: horizontal-tb;"><i>Re</i></span> ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ᠂ <br><span style="writing-mode: horizontal-tb;"><i>Im</i></span> ᠢᠨᠦ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠣᠯᠬᠤ ᠠᠭᠠᠳ <br><span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠢᠨᠦ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠳ᠋ᠦ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠬᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠮᠥᠨ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
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<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
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{| class="wikitable"<br />
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|[[File:Komplexe_addition.svg|thumb]]<br />
|style = "height: 220px;"|ᠬᠣᠶᠠᠷ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃ <br />
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= ᠬᠠᠰᠤᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠭᠡ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
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<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math><br />
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= ᠦᠷᠡᠵᠢᠬᠦ =<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃<ref>''Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И.'' Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — <abbr>М.</abbr>: Просвещение, 1998. — 288 с. — <nowiki>ISBN 5-09-008036-4</nowiki>.</ref> <br />
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<math>(a + bi) (c + di) = </math> <math>= ac + bci - bd + adi = </math> <math>= (ac - bd) + (bc + ad)i </math> <br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|<span style="writing-mode: horizontal-tb;"><i>2 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭ ᠠ᠋ ᠢᠢ (ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) <span style="writing-mode: horizontal-tb;"><i>3 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭᠠᠨ ᠳᠤ (ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃<br />
ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠢ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠣᠷᠤᠶ ᠲᠠᠢ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠵᠤ᠂ <span style="writing-mode: horizontal-tb;"><i>√5</i></span> ᠢᠢᠠᠷ ᠰᠤᠩᠭᠠᠨ ᠠ᠋᠃ ᠡᠨᠡ ᠢᠨᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠾᠢᠫᠣᠲ᠋ᠧᠨᠦᠽ ᠤᠨ<sup>hypotenuse</sup> ᠤᠷᠲᠤ ᠪᠣᠯᠠᠢ᠃ <br />
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= ᠬᠤᠪᠢᠶᠠᠬᠤ =<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠬᠤᠪᠢᠶᠠᠬᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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<math>\frac{a+bi}{c+di}=</math> <br />
<math>=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=</math> <br />
<math>=\frac{ac+bd}{c^2+d^2}+\left(\frac{bc-ad}{c^2+d^2}\right)i.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ<sup>quadrate</sup> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨᠡ᠋᠃<ref>Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01</ref><br />
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<math>(x+yi)^2=x^2-y^2 + 2xyi.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> <span style="writing-mode: horizontal-tb;"><i>(b ≠ 0)</i></span> ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ <br />
<math> \pm (\gamma + \delta i)</math> ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄<br />
<math>\gamma = \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠠ <math>\delta = (\sgn b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠣᠯᠤᠮᠤᠢ᠃<br />
<span style="writing-mode: horizontal-tb;"><i>sgn b</i></span> ᠭᠡᠭᠴᠢ ᠢᠨᠦ <math>\sgn b = \begin{cases} \ \ 1, & b > 0 \\ \ \ 0, & b = 0 \\ -1, & b < 0 \end{cases}</math> ᠭᠡᠵᠦ ᠲᠠᠢᠢᠯᠠᠭᠳᠠᠨᠠ᠃ <br />
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ᠡᠭᠦᠨ ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ ᠢᠢᠨ ᠲᠤᠬᠠᠢᠢᠲᠠ ᠳ᠋ᠤ <math> \pm (\gamma + \delta i)</math> ᠲᠣᠭᠠ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠦ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠲᠣᠭᠠ ᠢᠢ ᠭᠠᠷᠭᠠᠵᠤ ᠢᠷᠡᠬᠦ ᠶᠣᠰᠤᠲᠠᠢ᠃ <br />
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=ᠬᠡᠷᠡᠭᠯᠡᠭᠡ=<br />
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== ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ==<br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠳᠠᠪᠲᠠᠯᠲᠠ ᠪᠠᠷ ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠠᠢᠢᠭᠤᠯᠬᠤ᠃<br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠯ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠳ᠋ᠤ ᠡᠭᠦᠰᠬᠡᠳᠡᠭ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ<sup>fractal</sup> ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠵᠡᠰᠱᠢᠶ ᠡ᠋ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡ ᠬᠦ ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ <span style="writing-mode: horizontal-tb;"><i>z = 0</i></span> ᠡᠴᠡ ᠡᠬᠢᠯᠡᠭᠡᠳ ᠢᠲ᠋ᠧᠷᠠᠼ<sup>iteration</sup> ᠬᠢᠬᠦ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;"><i>f<sub>c</sub>(z)=z<sup>2</sup>+c</i></span> ᠹᠦᠨ᠍ᠻᠼ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠢᠢᠬᠤ ᠨᠥᠬᠦᠴᠡᠯ ᠪᠦᠬᠦᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠤ ᠣᠯᠠᠨᠯᠢᠬ ᠪᠠᠢᠢᠳᠠᠭ᠃ <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ ᠪᠠᠢᠢᠭᠠᠯᠢ ᠢᠢᠨ ᠭᠠᠢᠢᠬᠠᠮᠰᠢᠭᠲᠤ ᠳᠦᠷᠰᠦ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠦᠨ ᠰᠠᠢᠢᠬᠠᠨ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠶᠡᠷᠦ ᠪᠤᠰᠤ ᠢᠢᠨ ᠭᠣᠶᠤᠮᠰᠠᠭ ᠬᠣᠣᠰᠯᠠᠯ ᠢ ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠳᠡᠭ᠃<ref>ᠷᠡᠨᠴᠡᠨ ᠦ ᠡᠩᠬᠡᠪᠠᠲᠤ᠃ '''ᠫᠢᠲ᠋ᠾᠠᠭᠣᠷ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌'''᠃ ᠮᠣᠩᠭᠤᠯ ᠤᠯᠤᠯᠰ ᠊ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠪᠠ ᠲᠣᠭᠠᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠢᠢᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ https://imdt.ac.mn/c/1013874?content=1150891&fbclid=IwAR1HVqeXwT-h7dijj9EeIUzDvFustH99GAr9LHYQIU61XIByuBHCkSm10So 2020</ref> <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠵᠠᠭᠠᠭ ᠢ ᠳᠠᠪᠠᠬᠤ ᠳᠤ ᠪᠠᠨ ᠵᠦᠯᠢᠶ ᠠ᠋ ᠢᠢᠨ<sup>Julia</sup> ᠹᠷᠠᠻᠲᠠᠯ ᠤᠳ ᠦᠷᠭᠡᠯᠵᠢ ᠬᠣᠯᠪᠤᠯᠲᠠ ᠪᠠᠨ ᠠᠯᠳᠠᠵᠤ᠂ ᠹᠠᠲᠣᠤ ᠢᠢᠨ<sup>Fatou</sup> ᠲᠣᠭᠤᠰᠤ ᠪᠣᠯᠤᠨ ᠬᠦᠪᠢᠷᠠᠳᠠᠭ᠃<ref>ᠨᠡᠭᠡᠭᠡᠯᠲᠡᠲᠡᠢ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠳ᠃ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠳᠠᠯ᠃ ᠪᠢᠳᠡᠨ ᠦ ᠡᠷᠭᠢᠨ ᠲᠣᠭᠣᠷᠢᠨ ᠳ᠋ᠠᠬᠢ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠬᠡᠷᠬᠢᠨ ᠠᠵᠢᠯᠯᠠᠳᠠᠭ ᠪᠤᠢ᠃ ᠹᠷᠠᠻᠲᠯᠠ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠶᠢᠨ ᠹᠷᠠᠻᠲᠯᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢ᠌ᠭ᠌᠃ https://ultrait.ru/mn/smartphones/otkrytie-fraktalov-beskonechnost-fraktalov-kak-ustroen-mir.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/10/02</ref><br />
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== ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ==<br />
ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌<sup>quantum mechanics</sup> ᠠᠲ᠋ᠣᠮ ᠠᠴᠠ ᠵᠢᠵᠢᠭ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ᠂ ᠶᠠᠭᠤᠨ ᠤ ᠡᠮᠦᠨᠡ ᠪᠣᠽᠣᠨ<sup>boson</sup> (ᠵᠢᠱᠢᠶᠡ ᠨᠢ ᠹᠣᠲ᠋ᠣᠨ<sup>photon</sup> ᠪᠠ ᠹᠧᠷᠮᠢᠣᠨ ᠤ<sup>fermion</sup> (ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠨᠧᠦᠲ᠋ᠷᠣᠨ) ᠬᠥᠳᠡᠯᠭᠡᠭᠡᠨ᠂ ᠬᠠᠷᠢᠯᠴᠠᠨ ᠦᠢᠯᠡᠴᠢᠯᠡᠯ ᠢ ᠰᠤᠳᠤᠯᠳᠠᠭ᠃ <br />
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ᠴᠢᠩᠭᠢᠬᠦ ᠳᠦ ᠪᠡᠨ ᠲᠡᠳᠡᠭᠡᠷ ᠦᠨ ᠦᠢᠯᠡ ᠬᠥᠳᠡᠯᠦᠯ ᠦᠨ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭ᠌ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠯᠲᠠ ᠶᠢ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠤᠳᠬᠠ ᠪᠠᠷ ᠭᠠᠷᠭᠠᠳᠠᠭ᠃ <br />
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ᠴᠤᠬᠤᠮ ᠳᠠᠭᠠᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠦᠨᠳᠦᠰᠦ ᠰᠠᠭᠤᠷᠢ ᠪᠣᠯᠳᠠᠭ᠃ <br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠲᠤ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ <sup>Schrödinger</sup> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠠᠴᠢ ᠬᠣᠯᠪᠣᠭᠳᠠᠯ ᠨᠢ ᠰᠣᠩᠭᠣᠳᠠᠭ ᠹᠢᠽᠢᠻ ᠳ᠋ᠡᠬᠢ ᠨᠧᠧᠲ᠋ᠣᠨ᠍ ᠤ ᠬᠣᠶᠠᠳᠤᠭᠠᠷ ᠬᠠᠤᠯᠢ ᠶᠢᠨ ᠠᠳᠠᠯᠢ ᠶᠤᠮ᠃ <br />
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ᠠᠯᠢ ᠠᠯᠢ ᠨᠢ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠷᠢᠯᠠᠯ᠂ ᠢᠮᠫᠦ᠋ᠯᠰ ᠢ <sup>impuls</sup> ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭᠴᠢᠯᠠᠨ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠳᠠᠭ᠃ <br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠨᠢ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ<sup>function</sup> ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠬᠦ ᠳᠦ ᠲᠣᠬᠢᠷᠠᠮᠵᠢ ᠲᠠᠢ ᠤᠴᠢᠷ ᠠᠴᠠ ᠹᠢᠽᠢᠻ ᠦᠨ ᠡᠨᠡ ᠰᠠᠯᠪᠤᠷᠢ ᠳᠤ ᠵᠠᠶ᠋ᠢᠯᠠᠰᠢ ᠦᠭᠡᠢ ᠱᠠᠭᠠᠷᠳᠠᠯᠭᠠ ᠲᠠᠢ ᠶᠤᠮ᠃ <br />
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ᠴᠠᠭᠠᠰᠢᠯᠠᠪᠠᠯ᠂ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠢᠭᠤᠳ ᠨᠥᠯᠥᠭᠡ ᠪᠡᠷ ᠬᠢᠮᠢ ᠶᠢᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠬᠥᠭᠵᠢᠯ ᠡᠷᠴᠢᠮᠵᠢᠭᠰᠡᠨ᠃ <br />
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1927 ᠣᠨ ᠳᠤ ᠸᠠᠯᠲ᠋ᠧᠷ ᠾᠠᠢᠢᠲ᠋ᠯᠧᠷ<sup>Walter Heitler</sup> ᠂ ᠹᠷᠢᠼ ᠯᠣᠨ᠍ᠳᠣᠨ<sup>Fritz London</sup> ᠨᠠᠷ ᠸᠠᠯᠧᠨ᠍ᠲ ᠤᠨ<sup>valency</sup> ᠬᠣᠯᠪᠣᠭᠠᠰᠤ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠲᠣᠮᠢᠶᠠᠯᠠᠭᠰᠠᠨ᠃ <br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠨᠢᠭᠡ ᠭᠣᠣᠯ ᠠᠰᠠᠭᠤᠳᠠᠯ ᠪᠣᠯ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ ᠦᠨ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ ᠤᠯᠬᠤ ᠶᠠᠪᠤᠳᠠᠯ ᠶᠤᠮ᠃ <br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠭᠡᠳᠡᠭ ᠨᠢ ᠲᠣᠳᠣᠷᠬᠠᠢ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠵᠤᠷᠪᠤᠰ ᠲᠤ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠵᠤ ᠪᠣᠯᠬᠤ ᠪᠠᠶ᠋ᠢᠷᠢᠰᠢᠯ ᠤᠳ ᠤᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠲᠠᠷᠬᠠᠯᠲᠠ ᠶᠤᠮ᠄ <br />
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<small><math>\frac{-\hbar}{2m}\frac{\partial^2\Psi(x,t)}{\partial x^2} + V(x,t)\Psi(x,t) = i\hbar\frac{\partial \Psi(x,t)}{\partial t}</math></small><br />
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ᠻᠢᠨᠧᠲᠢᠭ<sup>kinetic</sup> ᠧᠨᠧᠷᠭᠢ<sup>energy</sup><br />
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ᠫᠣᠲ᠋ᠧᠨ᠍ᠼᠢᠶᠠᠯ<sup>potencial</sup> ᠧᠨᠧᠷᠭᠢ<br />
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ᠨᠡᠢᠢᠲᠡ ᠧᠨᠧᠷᠭᠢ<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠲᠦ ᠰᠢᠲᠦᠭᠰᠡᠨ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠠᠭᠤᠷᠢ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠨᠢᠭᠡ ᠪᠣᠯ ᠳᠡᠭᠡᠷᠡ ᠳᠤᠷᠠᠳᠤᠭᠰᠠᠨ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡᠬᠦ ᠲᠣᠮᠢᠶᠠ ᠶᠢ ᠠᠰᠢᠭᠯᠠᠭᠰᠠᠨ ᠢᠶ᠋ᠠᠷ ᠤᠰᠤ ᠲᠥᠷᠦᠭᠴᠢ ᠶᠢᠨ ᠮᠣᠯᠧᠻᠤᠯ<sup>molecule</sup> ᠳᠠᠬᠢ ᠬᠣᠶᠠᠷ ᠠᠲ᠋ᠣᠮ ᠨᠢ ᠻᠣᠸᠠᠯᠧᠨ᠍ᠲ<sup>covalent</sup> ᠬᠣᠯᠪᠣᠭᠠ ᠭᠡᠳᠡᠭ ᠢ ᠡᠭᠦᠰᠬᠡᠨ ᠧᠯᠧᠻᠲ᠋ᠷᠣᠨ ᠨᠤᠭᠤᠳ ᠢᠢᠠᠨ ᠬᠤᠪᠢᠶᠠᠯᠴᠠᠵᠤ ᠪᠠᠶ᠋ᠢᠳᠠᠭ ᠢ ᠪᠠᠲᠤᠯᠠᠭᠰᠠᠨ᠃<ref>Josiah Wu. Real Life Applications of Complex Numbers. 2020 https://issuu.com/harrowhongkong/docs/final_scientific_harrovian_issue_vi-i/s/11488755</ref><br />
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== ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠢᠢᠨ ᠣᠨᠤᠯ ==<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠶᠢ ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ᠂ ᠢᠯᠠᠩᠭᠤᠶᠠ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠨ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠶ᠋ᠢᠳᠠᠯ ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠭᠡᠨ ᠳᠤ ᠬᠡᠷᠡᠭᠯᠡᠳᠡᠭ᠃ <br />
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ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠳᠤ 《ᠰᠢᠰᠲ᠋ᠧᠮ》 ᠭᠡᠳᠡᠭ ᠦᠭᠡ ᠶᠢ ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ ᠪᠥᠭᠡᠳ ᠡᠨᠡ ᠨᠢ ᠵᠠᠪᠠᠯ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠢ ᠬᠡᠯᠡᠳᠡᠭ ᠦᠭᠡᠢ᠃ <br />
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ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠡᠭᠦᠨ ᠢ ᠬᠥᠷᠥᠩᠭᠡ ᠶᠢᠨ ᠵᠠᠬᠠ ᠵᠡᠭᠡᠯᠢ ᠳ᠋ᠡᠬᠢ ᠥᠭᠡᠷᠡᠴᠢᠯᠡᠯᠲᠡ᠂ ᠬᠢᠮᠢ ᠶᠢᠨ ᠤᠷᠪᠠᠯ᠂ ᠦᠢᠯᠡ ᠶᠠᠪᠤᠴᠠ ᠶᠢ ᠰᠤᠳᠤᠯᠬᠤ ᠳᠤ ᠬᠡᠷᠡᠭᠯᠡᠵᠦ ᠪᠣᠯᠤᠨᠠ᠃ <br />
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ᠲᠡᠭᠦᠨᠴᠢᠯᠡᠨ ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠷᠣᠪᠣᠲ ᠤᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠳᠤ<sup>technology</sup> ᠥᠷᠭᠡᠨ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ ᠲᠤᠯᠠ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠠᠨᠠᠯᠢᠽ ᠴᠤ ᠪᠠᠰᠠ ᠲᠡᠭᠦᠨ ᠳᠦ ᠬᠡᠷᠡᠭᠯᠡᠭᠳᠡᠨᠡ ᠭᠡᠰᠡᠨ ᠦᠭᠡ᠃<ref>Ujjvala Y. Gawarguru, Mitali K. Tibdewal, Rajashri A. Naphade, Rahul M. Jethwani. The Review of Introduction & Application of Complex Number in Engineering. 2nd National Conference Recent Innovations in Science and Engineering (NC-RISE 17). Volume: 5 Issue: 9. pp55 – 57. ISSN: 2321-8169. https://ijritcc.org/download/conferences/NC-RISE_17/Track_6_(ASH)/1506931102_02-10-2017.pdf</ref><br />
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ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠳᠤ ᠵᠠᠷᠢᠮ ᠳᠤ ᠪᠠᠨ ᠯᠠᠫ᠊ᠯᠠᠰ ᠤᠨ<sup>Laplace</sup> ᠬᠤᠪᠢᠷᠠᠭᠠᠯᠲᠠ ᠶᠢ ᠠᠰᠢᠭᠯᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠳ ᠢ<sup>system</sup> ᠴᠠᠭ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠮᠤᠵᠢ ᠠᠴᠠ ᠳᠠᠪᠲᠠᠮᠵᠢ ᠶᠢᠨ ᠮᠤᠵᠢ ᠳᠤ ᠰᠢᠯᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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ᠳᠠᠷᠠᠭᠠ ᠨᠢ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠨ ᠲᠡᠭ ᠪᠠ ᠲᠤᠶ᠋ᠢᠯ ᠤᠳ ᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠰᠢᠨᠵᠢᠯᠡᠨᠡ᠃<br />
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ᠢᠵᠠᠭᠤᠷ ᠤᠨ ᠮᠥᠷ ᠦᠨ (root locus) ᠠᠷᠭᠠ ᠪᠠᠷ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠳᠦ ᠲᠡᠭ ᠪᠠ ᠲᠤᠶ᠋ᠢᠯ ᠤᠳ ᠵᠡᠭᠦᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠠᠶ᠋ᠢᠨᠠ ᠤᠤ᠂ ᠡᠰᠡᠪᠡᠯ ᠪᠠᠷᠠᠭᠤᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠠᠶ᠋ᠢᠨᠠ ᠤᠤ ᠭᠡᠳᠡᠭ ᠨᠢ ᠴᠢᠬᠤᠯᠠ᠃ <br />
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ᠥᠭᠡᠷᠡ ᠪᠡᠷ ᠬᠡᠯᠡᠪᠡᠯ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠨᠢ ᠲᠡᠭ ᠡᠴᠡ ᠶᠡᠬᠡ ᠪᠠᠢᠢᠨᠠ ᠤᠤ ᠡᠰᠡᠪᠡᠯ ᠪᠠᠭᠠ ᠪᠠᠶ᠋ᠢᠨᠠ ᠤᠤ ᠭᠡᠳᠡᠭ ᠨᠢ ᠴᠢᠬᠤᠯᠠ ᠶᠤᠮ᠃ <br />
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ᠴᠠᠭ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠬᠤᠪᠢᠷᠠᠯᠲᠠ ᠦᠭᠡᠢ ᠱᠤᠭᠤᠮᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠨᠢ᠄<br />
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- ᠪᠠᠷᠠᠭᠤᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠲᠤᠶ᠋ᠢᠯ ᠲᠠᠢ ᠪᠣᠯ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠂<br />
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- ᠵᠡᠭᠦᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠦᠬᠦ ᠲᠤᠶ᠋ᠢᠯ ᠨᠢ ᠪᠠᠶ᠋ᠢᠪᠠᠯ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠂<br />
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- ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠲᠤᠶ᠋ᠢᠯ ᠲᠠᠢ ᠪᠣᠯ ᠶᠠᠯᠢ ᠦᠭᠡᠶ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠃<br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ - complex number - комплексное число<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ - imaginary unit - мнимая единица <br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭᠠ - imaginary number - чисто мнимое число<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ - imaginaty part - мнимая часть<br />
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ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠ - real number - действительное число<br />
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ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ - real part - действительная часть<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ - complex plane - комлексная плоскость<br />
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ᠲᠡᠩᠬᠡᠯᠢᠭ - axis - ось<br />
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ᠣᠯᠠᠨᠯᠢᠭ - set - множество<br />
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ᠡᠶᠡᠷᠭᠦ - positive - положительный<br />
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ᠰᠥᠷᠭᠦ - negative - отрицательный<br />
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ᠲᠡᠭᠰᠢᠳᠬᠡᠯ - equation - уравнение<br />
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ᠲᠣᠮᠢᠶ ᠠ᠋ - formula - формула<br />
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ᠡᠷᠭᠢᠯᠲᠡ - rotation - поворот<br />
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ᠮᠠᠭᠠᠳᠯᠠᠯ - probability - вероятность<br />
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ᠬᠡᠮ - degree - градус<br />
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ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠢᠢᠨ ᠣᠨᠤᠯ - control theory - теория управления<br />
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ᠠᠲ᠋ᠣᠮ - atom - атом<br />
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ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮ - elementary particle - элементарная частица<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ - wave function - волновая функция<br />
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ᠳᠠᠪᠲᠠᠮᠵᠢ - frequency - частота<br />
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ᠹᠷᠠᠻᠲ᠋ᠠᠯ - fractal - фрактал<br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ - quantum mechanics - квантовая механика<br />
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ᠰᠣᠩᠭᠣᠳᠠᠭ ᠹᠢᠽᠢᠻ - classical physics - классическая физика<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==</div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%BA%E1%A0%A3%E1%A0%AE%E1%A0%AB%E1%A0%8A%E1%A0%AF%E1%A0%A7%E1%A0%BA%E1%A0%B0_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%A0&diff=2755
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-10-10T15:20:26Z
<p>Angka: Spelling</p>
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'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ<sup>complex</sup> ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠭᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠥᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃ <br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠢᠢ ᠲᠥᠯᠦᠭᠡᠯᠡᠵᠦ ᠪᠤᠶ ᠠᠷᠭᠠᠨ᠋ᠳ᠋ ᠤᠨ<sup>Argand</sup> ᠳᠢᠶᠠᠭ᠋ᠷᠠᠮ ᠳᠡᠭᠡᠷ ᠡ᠋ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠸᠧᠺᠲ᠋ᠣᠷ<sup>vector</sup> ᠡᠭᠦᠰᠬᠡᠵᠦ ᠪᠤᠶ ᠬᠣᠣᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ <span style="writing-mode: horizontal-tb;"><i>(a, b)</i></span> ᠳᠦᠷᠰᠦᠯᠡᠨ ᠦᠵᠡᠭᠦᠯᠵᠦ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
ᠡᠭᠦᠨ ᠳ᠋ᠦ: <br><span style="writing-mode: horizontal-tb;"><i>Re</i></span> ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ᠂ <br><span style="writing-mode: horizontal-tb;"><i>Im</i></span> ᠢᠨᠦ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠣᠯᠬᠤ ᠠᠭᠠᠳ <br><span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠢᠨᠦ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠳ᠋ᠦ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠬᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠮᠥᠨ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
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<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
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{| class="wikitable"<br />
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|[[File:Komplexe_addition.svg|thumb]]<br />
|style = "height: 220px;"|ᠬᠣᠶᠠᠷ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃ <br />
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= ᠬᠠᠰᠤᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠭᠡ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
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<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math><br />
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= ᠦᠷᠡᠵᠢᠬᠦ =<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃<ref>''Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И.'' Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — <abbr>М.</abbr>: Просвещение, 1998. — 288 с. — <nowiki>ISBN 5-09-008036-4</nowiki>.</ref> <br />
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<math>(a + bi) (c + di) = </math> <math>= ac + bci - bd + adi = </math> <math>= (ac - bd) + (bc + ad)i </math> <br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|<span style="writing-mode: horizontal-tb;"><i>2 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭ ᠠ᠋ ᠢᠢ (ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) <span style="writing-mode: horizontal-tb;"><i>3 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭᠠᠨ ᠳᠤ (ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃<br />
ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠢ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠣᠷᠤᠶ ᠲᠠᠢ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠵᠤ᠂ <span style="writing-mode: horizontal-tb;"><i>√5</i></span> ᠢᠢᠠᠷ ᠰᠤᠩᠭᠠᠨ ᠠ᠋᠃ ᠡᠨᠡ ᠢᠨᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠾᠢᠫᠣᠲ᠋ᠧᠨᠦᠽ ᠤᠨ<sup>hypotenuse</sup> ᠤᠷᠲᠤ ᠪᠣᠯᠠᠢ᠃ <br />
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= ᠬᠤᠪᠢᠶᠠᠬᠤ =<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠬᠤᠪᠢᠶᠠᠬᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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<math>\frac{a+bi}{c+di}=</math> <br />
<math>=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=</math> <br />
<math>=\frac{ac+bd}{c^2+d^2}+\left(\frac{bc-ad}{c^2+d^2}\right)i.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ<sup>quadrate</sup> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨᠡ᠋᠃<ref>Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01</ref><br />
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<math>(x+yi)^2=x^2-y^2 + 2xyi.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> <span style="writing-mode: horizontal-tb;"><i>(b ≠ 0)</i></span> ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ <br />
<math> \pm (\gamma + \delta i)</math> ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄<br />
<math>\gamma = \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠠ <math>\delta = (\sgn b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠣᠯᠤᠮᠤᠢ᠃<br />
<span style="writing-mode: horizontal-tb;"><i>sgn b</i></span> ᠭᠡᠭᠴᠢ ᠢᠨᠦ <math>\sgn b = \begin{cases} \ \ 1, & b > 0 \\ \ \ 0, & b = 0 \\ -1, & b < 0 \end{cases}</math> ᠭᠡᠵᠦ ᠲᠠᠢᠢᠯᠠᠭᠳᠠᠨᠠ᠃ <br />
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ᠡᠭᠦᠨ ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ ᠢᠢᠨ ᠲᠤᠬᠠᠢᠢᠲᠠ ᠳ᠋ᠤ <math> \pm (\gamma + \delta i)</math> ᠲᠣᠭᠠ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠦ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠲᠣᠭᠠ ᠢᠢ ᠭᠠᠷᠭᠠᠵᠤ ᠢᠷᠡᠬᠦ ᠶᠣᠰᠤᠲᠠᠢ᠃ <br />
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=ᠬᠡᠷᠡᠭᠯᠡᠭᠡ=<br />
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== ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ==<br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠳᠠᠪᠲᠠᠯᠲᠠ ᠪᠠᠷ ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠠᠢᠢᠭᠤᠯᠬᠤ᠃<br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠯ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠳ᠋ᠤ ᠡᠭᠦᠰᠬᠡᠳᠡᠭ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ<sup>fractal</sup> ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠵᠡᠰᠱᠢᠶ ᠡ᠋ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡ ᠬᠦ ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ <span style="writing-mode: horizontal-tb;"><i>z = 0</i></span> ᠡᠴᠡ ᠡᠬᠢᠯᠡᠭᠡᠳ ᠢᠲ᠋ᠧᠷᠠᠼ ᠬᠢᠬᠦ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;"><i>f<sub>c</sub>(z)=z<sup>2</sup>+c</i></span> ᠹᠦᠨ᠍ᠻᠼ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠢᠢᠬᠤ ᠨᠥᠬᠦᠴᠡᠯ ᠪᠦᠬᠦᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠤ ᠣᠯᠠᠨᠯᠢᠬ ᠪᠠᠢᠢᠳᠠᠭ᠃ <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ ᠪᠠᠢᠢᠭᠠᠯᠢ ᠢᠢᠨ ᠭᠠᠢᠢᠬᠠᠮᠰᠢᠭᠲᠤ ᠳᠦᠷᠰᠦ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠦᠨ ᠰᠠᠢᠢᠬᠠᠨ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠶᠡᠷᠦ ᠪᠤᠰᠤ ᠢᠢᠨ ᠭᠣᠶᠤᠮᠰᠠᠭ ᠬᠣᠣᠰᠯᠠᠯ ᠢ ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠳᠡᠭ᠃<ref>ᠷᠡᠨᠴᠡᠨ ᠦ ᠡᠩᠬᠡᠪᠠᠲᠤ᠃ '''ᠫᠢᠲ᠋ᠾᠠᠭᠣᠷ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌'''᠃ ᠮᠣᠩᠭᠤᠯ ᠤᠯᠤᠯᠰ ᠊ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠪᠠ ᠲᠣᠭᠠᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠢᠢᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ https://imdt.ac.mn/c/1013874?content=1150891&fbclid=IwAR1HVqeXwT-h7dijj9EeIUzDvFustH99GAr9LHYQIU61XIByuBHCkSm10So 2020</ref> <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠵᠠᠭᠠᠭ ᠢ ᠳᠠᠪᠠᠬᠤ ᠳᠤ ᠪᠠᠨ ᠵᠦᠯᠢᠶ ᠠ᠋ ᠢᠢᠨ<sup>Julia</sup> ᠹᠷᠠᠻᠲᠠᠯ ᠤᠳ ᠦᠷᠭᠡᠯᠵᠢ ᠬᠣᠯᠪᠤᠯᠲᠠ ᠪᠠᠨ ᠠᠯᠳᠠᠵᠤ᠂ ᠹᠠᠲᠣᠤ ᠢᠢᠨ<sup>Fatou</sup> ᠲᠣᠭᠤᠰᠤ ᠪᠣᠯᠤᠨ ᠬᠦᠪᠢᠷᠠᠳᠠᠭ᠃<ref>ᠨᠡᠭᠡᠭᠡᠯᠲᠡᠲᠡᠢ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠳ᠃ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠳᠠᠯ᠃ ᠪᠢᠳᠡᠨ ᠦ ᠡᠷᠭᠢᠨ ᠲᠣᠭᠣᠷᠢᠨ ᠳ᠋ᠠᠬᠢ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠬᠡᠷᠬᠢᠨ ᠠᠵᠢᠯᠯᠠᠳᠠᠭ ᠪᠤᠢ᠃ ᠹᠷᠠᠻᠲᠯᠠ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠶᠢᠨ ᠹᠷᠠᠻᠲᠯᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢ᠌ᠭ᠌᠃ https://ultrait.ru/mn/smartphones/otkrytie-fraktalov-beskonechnost-fraktalov-kak-ustroen-mir.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/10/02</ref><br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|ᠳᠦᠷᠰᠦ ᠠᠮᠢᠯᠠᠭᠤᠯᠤᠨ ᠵᠦᠯᠢᠶ ᠠ᠋ ᠢᠢᠨ <sup>Julia</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠦᠵᠡᠭᠦᠯᠦᠭᠰᠡᠨ ᠢᠨᠦ᠃<br />
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== ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ==<br />
ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌<sup>quantum mechanics</sup> ᠠᠲ᠋ᠣᠮ ᠠᠴᠠ ᠵᠢᠵᠢᠭ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ᠂ ᠶᠠᠭᠤᠨ ᠤ ᠡᠮᠦᠨᠡ ᠪᠣᠽᠣᠨ<sup>boson</sup> (ᠵᠢᠱᠢᠶᠡ ᠨᠢ ᠹᠣᠲ᠋ᠣᠨ<sup>photon</sup>ᠪᠠ ᠹᠧᠷᠮᠢᠣᠨ ᠤ<sup>fermion</sup> (ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠨᠧᠦᠲ᠋ᠷᠣᠨ) ᠬᠥᠳᠡᠯᠭᠡᠭᠡᠨ᠂ ᠬᠠᠷᠢᠯᠴᠠᠨ ᠦᠢᠯᠡᠴᠢᠯᠡᠯ ᠢ ᠰᠤᠳᠤᠯᠳᠠᠭ᠃ <br />
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ᠴᠢᠩᠭᠢᠬᠦ ᠳᠦ ᠪᠡᠨ ᠲᠡᠳᠡᠭᠡᠷ ᠦᠨ ᠦᠢᠯᠡ ᠬᠥᠳᠡᠯᠦᠯ ᠦᠨ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭ᠌ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠯᠲᠠ ᠶᠢ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠤᠳᠬᠠ ᠪᠠᠷ ᠭᠠᠷᠭᠠᠳᠠᠭ᠃ <br />
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ᠴᠤᠬᠤᠮ ᠳᠠᠭᠠᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠦᠨᠳᠦᠰᠦ ᠰᠠᠭᠤᠷᠢ ᠪᠣᠯᠳᠠᠭ᠃ <br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠲᠤ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ <sup>Schrödinger</sup> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠠᠴᠢ ᠬᠣᠯᠪᠣᠭᠳᠠᠯ ᠨᠢ ᠰᠣᠩᠭᠣᠳᠠᠭ ᠹᠢᠽᠢᡘ ᠳ᠋ᠡᠬᠢ ᠨᠧᠧᠲ᠋ᠣᠨ᠍ ᠤ ᠬᠣᠶᠠᠳᠤᠭᠠᠷ ᠬᠠᠤᠯᠢ ᠶᠢᠨ ᠠᠳᠠᠯᠢ ᠶᠤᠮ᠃ <br />
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ᠠᠯᠢ ᠠᠯᠢ ᠨᠢ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠷᠢᠯᠠᠯ᠂ ᠢᠮᠫᠦ᠋ᠯᠰ ᠢ <sup>impuls</sup> ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭᠴᠢᠯᠠᠨ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠳᠠᠭ᠃ <br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠨᠢ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ<sup>function</sup> ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠬᠦ ᠳᠦ ᠲᠣᠬᠢᠷᠠᠮᠵᠢ ᠲᠠᠢ ᠤᠴᠢᠷ ᠠᠴᠠ ᠹᠢᠽᠢᠻ ᠦᠨ ᠡᠨᠡ ᠰᠠᠯᠪᠤᠷᠢ ᠳᠤ ᠵᠠᠶ᠋ᠢᠯᠠᠰᠢ ᠦᠭᠡᠢ ᠱᠠᠭᠠᠷᠳᠠᠯᠭᠠ ᠲᠠᠢ ᠶᠤᠮ᠃ <br />
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ᠴᠠᠭᠠᠰᠢᠯᠠᠪᠠᠯ᠂ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠢᠭᠤᠳ ᠨᠥᠯᠥᠭᠡ ᠪᠡᠷ ᠬᠢᠮᠢ ᠶᠢᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠬᠥᠭᠵᠢᠯ ᠡᠷᠴᠢᠮᠵᠢᠭᠰᠡᠨ᠃ <br />
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1927 ᠣᠨ ᠳᠤ ᠸᠠᠯᠲ᠋ᠧᠷ ᠾᠠᠢᠢᠲ᠋ᠯᠧᠷ<sup>Walter Heitler</sup> ᠂ ᠹᠷᠢᠼ ᠯᠣᠨ᠍ᠳᠣᠨ<sup>Fritz London</sup> ᠨᠠᠷ ᠸᠠᠯᠧᠨ᠍ᠲ ᠤᠨ ᠬᠣᠯᠪᠣᠭᠠᠰᠤ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠲᠣᠮᠢᠶᠠᠯᠠᠭᠰᠠᠨ᠃ <br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠨᠢᠭᠡ ᠭᠣᠣᠯ ᠠᠰᠠᠭᠤᠳᠠᠯ ᠪᠣᠯ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ ᠦᠨ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ ᠤᠯᠬᠤ ᠶᠠᠪᠤᠳᠠᠯ ᠶᠤᠮ᠃ <br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠭᠡᠳᠡᠭ ᠨᠢ ᠲᠣᠳᠣᠷᠬᠠᠢ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠵᠤᠷᠪᠤᠰ ᠲᠤ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠵᠤ ᠪᠣᠯᠬᠤ ᠪᠠᠶ᠋ᠢᠷᠢᠰᠢᠯ ᠤᠳ ᠤᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠲᠠᠷᠬᠠᠯᠲᠠ ᠶᠤᠮ᠄ <br />
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<small><math>\frac{-\hbar}{2m}\frac{\partial^2\Psi(x,t)}{\partial x^2} + V(x,t)\Psi(x,t) = i\hbar\frac{\partial \Psi(x,t)}{\partial t}</math></small><br />
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ᠻᠢᠨᠧᠲᠢᠭ<sup>kinetic</sup> ᠧᠨᠧᠷᠭᠢ<sup>energy</sup><br />
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ᠫᠣᠲ᠋ᠧᠨ᠍ᠼᠢᠶᠠᠯ<sup>potencial</sup> ᠧᠨᠧᠷᠭᠢ<br />
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ᠨᠡᠢᠢᠲᠡ ᠧᠨᠧᠷᠭᠢ<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠲᠦ ᠰᠢᠲᠦᠭᠰᠡᠨ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠠᠭᠤᠷᠢ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠨᠢᠭᠡ ᠪᠣᠯ ᠳᠡᠭᠡᠷᠡ ᠳᠤᠷᠠᠳᠤᠭᠰᠠᠨ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡᠬᠦ ᠲᠣᠮᠢᠶᠠ ᠶᠢ ᠠᠰᠢᠭᠯᠠᠭᠰᠠᠨ ᠢᠶ᠋ᠠᠷ ᠤᠰᠤ ᠲᠥᠷᠦᠭᠴᠢ ᠶᠢᠨ ᠮᠣᠯᠧᠻᠤᠯ ᠳᠠᠬᠢ ᠬᠣᠶᠠᠷ ᠠᠲ᠋ᠣᠮ ᠨᠢ ᠻᠣᠸᠠᠯᠧᠨ᠍ᠲ<sup>covalent</sup> ᠬᠣᠯᠪᠣᠭᠠ ᠭᠡᠳᠡᠭ ᠢ ᠡᠭᠦᠰᠬᠡᠨ ᠧᠯᠧᠻᠲ᠋ᠷᠣᠨ ᠨᠤᠭᠤᠳ ᠢᠢᠠᠨ ᠬᠤᠪᠢᠶᠠᠯᠴᠠᠵᠤ ᠪᠠᠶ᠋ᠢᠳᠠᠭ ᠢ ᠪᠠᠲᠤᠯᠠᠭᠰᠠᠨ᠃<ref>Josiah Wu. Real Life Applications of Complex Numbers. 2020 https://issuu.com/harrowhongkong/docs/final_scientific_harrovian_issue_vi-i/s/11488755</ref><br />
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== ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠢᠢᠨ ᠣᠨᠤᠯ ==<br />
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{| class="wikitable"<br />
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|style = "height: 340px;"|ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠢᠢᠨ ᠲᠣᠭᠲᠠᠯᠴᠠᠭᠠᠨ ᠤ ᠡᠩ ᠦᠨ ᠪᠦᠳᠦᠭᠦᠪᠴᠢ᠃ <br><br><span style="writing-mode: horizontal-tb;"><i>X(s)</i></span> ᠣᠷᠤᠯᠲᠠ ᠢᠢᠨ ᠮᠡᠳᠡᠭᠡᠯᠡᠯ᠂ <br><br><span style="writing-mode: horizontal-tb;"><i>Y(s)</i></span> ᠭᠠᠷᠤᠯᠲᠠ ᠢᠢᠨ ᠮᠡᠳᠡᠭᠡᠯᠡᠯ᠂ <br><br><span style="writing-mode: horizontal-tb;"><i>G(s)</i></span> ᠤᠷᠤᠭᠰᠢ ᠳᠠᠮᠵᠢᠭᠤᠯᠬᠤ ᠹᠦᠨ᠍ᠻᠼ<sup>function</sup>᠂ <br><br><span style="writing-mode: horizontal-tb;"><i>H(s)</i></span> ᠡᠷᠭᠢᠬᠦ ᠬᠣᠯᠪᠤᠭᠠ<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠶᠢ ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ᠂ ᠢᠯᠠᠩᠭᠤᠶᠠ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠨ ᠲᠣᠭᠲᠠᠪᠤᠷᠢᠲᠠᠢ ᠪᠠᠶᠢᠳᠠᠯ ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠭᠡᠨ ᠳᠤ ᠬᠡᠷᠡᠭᠯᠡᠳᠡᠭ᠃ <br />
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ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠳᠤ 《ᠰᠢᠰᠲ᠋ᠧᠮ》 ᠭᠡᠳᠡᠭ ᠦᠭᠡ ᠶᠢ ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ ᠪᠥᠭᠡᠳ ᠡᠨᠡ ᠨᠢ ᠵᠠᠪᠠᠯ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠢ ᠬᠡᠯᠡᠳᠡᠭ ᠦᠭᠡᠢ᠃ <br />
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ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠡᠭᠦᠨ ᠢ ᠬᠥᠷᠥᠩᠭᠡ ᠶᠢᠨ ᠵᠠᠬᠠ ᠵᠡᠭᠡᠯᠢ ᠳ᠋ᠡᠬᠢ ᠥᠭᠡᠷᠡᠴᠢᠯᠡᠯᠲᠡ᠂ ᠬᠢᠮᠢ ᠶᠢᠨ ᠤᠷᠪᠠᠯ᠂ ᠦᠢᠯᠡ ᠶᠠᠪᠤᠴᠠ ᠶᠢ ᠰᠤᠳᠤᠯᠬᠤ ᠳᠤ ᠬᠡᠷᠡᠭᠯᠡᠵᠦ ᠪᠣᠯᠤᠨᠠ᠃ <br />
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ᠲᠡᠭᠦᠨᠴᠢᠯᠡᠨ ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠷᠣᠪᠣᠲ ᠤᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠳᠤ ᠥᠷᠭᠡᠨ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ ᠲᠤᠯᠠ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠠᠨᠠᠯᠢᠽ ᠴᠤ ᠪᠠᠰᠠ ᠲᠡᠭᠦᠨ ᠳᠦ ᠬᠡᠷᠡᠭᠯᠡᠭᠳᠡᠨᠡ ᠭᠡᠰᠡᠨ ᠦᠭᠡ᠃<ref>Ujjvala Y. Gawarguru, Mitali K. Tibdewal, Rajashri A. Naphade, Rahul M. Jethwani. The Review of Introduction & Application of Complex Number in Engineering. 2nd National Conference Recent Innovations in Science and Engineering (NC-RISE 17). Volume: 5 Issue: 9. pp55 – 57. ISSN: 2321-8169. https://ijritcc.org/download/conferences/NC-RISE_17/Track_6_(ASH)/1506931102_02-10-2017.pdf</ref><br />
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ᠬᠢᠨᠠᠯᠲᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠳᠤ ᠵᠠᠷᠢᠮ ᠳᠤ ᠪᠠᠨ ᠯᠠᠫ᠊ᠯᠠᠰ ᠤᠨ<sup>Laplace</sup> ᠬᠤᠪᠢᠷᠠᠭᠠᠯᠲᠠ ᠶᠢ ᠠᠰᠢᠭᠯᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠳ ᠢ<sup>system</sup> ᠴᠠᠭ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠮᠤᠵᠢ ᠠᠴᠠ ᠳᠠᠪᠲᠠᠮᠵᠢ ᠶᠢᠨ ᠮᠤᠵᠢ ᠳᠤ ᠰᠢᠯᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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ᠳᠠᠷᠠᠭᠠ ᠨᠢ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠨ ᠲᠡᠭ ᠪᠠ ᠲᠤᠶ᠋ᠢᠯ ᠤᠳ ᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠰᠢᠨᠵᠢᠯᠡᠨᠡ᠃<br />
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ᠢᠵᠠᠭᠤᠷ ᠤᠨ ᠮᠥᠷ ᠦᠨ ᠠᠷᠭᠠ ᠪᠠᠷ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠳᠦ ᠲᠡᠭ ᠪᠠ ᠲᠤᠶ᠋ᠢᠯ ᠤᠳ ᠵᠡᠭᠦᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠠᠶ᠋ᠢᠨᠠ ᠤᠤ᠂ ᠡᠰᠡᠪᠡᠯ ᠪᠠᠷᠠᠭᠤᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠠᠶ᠋ᠢᠨᠠ ᠤᠤ ᠭᠡᠳᠡᠭ ᠨᠢ ᠴᠢᠬᠤᠯᠠ᠃ <br />
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ᠥᠭᠡᠷᠡ ᠪᠡᠷ ᠬᠡᠯᠡᠪᠡᠯ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠨᠢ ᠲᠡᠭ ᠡᠴᠡ ᠶᠡᠬᠡ ᠪᠠᠢᠢᠨᠠ ᠤᠤ ᠡᠰᠡᠪᠡᠯ ᠪᠠᠭᠠ ᠪᠠᠶ᠋ᠢᠨᠠ ᠤᠤ ᠭᠡᠳᠡᠭ ᠨᠢ ᠴᠢᠬᠤᠯᠠ ᠶᠤᠮ᠃ ᠬᠡᠷᠪᠡ ᠴᠠᠭ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠬᠤᠪᠢᠷᠠᠯᠲᠠ ᠦᠭᠡᠢ ᠱᠤᠭᠤᠮᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠨᠢ᠄<br />
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- ᠪᠠᠷᠠᠭᠤᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠲᠤᠶ᠋ᠢᠯ ᠲᠠᠢ ᠪᠣᠯ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠂<br />
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- ᠵᠡᠭᠦᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠦᠬᠦ ᠲᠤᠶ᠋ᠢᠯ ᠨᠢ ᠪᠠᠶ᠋ᠢᠪᠠᠯ ᠲᠣᠭᠲᠠᠪᠤᠷᠢᠲᠠᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠂<br />
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- ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠲᠤᠶ᠋ᠢᠯ ᠲᠠᠢ ᠪᠣᠯ ᠶᠠᠯᠢᠭᠦᠢ ᠲᠣᠭᠲᠠᠪᠤᠷᠢᠲᠠᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠃<br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ - complex number - комплексное число<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ - imaginary unit - мнимая единица <br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭᠠ - imaginary number - чисто мнимое число<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ - imaginaty part - мнимая часть<br />
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ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠ - real number - действительное число<br />
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ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ - real part - действительная часть<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ - complex plane - комлексная плоскость<br />
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ᠲᠡᠩᠬᠡᠯᠢᠭ - axis - ось<br />
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ᠣᠯᠠᠨᠯᠢᠭ - set - множество<br />
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ᠡᠶᠡᠷᠭᠦ - positive - положительный<br />
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ᠰᠥᠷᠭᠦ - negative - отрицательный<br />
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ᠲᠡᠭᠰᠢᠳᠬᠡᠯ - equation - уравнение<br />
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ᠡᠷᠭᠢᠯᠲᠡ - rotation - поворот<br />
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ᠬᠡᠮ - degree - градус<br />
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ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠢᠢᠨ ᠣᠨᠤᠯ - control theory - теория управления<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==</div>
Angka
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-10-10T14:31:48Z
<p>Angka: Edit</p>
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'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ<sup>complex</sup> ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠯᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃ <br />
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|style = "height: 225px;"|ᠺᠣᠮᠫᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠢᠢ ᠲᠥᠯᠦᠭᠡᠯᠡᠵᠦ ᠪᠤᠶ ᠠᠷᠭᠠᠨ᠋ᠳ᠋ ᠤᠨ<sup>Argand</sup> ᠳᠢᠶᠠᠭ᠋ᠷᠠᠮ ᠳᠡᠭᠡᠷ ᠡ᠋ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠸᠧᠺᠲ᠋ᠣᠷ<sup>vector</sup> ᠡᠭᠦᠰᠬᠡᠵᠦ ᠪᠤᠶ ᠬᠣᠣᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ <span style="writing-mode: horizontal-tb;"><i>(a, b)</i></span> ᠳᠦᠷᠰᠦᠯᠡᠨ ᠦᠵᠡᠭᠦᠯᠵᠦ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
ᠡᠭᠦᠨ ᠳ᠋ᠦ: <br><span style="writing-mode: horizontal-tb;"><i>Re</i></span> ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ᠂ <br><span style="writing-mode: horizontal-tb;"><i>Im</i></span> ᠢᠨᠦ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠣᠯᠬᠤ ᠠᠭᠠᠳ <br><span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠢᠨᠦ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠳ᠋ᠦ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠬᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠮᠥᠨ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
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<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
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|style = "height: 220px;"|ᠬᠣᠶᠠᠷ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃ <br />
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= ᠬᠠᠰᠤᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠳᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
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<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math><br />
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= ᠦᠷᠡᠵᠢᠬᠦ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃<ref>''Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И.'' Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — <abbr>М.</abbr>: Просвещение, 1998. — 288 с. — <nowiki>ISBN 5-09-008036-4</nowiki>.</ref> <br />
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<math>(a + bi) (c + di) = </math> <math>= ac + bci - bd + adi = </math> <math>= (ac - bd) + (bc + ad)i </math> <br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|<span style="writing-mode: horizontal-tb;"><i>2 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭ ᠠ᠋ ᠢᠢ (ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠬᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>3 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭᠠᠨ ᠳᠤ (ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃<br />
ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠢ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠣᠷᠤᠶ ᠲᠠᠢ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠵᠤ᠂ <span style="writing-mode: horizontal-tb;"><i>√5</i></span> ᠢᠢᠠᠷ ᠰᠤᠩᠭᠠᠨ ᠠ᠋᠃ ᠡᠨᠡ ᠢᠨᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠾᠢᠫᠣᠲ᠋ᠧᠨᠦᠽ ᠤᠨ<sup>hypotenuse</sup> ᠤᠷᠲᠤ ᠪᠣᠯᠠᠢ᠃ <br />
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= ᠬᠤᠪᠢᠶᠠᠬᠤ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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<math>\frac{a+bi}{c+di}=</math> <br />
<math>=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=</math> <br />
<math>=\frac{ac+bd}{c^2+d^2}+\left(\frac{bc-ad}{c^2+d^2}\right)i.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ<sup>quadrate</sup> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨᠡ᠋᠃<ref>Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01</ref><br />
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<math>(x+yi)^2=x^2-y^2 + 2xyi.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> <span style="writing-mode: horizontal-tb;"><i>(b ≠ 0)</i></span> ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ <br />
<math> \pm (\gamma + \delta i)</math> ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄<br />
<math>\gamma = \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠠ <math>\delta = (\sgn b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠣᠯᠤᠮᠤᠢ᠃<br />
<span style="writing-mode: horizontal-tb;"><i>sgn b</i></span> ᠭᠡᠭᠴᠢ ᠢᠨᠦ <math>\sgn b = \begin{cases} \ \ 1, & b > 0 \\ \ \ 0, & b = 0 \\ -1, & b < 0 \end{cases}</math> ᠭᠡᠵᠦ ᠲᠠᠢᠢᠯᠠᠭᠳᠠᠨᠠ᠃ <br />
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ᠡᠭᠦᠨ ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ ᠢᠢ ᠲᠤᠬᠠᠢᠢᠲᠠ ᠳ᠋ᠤ <math> \pm (\gamma + \delta i)</math> ᠲᠣᠭᠠ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠦ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠲᠣᠭᠠ ᠢᠢ ᠭᠠᠷᠭᠠᠵᠤ ᠢᠷᠡᠬᠦ ᠶᠣᠰᠤᠲᠠᠢ᠃ <br />
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=ᠬᠡᠷᠡᠭᠯᠡᠭᠡ=<br />
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== ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ==<br />
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|style = "height: 225px;"|ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠳᠠᠪᠲᠠᠯᠲᠠ ᠪᠠᠷ ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠠᠢᠢᠭᠤᠯᠬᠤ᠃<br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠯ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠳ᠋ᠤ ᠡᠭᠦᠰᠬᠡᠳᠡᠭ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ<sup>fractal</sup> ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠵᠡᠰᠱᠢᠶ ᠡ᠋ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡ ᠬᠦ ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ <span style="writing-mode: horizontal-tb;"><i>z = 0</i></span> ᠡᠴᠡ ᠡᠬᠢᠯᠡᠭᠡᠳ ᠢᠲ᠋ᠧᠷᠠᠼ ᠬᠢᠬᠦ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;"><i>f<sub>c</sub>(z)=z<sup>2</sup>+c</i></span> ᠹᠦᠨ᠍ᠻᠼ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠢᠢᠬᠤ ᠨᠥᠬᠦᠴᠡᠯ ᠪᠦᠬᠦᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠤ ᠣᠯᠠᠨᠯᠢᠬ ᠪᠠᠢᠢᠳᠠᠭ᠃ <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ ᠪᠠᠢᠢᠭᠠᠯᠢ ᠢᠢᠨ ᠭᠠᠢᠢᠬᠠᠮᠰᠢᠭᠲᠤ ᠳᠦᠷᠰᠦ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠦᠨ ᠰᠠᠢᠢᠬᠠᠨ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠶᠡᠷᠦ ᠪᠤᠰᠤ ᠢᠢᠨ ᠭᠣᠶᠤᠮᠰᠠᠭ ᠬᠣᠣᠰᠯᠠᠯ ᠢ ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠳᠡᠭ᠃<ref>ᠷᠡᠨᠴᠡᠨ ᠦ ᠡᠩᠬᠡᠪᠠᠲᠤ᠃ '''ᠫᠢᠲ᠋ᠾᠠᠭᠣᠷ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌'''᠃ ᠮᠣᠩᠭᠤᠯ ᠤᠯᠤᠯᠰ ᠊ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠪᠠ ᠲᠣᠭᠠᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠢᠢᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ https://imdt.ac.mn/c/1013874?content=1150891&fbclid=IwAR1HVqeXwT-h7dijj9EeIUzDvFustH99GAr9LHYQIU61XIByuBHCkSm10So 2020</ref> <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠵᠠᠭᠠᠭ ᠢ ᠳᠠᠪᠠᠬᠤ ᠳᠤ ᠪᠠᠨ ᠵᠦᠯᠢᠶ ᠠ᠋ ᠢᠢᠨ<sup>Julia</sup> ᠹᠷᠠᠻᠲᠠᠯ ᠤᠳ ᠦᠷᠭᠡᠯᠵᠢ ᠬᠣᠯᠪᠤᠯᠲᠠ ᠪᠠᠨ ᠠᠯᠳᠠᠵᠤ᠂ ᠹᠠᠲᠣᠤ ᠢᠢᠨ<sup>Fatou</sup> ᠲᠣᠭᠤᠰᠤ ᠪᠣᠯᠤᠨ ᠬᠦᠪᠢᠷᠠᠳᠠᠭ᠃<ref>ᠨᠡᠭᠡᠭᠡᠯᠲᠡᠲᠡᠢ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠳ᠃ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠳᠠᠯ᠃ ᠪᠢᠳᠡᠨ ᠦ ᠡᠷᠭᠢᠨ ᠲᠣᠭᠣᠷᠢᠨ ᠳ᠋ᠠᠬᠢ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠬᠡᠷᠬᠢᠨ ᠠᠵᠢᠯᠯᠠᠳᠠᠭ ᠪᠤᠢ᠃ ᠹᠷᠠᠻᠲᠯᠠ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠶᠢᠨ ᠹᠷᠠᠻᠲᠯᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢ᠌ᠭ᠌᠃ https://ultrait.ru/mn/smartphones/otkrytie-fraktalov-beskonechnost-fraktalov-kak-ustroen-mir.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/10/02</ref><br />
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== ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ==<br />
ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌<sup>quantum mechanics</sup> ᠠᠲ᠋ᠣᠮ ᠠᠴᠠ ᠵᠢᠵᠢᠭ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ᠂ ᠶᠠᠭᠤᠨ ᠤ ᠡᠮᠦᠨᠡ ᠪᠣᠽᠣᠨ<sup>boson</sup> (ᠵᠢᠱᠢᠶᠡ ᠨᠢ ᠹᠣᠲ᠋ᠣᠨ<sup>photon</sup>ᠪᠠ ᠹᠧᠷᠮᠢᠣᠨ ᠤ<sup>fermion</sup> (ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠨᠧᠦᠲ᠋ᠷᠣᠨ) ᠬᠥᠳᠡᠯᠭᠡᠭᠡᠨ᠂ ᠬᠠᠷᠢᠯᠴᠠᠨ ᠦᠢᠯᠡᠴᠢᠯᠡᠯ ᠢ ᠰᠤᠳᠤᠯᠳᠠᠭ᠃ <br />
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ᠴᠢᠩᠭᠢᠬᠦ ᠳᠦ ᠪᠡᠨ ᠲᠡᠳᠡᠭᠡᠷ ᠦᠨ ᠦᠢᠯᠡ ᠬᠥᠳᠡᠯᠦᠯ ᠦᠨ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭ᠌ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠯᠲᠠ ᠶᠢ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠤᠳᠬᠠ ᠪᠠᠷ ᠭᠠᠷᠭᠠᠳᠠᠭ᠃ <br />
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ᠴᠤᠬᠤᠮ ᠳᠠᠭᠠᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠦᠨᠳᠦᠰᠦ ᠰᠠᠭᠤᠷᠢ ᠪᠣᠯᠳᠠᠭ᠃ <br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠲᠤ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ <sup>Schrödinger</sup> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠠᠴᠢ ᠬᠣᠯᠪᠣᠭᠳᠠᠯ ᠨᠢ ᠰᠣᠩᠭᠣᠳᠠᠭ ᠹᠢᠽᠢᡘ ᠳ᠋ᠡᠬᠢ ᠨᠧᠧᠲ᠋ᠣᠨ᠍ ᠤ ᠬᠣᠶᠠᠳᠤᠭᠠᠷ ᠬᠠᠤᠯᠢ ᠶᠢᠨ ᠠᠳᠠᠯᠢ ᠶᠤᠮ᠃ <br />
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ᠠᠯᠢ ᠠᠯᠢ ᠨᠢ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠷᠢᠯᠠᠯ᠂ ᠢᠮᠫᠦ᠋ᠯᠰ ᠢ <sup>impuls</sup> ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭᠴᠢᠯᠠᠨ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠳᠠᠭ᠃ <br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠨᠢ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ<sup>function</sup> ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠬᠦ ᠳᠦ ᠲᠣᠬᠢᠷᠠᠮᠵᠢ ᠲᠠᠢ ᠤᠴᠢᠷ ᠠᠴᠠ ᠹᠢᠽᠢᠻ ᠦᠨ ᠡᠨᠡ ᠰᠠᠯᠪᠤᠷᠢ ᠳᠤ ᠵᠠᠶ᠋ᠢᠯᠠᠰᠢ ᠦᠭᠡᠢ ᠱᠠᠭᠠᠷᠳᠠᠯᠭᠠ ᠲᠠᠢ ᠶᠤᠮ᠃ <br />
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ᠴᠠᠭᠠᠰᠢᠯᠠᠪᠠᠯ᠂ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠢᠭᠤᠳ ᠨᠥᠯᠥᠭᠡ ᠪᠡᠷ ᠬᠢᠮᠢ ᠶᠢᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠬᠥᠭᠵᠢᠯ ᠡᠷᠴᠢᠮᠵᠢᠭᠰᠡᠨ᠃ <br />
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1927 ᠣᠨ ᠳᠤ ᠸᠠᠯᠲ᠋ᠧᠷ ᠾᠠᠢᠢᠲ᠋ᠯᠧᠷ<sup>Walter Heitler</sup> ᠂ ᠹᠷᠢᠼ ᠯᠣᠨ᠍ᠳᠣᠨ<sup>Fritz London</sup> ᠨᠠᠷ ᠸᠠᠯᠧᠨ᠍ᠲ ᠤᠨ ᠬᠣᠯᠪᠣᠭᠠᠰᠤ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠲᠣᠮᠢᠶᠠᠯᠠᠭᠰᠠᠨ᠃ <br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠨᠢᠭᠡ ᠭᠣᠣᠯ ᠠᠰᠠᠭᠤᠳᠠᠯ ᠪᠣᠯ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ ᠦᠨ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ ᠤᠯᠬᠤ ᠶᠠᠪᠤᠳᠠᠯ ᠶᠤᠮ᠃ <br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠭᠡᠳᠡᠭ ᠨᠢ ᠲᠣᠳᠣᠷᠬᠠᠢ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠵᠤᠷᠪᠤᠰ ᠲᠤ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠵᠤ ᠪᠣᠯᠬᠤ ᠪᠠᠶ᠋ᠢᠷᠢᠰᠢᠯ ᠤᠳ ᠤᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠲᠠᠷᠬᠠᠯᠲᠠ ᠶᠤᠮ᠄ <br />
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<small><math>\frac{-\hbar}{2m}\frac{\partial^2\Psi(x,t)}{\partial x^2} + V(x,t)\Psi(x,t) = i\hbar\frac{\partial \Psi(x,t)}{\partial t}</math></small><br />
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ᠻᠢᠨᠧᠲᠢᠭ<sup>kinetic</sup> ᠧᠨᠧᠷᠭᠢ<sup>energy</sup><br />
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ᠫᠣᠲ᠋ᠧᠨ᠍ᠼᠢᠶᠠᠯ<sup>potencial</sup> ᠧᠨᠧᠷᠭᠢ<br />
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ᠨᠡᠢᠢᠲᠡ ᠧᠨᠧᠷᠭᠢ<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠲᠦ ᠰᠢᠲᠦᠭᠰᠡᠨ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠠᠭᠤᠷᠢ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠨᠢᠭᠡ ᠪᠣᠯ ᠳᠡᠭᠡᠷᠡ ᠳᠤᠷᠠᠳᠤᠭᠰᠠᠨ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡᠬᠦ ᠲᠣᠮᠢᠶᠠ ᠶᠢ ᠠᠰᠢᠭᠯᠠᠭᠰᠠᠨ ᠢᠶ᠋ᠠᠷ ᠤᠰᠤ ᠲᠥᠷᠦᠭᠴᠢ ᠶᠢᠨ ᠮᠣᠯᠧᠻᠤᠯ ᠳᠠᠬᠢ ᠬᠣᠶᠠᠷ ᠠᠲ᠋ᠣᠮ ᠨᠢ ᠻᠣᠸᠠᠯᠧᠨ᠍ᠲ<sup>covalent</sup> ᠬᠣᠯᠪᠣᠭᠠ ᠭᠡᠳᠡᠭ ᠢ ᠡᠭᠦᠰᠬᠡᠨ ᠧᠯᠧᠻᠲ᠋ᠷᠣᠨ ᠨᠤᠭᠤᠳ ᠢᠢᠠᠨ ᠬᠤᠪᠢᠶᠠᠯᠴᠠᠵᠤ ᠪᠠᠶ᠋ᠢᠳᠠᠭ ᠢ ᠪᠠᠲᠤᠯᠠᠭᠰᠠᠨ᠃<ref>Josiah Wu. Real Life Applications of Complex Numbers. 2020 https://issuu.com/harrowhongkong/docs/final_scientific_harrovian_issue_vi-i/s/11488755</ref><br />
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== ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠢᠢᠨ ᠣᠨᠤᠯ ==<br />
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|style = "height: 340px;"|ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠢᠢᠨ ᠲᠣᠭᠲᠠᠯᠴᠠᠭᠠᠨ ᠤ ᠡᠩ ᠦᠨ ᠪᠦᠳᠦᠭᠦᠪᠴᠢ᠃ <br><br><span style="writing-mode: horizontal-tb;"><i>X(s)</i></span> ᠣᠷᠤᠯᠲᠠ ᠢᠢᠨ ᠮᠡᠳᠡᠭᠡᠯᠡᠯ᠂ <br><br><span style="writing-mode: horizontal-tb;"><i>Y(s)</i></span> ᠭᠠᠷᠤᠯᠲᠠ ᠢᠢᠨ ᠮᠡᠳᠡᠭᠡᠯᠡᠯ᠂ <br><br><span style="writing-mode: horizontal-tb;"><i>G(s)</i></span> ᠤᠷᠤᠭᠰᠢ ᠳᠠᠮᠵᠢᠭᠤᠯᠬᠤ ᠹᠦᠨ᠍ᠻᠼ<sup>function</sup>᠂ <br><br><span style="writing-mode: horizontal-tb;"><i>H(s)</i></span> ᠡᠷᠭᠢᠬᠦ ᠬᠣᠯᠪᠤᠭᠠ<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠶᠢ ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ᠂ ᠢᠯᠠᠩᠭᠤᠶᠠ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠨ ᠲᠣᠭᠲᠠᠪᠤᠷᠢᠲᠠᠢ ᠪᠠᠶᠢᠳᠠᠯ ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠭᠡᠨ ᠳᠤ ᠬᠡᠷᠡᠭᠯᠡᠳᠡᠭ᠃ <br />
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ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠳᠤ 《ᠰᠢᠰᠲ᠋ᠧᠮ》 ᠭᠡᠳᠡᠭ ᠦᠭᠡ ᠶᠢ ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ ᠪᠥᠭᠡᠳ ᠡᠨᠡ ᠨᠢ ᠵᠠᠪᠠᠯ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠢ ᠬᠡᠯᠡᠳᠡᠭ ᠦᠭᠡᠢ᠃ <br />
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ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠡᠭᠦᠨ ᠢ ᠬᠥᠷᠥᠩᠭᠡ ᠶᠢᠨ ᠵᠠᠬᠠ ᠵᠡᠭᠡᠯᠢ ᠳ᠋ᠡᠬᠢ ᠥᠭᠡᠷᠡᠴᠢᠯᠡᠯᠲᠡ᠂ ᠬᠢᠮᠢ ᠶᠢᠨ ᠤᠷᠪᠠᠯ᠂ ᠦᠢᠯᠡ ᠶᠠᠪᠤᠴᠠ ᠶᠢ ᠰᠤᠳᠤᠯᠬᠤ ᠳᠤ ᠬᠡᠷᠡᠭᠯᠡᠵᠦ ᠪᠣᠯᠤᠨᠠ᠃ <br />
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ᠲᠡᠭᠦᠨᠴᠢᠯᠡᠨ ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠷᠣᠪᠣᠲ ᠤᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠳᠤ ᠥᠷᠭᠡᠨ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ ᠲᠤᠯᠠ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠠᠨᠠᠯᠢᠽ ᠴᠤ ᠪᠠᠰᠠ ᠲᠡᠭᠦᠨ ᠳᠦ ᠬᠡᠷᠡᠭᠯᠡᠭᠳᠡᠨᠡ ᠭᠡᠰᠡᠨ ᠦᠭᠡ᠃<ref>Ujjvala Y. Gawarguru, Mitali K. Tibdewal, Rajashri A. Naphade, Rahul M. Jethwani. The Review of Introduction & Application of Complex Number in Engineering. 2nd National Conference Recent Innovations in Science and Engineering (NC-RISE 17). Volume: 5 Issue: 9. pp55 – 57. ISSN: 2321-8169. https://ijritcc.org/download/conferences/NC-RISE_17/Track_6_(ASH)/1506931102_02-10-2017.pdf</ref><br />
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ᠬᠢᠨᠠᠯᠲᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠳᠤ ᠵᠠᠷᠢᠮ ᠳᠤ ᠪᠠᠨ ᠯᠠᠫ᠊ᠯᠠᠰ ᠤᠨ<sup>Laplace</sup> ᠬᠤᠪᠢᠷᠠᠭᠠᠯᠲᠠ ᠶᠢ ᠠᠰᠢᠭᠯᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠳ ᠢ<sup>system</sup> ᠴᠠᠭ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠮᠤᠵᠢ ᠠᠴᠠ ᠳᠠᠪᠲᠠᠮᠵᠢ ᠶᠢᠨ ᠮᠤᠵᠢ ᠳᠤ ᠰᠢᠯᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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ᠳᠠᠷᠠᠭᠠ ᠨᠢ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠨ ᠲᠡᠭ ᠪᠠ ᠲᠤᠶ᠋ᠢᠯ ᠤᠳ ᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠰᠢᠨᠵᠢᠯᠡᠨᠡ᠃<br />
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ᠢᠵᠠᠭᠤᠷ ᠤᠨ ᠮᠥᠷ ᠦᠨ ᠠᠷᠭᠠ ᠪᠠᠷ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠳᠦ ᠲᠡᠭ ᠪᠠ ᠲᠤᠶ᠋ᠢᠯ ᠤᠳ ᠵᠡᠭᠦᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠠᠶ᠋ᠢᠨᠠ ᠤᠤ᠂ ᠡᠰᠡᠪᠡᠯ ᠪᠠᠷᠠᠭᠤᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠠᠶ᠋ᠢᠨᠠ ᠤᠤ ᠭᠡᠳᠡᠭ ᠨᠢ ᠴᠢᠬᠤᠯᠠ᠃ <br />
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ᠥᠭᠡᠷᠡ ᠪᠡᠷ ᠬᠡᠯᠡᠪᠡᠯ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠨᠢ ᠲᠡᠭ ᠡᠴᠡ ᠶᠡᠬᠡ ᠪᠠᠢᠢᠨᠠ ᠤᠤ ᠡᠰᠡᠪᠡᠯ ᠪᠠᠭᠠ ᠪᠠᠶ᠋ᠢᠨᠠ ᠤᠤ ᠭᠡᠳᠡᠭ ᠨᠢ ᠴᠢᠬᠤᠯᠠ ᠶᠤᠮ᠃ ᠬᠡᠷᠪᠡ ᠴᠠᠭ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠬᠤᠪᠢᠷᠠᠯᠲᠠ ᠦᠭᠡᠢ ᠱᠤᠭᠤᠮᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠨᠢ᠄<br />
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- ᠪᠠᠷᠠᠭᠤᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠲᠤᠶ᠋ᠢᠯ ᠲᠠᠢ ᠪᠣᠯ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠂<br />
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- ᠵᠡᠭᠦᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠦᠬᠦ ᠲᠤᠶ᠋ᠢᠯ ᠨᠢ ᠪᠠᠶ᠋ᠢᠪᠠᠯ ᠲᠣᠭᠲᠠᠪᠤᠷᠢᠲᠠᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠂<br />
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- ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠲᠤᠶ᠋ᠢᠯ ᠲᠠᠢ ᠪᠣᠯ ᠶᠠᠯᠢᠭᠦᠢ ᠲᠣᠭᠲᠠᠪᠤᠷᠢᠲᠠᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠃<br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ - complex number - комплексное число<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ - imaginary unit - мнимая единица <br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭᠠ - imaginary number - чисто мнимое число<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ - imaginaty part - мнимая часть<br />
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ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠ - real number - действительное число<br />
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ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ - real part - действительная часть<br />
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ᠲᠡᠩᠬᠡᠯᠢᠭ - axis - ось<br />
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ᠣᠯᠠᠨᠯᠢᠭ - set - множество<br />
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ᠡᠶᠡᠷᠭᠦ - positive - положительный<br />
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ᠰᠥᠷᠭᠦ - negative - отрицательный<br />
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ᠲᠡᠭᠰᠢᠳᠬᠡᠯ - equation - уравнение<br />
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ᠡᠷᠭᠢᠯᠲᠡ - rotation - поворот<br />
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ᠬᠡᠮ - degree - градус<br />
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ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠢᠢᠨ ᠣᠨᠤᠯ - control theory - теория управления<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==</div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%BA%E1%A0%A3%E1%A0%AE%E1%A0%AB%E1%A0%8A%E1%A0%AF%E1%A0%A7%E1%A0%BA%E1%A0%B0_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%A0&diff=2753
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-10-10T10:14:20Z
<p>Angka: Images</p>
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'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ<sup>complex</sup> ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠯᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃ <br />
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|style = "height: 225px;"|ᠺᠣᠮᠫᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠢᠢ ᠲᠥᠯᠦᠭᠡᠯᠡᠵᠦ ᠪᠤᠶ ᠠᠷᠭᠠᠨ᠋ᠳ᠋ ᠤᠨ<sup>Argand</sup> ᠳᠢᠶᠠᠭ᠋ᠷᠠᠮ ᠳᠡᠭᠡᠷ ᠡ᠋ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠸᠧᠺᠲ᠋ᠣᠷ<sup>vector</sup> ᠡᠭᠦᠰᠬᠡᠵᠦ ᠪᠤᠶ ᠬᠣᠣᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ <span style="writing-mode: horizontal-tb;"><i>(a, b)</i></span> ᠳᠦᠷᠰᠦᠯᠡᠨ ᠦᠵᠡᠭᠦᠯᠵᠦ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
ᠡᠭᠦᠨ ᠳ᠋ᠦ: <br><span style="writing-mode: horizontal-tb;"><i>Re</i></span> ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ᠂ <br><span style="writing-mode: horizontal-tb;"><i>Im</i></span> ᠢᠨᠦ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠣᠯᠬᠤ ᠠᠭᠠᠳ <br><span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠢᠨᠦ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠳ᠋ᠦ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠬᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠮᠥᠨ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
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<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
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{| class="wikitable"<br />
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|style = "height: 220px;"|ᠬᠣᠶᠠᠷ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃ <br />
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= ᠬᠠᠰᠤᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠳᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
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<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math><br />
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= ᠦᠷᠡᠵᠢᠬᠦ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃<ref>''Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И.'' Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — <abbr>М.</abbr>: Просвещение, 1998. — 288 с. — <nowiki>ISBN 5-09-008036-4</nowiki>.</ref> <br />
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<math>(a + bi) (c + di) = </math> <math>= ac + bci - bd + adi = </math> <math>= (ac - bd) + (bc + ad)i </math> <br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|<span style="writing-mode: horizontal-tb;"><i>2 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭ ᠠ᠋ ᠢᠢ (ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠬᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>3 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭᠠᠨ ᠳᠤ (ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃<br />
ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠢ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠣᠷᠤᠶ ᠲᠠᠢ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠵᠤ᠂ <span style="writing-mode: horizontal-tb;"><i>√5</i></span> ᠢᠢᠠᠷ ᠰᠤᠩᠭᠠᠨ ᠠ᠋᠃ ᠡᠨᠡ ᠢᠨᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠾᠢᠫᠣᠲ᠋ᠧᠨᠦᠽ ᠤᠨ<sup>hypotenuse</sup> ᠤᠷᠲᠤ ᠪᠣᠯᠠᠢ᠃ <br />
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= ᠬᠤᠪᠢᠶᠠᠬᠤ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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<math>\frac{a+bi}{c+di}=</math> <br />
<math>=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=</math> <br />
<math>=\frac{ac+bd}{c^2+d^2}+\left(\frac{bc-ad}{c^2+d^2}\right)i.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ<sup>quadrate</sup> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨᠡ᠋᠃<ref>Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01</ref><br />
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<math>(x+yi)^2=x^2-y^2 + 2xyi.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> <span style="writing-mode: horizontal-tb;"><i>(b ≠ 0)</i></span> ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ <br />
<math> \pm (\gamma + \delta i)</math> ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄<br />
<math>\gamma = \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠠ <math>\delta = (\sgn b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠣᠯᠤᠮᠤᠢ᠃<br />
<span style="writing-mode: horizontal-tb;"><i>sgn b</i></span> ᠭᠡᠭᠴᠢ ᠢᠨᠦ <math>\sgn b = \begin{cases} \ \ 1, & b > 0 \\ \ \ 0, & b = 0 \\ -1, & b < 0 \end{cases}</math> ᠭᠡᠵᠦ ᠲᠠᠢᠢᠯᠠᠭᠳᠠᠨᠠ᠃ <br />
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ᠡᠭᠦᠨ ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ ᠢᠢ ᠲᠤᠬᠠᠢᠢᠲᠠ ᠳ᠋ᠤ <math> \pm (\gamma + \delta i)</math> ᠲᠣᠭᠠ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠦ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠲᠣᠭᠠ ᠢᠢ ᠭᠠᠷᠭᠠᠵᠤ ᠢᠷᠡᠬᠦ ᠶᠣᠰᠤᠲᠠᠢ᠃ <br />
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=ᠬᠡᠷᠡᠭᠯᠡᠭᠡ=<br />
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== ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ==<br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠳᠠᠪᠲᠠᠯᠲᠠ ᠪᠠᠷ ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠠᠢᠢᠭᠤᠯᠬᠤ᠃<br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠯ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠳ᠋ᠤ ᠡᠭᠦᠰᠬᠡᠳᠡᠭ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ<sup>fractal</sup> ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠵᠡᠰᠱᠢᠶ ᠡ᠋ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡ ᠬᠦ ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ <span style="writing-mode: horizontal-tb;"><i>z = 0</i></span> ᠡᠴᠡ ᠡᠬᠢᠯᠡᠭᠡᠳ ᠢᠲ᠋ᠧᠷᠠᠼ ᠬᠢᠬᠦ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;"><i>f<sub>c</sub>(z)=z<sup>2</sup>+c</i></span> ᠹᠦᠨ᠍ᠻᠼ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠢᠢᠬᠤ ᠨᠥᠬᠦᠴᠡᠯ ᠪᠦᠬᠦᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠤ ᠣᠯᠠᠨᠯᠢᠬ ᠪᠠᠢᠢᠳᠠᠭ᠃ <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ ᠪᠠᠢᠢᠭᠠᠯᠢ ᠢᠢᠨ ᠭᠠᠢᠢᠬᠠᠮᠰᠢᠭᠲᠤ ᠳᠦᠷᠰᠦ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠦᠨ ᠰᠠᠢᠢᠬᠠᠨ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠶᠡᠷᠦ ᠪᠤᠰᠤ ᠢᠢᠨ ᠭᠣᠶᠤᠮᠰᠠᠭ ᠬᠣᠣᠰᠯᠠᠯ ᠢ ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠳᠡᠭ᠃<ref>ᠷᠡᠨᠴᠡᠨ ᠦ ᠡᠩᠬᠡᠪᠠᠲᠤ᠃ '''ᠫᠢᠲ᠋ᠾᠠᠭᠣᠷ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌'''᠃ ᠮᠣᠩᠭᠤᠯ ᠤᠯᠤᠯᠰ ᠊ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠪᠠ ᠲᠣᠭᠠᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠢᠢᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ https://imdt.ac.mn/c/1013874?content=1150891&fbclid=IwAR1HVqeXwT-h7dijj9EeIUzDvFustH99GAr9LHYQIU61XIByuBHCkSm10So 2020</ref> <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠵᠠᠭᠠᠭ ᠢ ᠳᠠᠪᠠᠬᠤ ᠳᠤ ᠪᠠᠨ ᠵᠦᠯᠢᠶ ᠠ᠋ ᠢᠢᠨ<sup>Julia</sup> ᠹᠷᠠᠻᠲᠠᠯ ᠤᠳ ᠦᠷᠭᠡᠯᠵᠢ ᠬᠣᠯᠪᠤᠯᠲᠠ ᠪᠠᠨ ᠠᠯᠳᠠᠵᠤ᠂ ᠹᠠᠲᠣᠤ ᠢᠢᠨ<sup>Fatou</sup> ᠲᠣᠭᠤᠰᠤ ᠪᠣᠯᠤᠨ ᠬᠦᠪᠢᠷᠠᠳᠠᠭ᠃<ref>ᠨᠡᠭᠡᠭᠡᠯᠲᠡᠲᠡᠢ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠳ᠃ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠳᠠᠯ᠃ ᠪᠢᠳᠡᠨ ᠦ ᠡᠷᠭᠢᠨ ᠲᠣᠭᠣᠷᠢᠨ ᠳ᠋ᠠᠬᠢ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠬᠡᠷᠬᠢᠨ ᠠᠵᠢᠯᠯᠠᠳᠠᠭ ᠪᠤᠢ᠃ ᠹᠷᠠᠻᠲᠯᠠ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠶᠢᠨ ᠹᠷᠠᠻᠲᠯᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢ᠌ᠭ᠌᠃ https://ultrait.ru/mn/smartphones/otkrytie-fraktalov-beskonechnost-fraktalov-kak-ustroen-mir.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/10/02</ref><br />
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== ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ==<br />
ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌<sup>quantum mechanics</sup> ᠠᠲ᠋ᠣᠮ ᠠᠴᠠ ᠵᠢᠵᠢᠭ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ᠂ ᠶᠠᠭᠤᠨ ᠤ ᠡᠮᠦᠨᠡ ᠪᠣᠽᠣᠨ<sup>boson</sup> (ᠵᠢᠱᠢᠶᠡ ᠨᠢ ᠹᠣᠲ᠋ᠣᠨ<sup>photon</sup>ᠪᠠ ᠹᠧᠷᠮᠢᠣᠨ ᠤ<sup>fermion</sup> (ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠨᠧᠦᠲ᠋ᠷᠣᠨ) ᠬᠥᠳᠡᠯᠭᠡᠭᠡᠨ᠂ ᠬᠠᠷᠢᠯᠴᠠᠨ ᠦᠢᠯᠡᠴᠢᠯᠡᠯ ᠢ ᠰᠤᠳᠤᠯᠳᠠᠭ᠃ <br />
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ᠴᠢᠩᠭᠢᠬᠦ ᠳᠦ ᠪᠡᠨ ᠲᠡᠳᠡᠭᠡᠷ ᠦᠨ ᠦᠢᠯᠡ ᠬᠥᠳᠡᠯᠦᠯ ᠦᠨ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭ᠌ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠯᠲᠠ ᠶᠢ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠤᠳᠬᠠ ᠪᠠᠷ ᠭᠠᠷᠭᠠᠳᠠᠭ᠃ <br />
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ᠴᠤᠬᠤᠮ ᠳᠠᠭᠠᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠦᠨᠳᠦᠰᠦ ᠰᠠᠭᠤᠷᠢ ᠪᠣᠯᠳᠠᠭ᠃ <br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠲᠤ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ <sup>Schrödinger</sup> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠠᠴᠢ ᠬᠣᠯᠪᠣᠭᠳᠠᠯ ᠨᠢ ᠰᠣᠩᠭᠣᠳᠠᠭ ᠹᠢᠽᠢᡘ ᠳ᠋ᠡᠬᠢ ᠨᠧᠧᠲ᠋ᠣᠨ᠍ ᠤ ᠬᠣᠶᠠᠳᠤᠭᠠᠷ ᠬᠠᠤᠯᠢ ᠶᠢᠨ ᠠᠳᠠᠯᠢ ᠶᠤᠮ᠃ <br />
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ᠠᠯᠢ ᠠᠯᠢ ᠨᠢ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠷᠢᠯᠠᠯ᠂ ᠢᠮᠫᠦ᠋ᠯᠰ ᠢ <sup>impuls</sup> ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭᠴᠢᠯᠠᠨ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠳᠠᠭ᠃ <br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠨᠢ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ<sup>function</sup> ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠬᠦ ᠳᠦ ᠲᠣᠬᠢᠷᠠᠮᠵᠢ ᠲᠠᠢ ᠤᠴᠢᠷ ᠠᠴᠠ ᠹᠢᠽᠢᠻ ᠦᠨ ᠡᠨᠡ ᠰᠠᠯᠪᠤᠷᠢ ᠳᠤ ᠵᠠᠶ᠋ᠢᠯᠠᠰᠢ ᠦᠭᠡᠢ ᠱᠠᠭᠠᠷᠳᠠᠯᠭᠠ ᠲᠠᠢ ᠶᠤᠮ᠃ <br />
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ᠴᠠᠭᠠᠰᠢᠯᠠᠪᠠᠯ᠂ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠢᠭᠤᠳ ᠨᠥᠯᠥᠭᠡ ᠪᠡᠷ ᠬᠢᠮᠢ ᠶᠢᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠬᠥᠭᠵᠢᠯ ᠡᠷᠴᠢᠮᠵᠢᠭᠰᠡᠨ᠃ <br />
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1927 ᠣᠨ ᠳᠤ ᠸᠠᠯᠲ᠋ᠧᠷ ᠾᠠᠢᠢᠲ᠋ᠯᠧᠷ<sup>Walter Heitler</sup> ᠂ ᠹᠷᠢᠼ ᠯᠣᠨ᠍ᠳᠣᠨ<sup>Fritz London</sup> ᠨᠠᠷ ᠸᠠᠯᠧᠨ᠍ᠲ ᠤᠨ ᠬᠣᠯᠪᠣᠭᠠᠰᠤ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠲᠣᠮᠢᠶᠠᠯᠠᠭᠰᠠᠨ᠃ <br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠨᠢᠭᠡ ᠭᠣᠣᠯ ᠠᠰᠠᠭᠤᠳᠠᠯ ᠪᠣᠯ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ ᠦᠨ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ ᠤᠯᠬᠤ ᠶᠠᠪᠤᠳᠠᠯ ᠶᠤᠮ᠃ <br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠭᠡᠳᠡᠭ ᠨᠢ ᠲᠣᠳᠣᠷᠬᠠᠢ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠵᠤᠷᠪᠤᠰ ᠲᠤ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠵᠤ ᠪᠣᠯᠬᠤ ᠪᠠᠶ᠋ᠢᠷᠢᠰᠢᠯ ᠤᠳ ᠤᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠲᠠᠷᠬᠠᠯᠲᠠ ᠶᠤᠮ᠄ <br />
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<small><math>\frac{-\hbar}{2m}\frac{\partial^2\Psi(x,t)}{\partial x^2} + V(x,t)\Psi(x,t) = i\hbar\frac{\partial \Psi(x,t)}{\partial t}</math></small><br />
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ᠻᠢᠨᠧᠲᠢᠭ<sup>kinetic</sup> ᠧᠨᠧᠷᠭᠢ<sup>energy</sup><br />
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ᠫᠣᠲ᠋ᠧᠨ᠍ᠼᠢᠶᠠᠯ<sup>potencial</sup> ᠧᠨᠧᠷᠭᠢ<br />
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ᠨᠡᠢᠢᠲᠡ ᠧᠨᠧᠷᠭᠢ<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠲᠦ ᠰᠢᠲᠦᠭᠰᠡᠨ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠠᠭᠤᠷᠢ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠨᠢᠭᠡ ᠪᠣᠯ ᠳᠡᠭᠡᠷᠡ ᠳᠤᠷᠠᠳᠤᠭᠰᠠᠨ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡᠬᠦ ᠲᠣᠮᠢᠶᠠ ᠶᠢ ᠠᠰᠢᠭᠯᠠᠭᠰᠠᠨ ᠢᠶ᠋ᠠᠷ ᠤᠰᠤ ᠲᠥᠷᠦᠭᠴᠢ ᠶᠢᠨ ᠮᠣᠯᠧᠻᠤᠯ ᠳᠠᠬᠢ ᠬᠣᠶᠠᠷ ᠠᠲ᠋ᠣᠮ ᠨᠢ ᠻᠣᠸᠠᠯᠧᠨ᠍ᠲ<sup>covalent</sup> ᠬᠣᠯᠪᠣᠭᠠ ᠭᠡᠳᠡᠭ ᠢ ᠡᠭᠦᠰᠬᠡᠨ ᠧᠯᠧᠻᠲ᠋ᠷᠣᠨ ᠨᠤᠭᠤᠳ ᠢᠢᠠᠨ ᠬᠤᠪᠢᠶᠠᠯᠴᠠᠵᠤ ᠪᠠᠶ᠋ᠢᠳᠠᠭ ᠢ ᠪᠠᠲᠤᠯᠠᠭᠰᠠᠨ᠃<ref>Josiah Wu. Real Life Applications of Complex Numbers. 2020 https://issuu.com/harrowhongkong/docs/final_scientific_harrovian_issue_vi-i/s/11488755</ref><br />
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== ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠢᠢᠨ ᠣᠨᠤᠯ ==<br />
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{| class="wikitable"<br />
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|style = "height: 340px;"|ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠢᠢᠨ ᠲᠣᠭᠲᠠᠯᠴᠠᠭᠠᠨ ᠤ ᠡᠩ ᠦᠨ ᠪᠦᠳᠦᠭᠦᠪᠴᠢ᠃ <br><br><span style="writing-mode: horizontal-tb;"><i>X(s)</i></span> ᠣᠷᠤᠯᠲᠠ ᠢᠢᠨ ᠮᠡᠳᠡᠭᠡᠯᠡᠯ᠂ <br><br><span style="writing-mode: horizontal-tb;"><i>Y(s)</i></span> ᠭᠠᠷᠤᠯᠲᠠ ᠢᠢᠨ ᠮᠡᠳᠡᠭᠡᠯᠡᠯ᠂ <br><br><span style="writing-mode: horizontal-tb;"><i>G(s)</i></span> ᠤᠷᠤᠭᠰᠢ ᠳᠠᠮᠵᠢᠭᠤᠯᠬᠤ ᠹᠦᠨ᠍ᠻᠼ<sup>function</sup>᠂ <br><br><span style="writing-mode: horizontal-tb;"><i>H(s)</i></span> ᠡᠷᠭᠢᠬᠦ ᠬᠣᠯᠪᠤᠭᠠ<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠶᠢ ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ᠂ ᠢᠯᠠᠩᠭᠤᠶᠠ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠨ ᠲᠣᠭᠲᠠᠪᠤᠷᠢᠲᠠᠢ ᠪᠠᠶᠢᠳᠠᠯ ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠭᠡᠨ ᠳᠤ ᠬᠡᠷᠡᠭᠯᠡᠳᠡᠭ᠃ <br />
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ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠳᠤ 《ᠰᠢᠰᠲ᠋ᠧᠮ》 ᠭᠡᠳᠡᠭ ᠦᠭᠡ ᠶᠢ ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ ᠪᠥᠭᠡᠳ ᠡᠨᠡ ᠨᠢ ᠵᠠᠪᠠᠯ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠢ ᠬᠡᠯᠡᠳᠡᠭ ᠦᠭᠡᠢ᠃ <br />
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ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠡᠭᠦᠨ ᠢ ᠬᠥᠷᠥᠩᠭᠡ ᠶᠢᠨ ᠵᠠᠬᠠ ᠵᠡᠭᠡᠯᠢ ᠳ᠋ᠡᠬᠢ ᠥᠭᠡᠷᠡᠴᠢᠯᠡᠯᠲᠡ᠂ ᠬᠢᠮᠢ ᠶᠢᠨ ᠤᠷᠪᠠᠯ᠂ ᠦᠢᠯᠡ ᠶᠠᠪᠤᠴᠠ ᠶᠢ ᠰᠤᠳᠤᠯᠬᠤ ᠳᠤ ᠬᠡᠷᠡᠭᠯᠡᠵᠦ ᠪᠣᠯᠤᠨᠠ᠃ <br />
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ᠲᠡᠭᠦᠨᠴᠢᠯᠡᠨ ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠷᠣᠪᠣᠲ ᠤᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠳᠤ ᠥᠷᠭᠡᠨ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ ᠲᠤᠯᠠ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠠᠨᠠᠯᠢᠽ ᠴᠤ ᠪᠠᠰᠠ ᠲᠡᠭᠦᠨ ᠳᠦ ᠬᠡᠷᠡᠭᠯᠡᠭᠳᠡᠨᠡ ᠭᠡᠰᠡᠨ ᠦᠭᠡ᠃<ref>Ujjvala Y. Gawarguru, Mitali K. Tibdewal, Rajashri A. Naphade, Rahul M. Jethwani. The Review of Introduction & Application of Complex Number in Engineering. 2nd National Conference Recent Innovations in Science and Engineering (NC-RISE 17). Volume: 5 Issue: 9. pp55 – 57. ISSN: 2321-8169. https://ijritcc.org/download/conferences/NC-RISE_17/Track_6_(ASH)/1506931102_02-10-2017.pdf</ref><br />
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ᠬᠢᠨᠠᠯᠲᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠳᠤ ᠵᠠᠷᠢᠮ ᠳᠤ ᠪᠠᠨ ᠯᠠᠫ᠊ᠯᠠᠰ ᠤᠨ<sup>Laplace</sup> ᠬᠤᠪᠢᠷᠠᠭᠠᠯᠲᠠ ᠶᠢ ᠠᠰᠢᠭᠯᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠳ ᠢ<sup>system</sup> ᠴᠠᠭ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠮᠤᠵᠢ ᠠᠴᠠ ᠳᠠᠪᠲᠠᠮᠵᠢ ᠶᠢᠨ ᠮᠤᠵᠢ ᠳᠤ ᠰᠢᠯᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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ᠳᠠᠷᠠᠭᠠ ᠨᠢ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠨ ᠲᠡᠭ ᠪᠠ ᠲᠤᠶ᠋ᠢᠯ ᠤᠳ ᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠰᠢᠨᠵᠢᠯᠡᠨᠡ᠃<br />
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ᠢᠵᠠᠭᠤᠷ ᠤᠨ ᠮᠥᠷ ᠦᠨ ᠠᠷᠭᠠ ᠪᠠᠷ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠳᠦ ᠲᠡᠭ ᠪᠠ ᠲᠤᠶ᠋ᠢᠯ ᠤᠳ ᠵᠡᠭᠦᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠠᠶ᠋ᠢᠨᠠ ᠤᠤ᠂ ᠡᠰᠡᠪᠡᠯ ᠪᠠᠷᠠᠭᠤᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠠᠶ᠋ᠢᠨᠠ ᠤᠤ ᠭᠡᠳᠡᠭ ᠨᠢ ᠴᠢᠬᠤᠯᠠ᠃ <br />
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ᠥᠭᠡᠷᠡ ᠪᠡᠷ ᠬᠡᠯᠡᠪᠡᠯ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠨᠢ ᠲᠡᠭ ᠡᠴᠡ ᠶᠡᠬᠡ ᠪᠠᠢᠢᠨᠠ ᠤᠤ ᠡᠰᠡᠪᠡᠯ ᠪᠠᠭᠠ ᠪᠠᠶ᠋ᠢᠨᠠ ᠤᠤ ᠭᠡᠳᠡᠭ ᠨᠢ ᠴᠢᠬᠤᠯᠠ ᠶᠤᠮ᠃ ᠬᠡᠷᠪᠡ ᠴᠠᠭ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠬᠤᠪᠢᠷᠠᠯᠲᠠ ᠦᠭᠡᠢ ᠱᠤᠭᠤᠮᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠨᠢ᠄<br />
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- ᠪᠠᠷᠠᠭᠤᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠲᠤᠶ᠋ᠢᠯ ᠲᠠᠢ ᠪᠣᠯ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠂<br />
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- ᠵᠡᠭᠦᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠦᠬᠦ ᠲᠤᠶ᠋ᠢᠯ ᠨᠢ ᠪᠠᠶ᠋ᠢᠪᠠᠯ ᠲᠣᠭᠲᠠᠪᠤᠷᠢᠲᠠᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠂<br />
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- ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠲᠤᠶ᠋ᠢᠯ ᠲᠠᠢ ᠪᠣᠯ ᠶᠠᠯᠢᠭᠦᠢ ᠲᠣᠭᠲᠠᠪᠤᠷᠢᠲᠠᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠃<br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==</div>
Angka
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-10-10T09:52:36Z
<p>Angka: Layout</p>
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'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ<sup>complex</sup> ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠯᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃ <br />
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|style = "height: 225px;"|ᠺᠣᠮᠫᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠢᠢ ᠲᠥᠯᠦᠭᠡᠯᠡᠵᠦ ᠪᠤᠶ ᠠᠷᠭᠠᠨ᠋ᠳ᠋ ᠤᠨ<sup>Argand</sup> ᠳᠢᠶᠠᠭ᠋ᠷᠠᠮ ᠳᠡᠭᠡᠷ ᠡ᠋ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠸᠧᠺᠲ᠋ᠣᠷ<sup>vector</sup> ᠡᠭᠦᠰᠬᠡᠵᠦ ᠪᠤᠶ ᠬᠣᠣᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ <span style="writing-mode: horizontal-tb;"><i>(a, b)</i></span> ᠳᠦᠷᠰᠦᠯᠡᠨ ᠦᠵᠡᠭᠦᠯᠵᠦ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
ᠡᠭᠦᠨ ᠳ᠋ᠦ: <br><span style="writing-mode: horizontal-tb;"><i>Re</i></span> ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ᠂ <br><span style="writing-mode: horizontal-tb;"><i>Im</i></span> ᠢᠨᠦ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠣᠯᠬᠤ ᠠᠭᠠᠳ <br><span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠢᠨᠦ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠳ᠋ᠦ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠬᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠮᠥᠨ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
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<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
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|style = "height: 220px;"|ᠬᠣᠶᠠᠷ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃ <br />
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= ᠬᠠᠰᠤᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠳᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
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<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math><br />
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= ᠦᠷᠡᠵᠢᠬᠦ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃<ref>''Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И.'' Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — <abbr>М.</abbr>: Просвещение, 1998. — 288 с. — <nowiki>ISBN 5-09-008036-4</nowiki>.</ref> <br />
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<math>(a + bi) (c + di) = </math> <math>= ac + bci - bd + adi = </math> <math>= (ac - bd) + (bc + ad)i </math> <br />
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|style = "height: 225px;"|<span style="writing-mode: horizontal-tb;"><i>2 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭ ᠠ᠋ ᠢᠢ (ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠬᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>3 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭᠠᠨ ᠳᠤ (ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃<br />
ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠢ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠣᠷᠤᠶ ᠲᠠᠢ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠵᠤ᠂ <span style="writing-mode: horizontal-tb;"><i>√5</i></span> ᠢᠢᠠᠷ ᠰᠤᠩᠭᠠᠨ ᠠ᠋᠃ ᠡᠨᠡ ᠢᠨᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠾᠢᠫᠣᠲ᠋ᠧᠨᠦᠽ ᠤᠨ<sup>hypotenuse</sup> ᠤᠷᠲᠤ ᠪᠣᠯᠠᠢ᠃ <br />
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= ᠬᠤᠪᠢᠶᠠᠬᠤ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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<math>\frac{a+bi}{c+di}=</math> <br />
<math>=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=</math> <br />
<math>=\frac{ac+bd}{c^2+d^2}+\left(\frac{bc-ad}{c^2+d^2}\right)i.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ<sup>quadrate</sup> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨᠡ᠋᠃<ref>Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01</ref><br />
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<math>(x+yi)^2=x^2-y^2 + 2xyi.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> <span style="writing-mode: horizontal-tb;"><i>(b ≠ 0)</i></span> ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ <br />
<math> \pm (\gamma + \delta i)</math> ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄<br />
<math>\gamma = \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠠ <math>\delta = (\sgn b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠣᠯᠤᠮᠤᠢ᠃<br />
<span style="writing-mode: horizontal-tb;"><i>sgn b</i></span> ᠭᠡᠭᠴᠢ ᠢᠨᠦ <math>\sgn b = \begin{cases} \ \ 1, & b > 0 \\ \ \ 0, & b = 0 \\ -1, & b < 0 \end{cases}</math> ᠭᠡᠵᠦ ᠲᠠᠢᠢᠯᠠᠭᠳᠠᠨᠠ᠃ <br />
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ᠡᠭᠦᠨ ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ ᠢᠢ ᠲᠤᠬᠠᠢᠢᠲᠠ ᠳ᠋ᠤ <math> \pm (\gamma + \delta i)</math> ᠲᠣᠭᠠ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠦ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠲᠣᠭᠠ ᠢᠢ ᠭᠠᠷᠭᠠᠵᠤ ᠢᠷᠡᠬᠦ ᠶᠣᠰᠤᠲᠠᠢ᠃ <br />
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=ᠬᠡᠷᠡᠭᠯᠡᠭᠡ=<br />
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== ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ==<br />
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|style = "height: 225px;"|ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠳᠠᠪᠲᠠᠯᠲᠠ ᠪᠠᠷ ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠠᠢᠢᠭᠤᠯᠬᠤ᠃<br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠯ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠳ᠋ᠤ ᠡᠭᠦᠰᠬᠡᠳᠡᠭ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ<sup>fractal</sup> ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠵᠡᠰᠱᠢᠶ ᠡ᠋ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡ ᠬᠦ ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ <span style="writing-mode: horizontal-tb;"><i>z = 0</i></span> ᠡᠴᠡ ᠡᠬᠢᠯᠡᠭᠡᠳ ᠢᠲ᠋ᠧᠷᠠᠼ ᠬᠢᠬᠦ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;"><i>f<sub>c</sub>(z)=z<sup>2</sup>+c</i></span> ᠹᠦᠨ᠍ᠻᠼ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠢᠢᠬᠤ ᠨᠥᠬᠦᠴᠡᠯ ᠪᠦᠬᠦᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠤ ᠣᠯᠠᠨᠯᠢᠬ ᠪᠠᠢᠢᠳᠠᠭ᠃ <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ ᠪᠠᠢᠢᠭᠠᠯᠢ ᠢᠢᠨ ᠭᠠᠢᠢᠬᠠᠮᠰᠢᠭᠲᠤ ᠳᠦᠷᠰᠦ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠦᠨ ᠰᠠᠢᠢᠬᠠᠨ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠶᠡᠷᠦ ᠪᠤᠰᠤ ᠢᠢᠨ ᠭᠣᠶᠤᠮᠰᠠᠭ ᠬᠣᠣᠰᠯᠠᠯ ᠢ ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠳᠡᠭ᠃<ref>ᠷᠡᠨᠴᠡᠨ ᠦ ᠡᠩᠬᠡᠪᠠᠲᠤ᠃ '''ᠫᠢᠲ᠋ᠾᠠᠭᠣᠷ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌'''᠃ ᠮᠣᠩᠭᠤᠯ ᠤᠯᠤᠯᠰ ᠊ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠪᠠ ᠲᠣᠭᠠᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠢᠢᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ https://imdt.ac.mn/c/1013874?content=1150891&fbclid=IwAR1HVqeXwT-h7dijj9EeIUzDvFustH99GAr9LHYQIU61XIByuBHCkSm10So 2020</ref> <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠵᠠᠭᠠᠭ ᠢ ᠳᠠᠪᠠᠬᠤ ᠳᠤ ᠪᠠᠨ ᠵᠦᠯᠢᠶ ᠠ᠋ ᠢᠢᠨ<sup>Julia</sup> ᠹᠷᠠᠻᠲᠠᠯ ᠤᠳ ᠦᠷᠭᠡᠯᠵᠢ ᠬᠣᠯᠪᠤᠯᠲᠠ ᠪᠠᠨ ᠠᠯᠳᠠᠵᠤ᠂ ᠹᠠᠲᠣᠤ ᠢᠢᠨ<sup>Fatou</sup> ᠲᠣᠭᠤᠰᠤ ᠪᠣᠯᠤᠨ ᠬᠦᠪᠢᠷᠠᠳᠠᠭ᠃<ref>ᠨᠡᠭᠡᠭᠡᠯᠲᠡᠲᠡᠢ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠳ᠃ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠳᠠᠯ᠃ ᠪᠢᠳᠡᠨ ᠦ ᠡᠷᠭᠢᠨ ᠲᠣᠭᠣᠷᠢᠨ ᠳ᠋ᠠᠬᠢ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠬᠡᠷᠬᠢᠨ ᠠᠵᠢᠯᠯᠠᠳᠠᠭ ᠪᠤᠢ᠃ ᠹᠷᠠᠻᠲᠯᠠ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠶᠢᠨ ᠹᠷᠠᠻᠲᠯᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢ᠌ᠭ᠌᠃ https://ultrait.ru/mn/smartphones/otkrytie-fraktalov-beskonechnost-fraktalov-kak-ustroen-mir.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/10/02</ref><br />
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|style = "height: 225px;"|ᠳᠦᠷᠰᠦ ᠠᠮᠢᠯᠠᠭᠤᠯᠤᠨ ᠵᠦᠯᠢᠶ ᠠ᠋ ᠢᠢᠨ <sup>Julia</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠦᠵᠡᠭᠦᠯᠦᠭᠰᠡᠨ ᠢᠨᠦ᠃<br />
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== ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ==<br />
ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌<sup>quantum mechanics</sup> ᠠᠲ᠋ᠣᠮ ᠠᠴᠠ ᠵᠢᠵᠢᠭ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ᠂ ᠶᠠᠭᠤᠨ ᠤ ᠡᠮᠦᠨᠡ ᠪᠣᠽᠣᠨ<sup>boson</sup> (ᠵᠢᠱᠢᠶᠡ ᠨᠢ ᠹᠣᠲ᠋ᠣᠨ<sup>photon</sup>ᠪᠠ ᠹᠧᠷᠮᠢᠣᠨ ᠤ<sup>fermion</sup> (ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠨᠧᠦᠲ᠋ᠷᠣᠨ) ᠬᠥᠳᠡᠯᠭᠡᠭᠡᠨ᠂ ᠬᠠᠷᠢᠯᠴᠠᠨ ᠦᠢᠯᠡᠴᠢᠯᠡᠯ ᠢ ᠰᠤᠳᠤᠯᠳᠠᠭ᠃ <br />
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ᠴᠢᠩᠭᠢᠬᠦ ᠳᠦ ᠪᠡᠨ ᠲᠡᠳᠡᠭᠡᠷ ᠦᠨ ᠦᠢᠯᠡ ᠬᠥᠳᠡᠯᠦᠯ ᠦᠨ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭ᠌ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠯᠲᠠ ᠶᠢ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠤᠳᠬᠠ ᠪᠠᠷ ᠭᠠᠷᠭᠠᠳᠠᠭ᠃ <br />
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ᠴᠤᠬᠤᠮ ᠳᠠᠭᠠᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠦᠨᠳᠦᠰᠦ ᠰᠠᠭᠤᠷᠢ ᠪᠣᠯᠳᠠᠭ᠃ <br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠲᠤ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ <sup>Schrödinger</sup> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠠᠴᠢ ᠬᠣᠯᠪᠣᠭᠳᠠᠯ ᠨᠢ ᠰᠣᠩᠭᠣᠳᠠᠭ ᠹᠢᠽᠢᡘ ᠳ᠋ᠡᠬᠢ ᠨᠧᠧᠲ᠋ᠣᠨ᠍ ᠤ ᠬᠣᠶᠠᠳᠤᠭᠠᠷ ᠬᠠᠤᠯᠢ ᠶᠢᠨ ᠠᠳᠠᠯᠢ ᠶᠤᠮ᠃ <br />
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ᠠᠯᠢ ᠠᠯᠢ ᠨᠢ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠷᠢᠯᠠᠯ᠂ ᠢᠮᠫᠦ᠋ᠯᠰ ᠢ <sup>impuls</sup> ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭᠴᠢᠯᠠᠨ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠳᠠᠭ᠃ <br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠨᠢ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ<sup>function</sup> ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠬᠦ ᠳᠦ ᠲᠣᠬᠢᠷᠠᠮᠵᠢ ᠲᠠᠢ ᠤᠴᠢᠷ ᠠᠴᠠ ᠹᠢᠽᠢᠻ ᠦᠨ ᠡᠨᠡ ᠰᠠᠯᠪᠤᠷᠢ ᠳᠤ ᠵᠠᠶ᠋ᠢᠯᠠᠰᠢ ᠦᠭᠡᠢ ᠱᠠᠭᠠᠷᠳᠠᠯᠭᠠ ᠲᠠᠢ ᠶᠤᠮ᠃ <br />
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ᠴᠠᠭᠠᠰᠢᠯᠠᠪᠠᠯ᠂ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠢᠭᠤᠳ ᠨᠥᠯᠥᠭᠡ ᠪᠡᠷ ᠬᠢᠮᠢ ᠶᠢᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠬᠥᠭᠵᠢᠯ ᠡᠷᠴᠢᠮᠵᠢᠭᠰᠡᠨ᠃ <br />
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1927 ᠣᠨ ᠳᠤ ᠸᠠᠯᠲ᠋ᠧᠷ ᠾᠠᠢᠢᠲ᠋ᠯᠧᠷ<sup>Walter Heitler</sup> ᠂ ᠹᠷᠢᠼ ᠯᠣᠨ᠍ᠳᠣᠨ<sup>Fritz London</sup> ᠨᠠᠷ ᠸᠠᠯᠧᠨ᠍ᠲ ᠤᠨ ᠬᠣᠯᠪᠣᠭᠠᠰᠤ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠲᠣᠮᠢᠶᠠᠯᠠᠭᠰᠠᠨ᠃ <br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠨᠢᠭᠡ ᠭᠣᠣᠯ ᠠᠰᠠᠭᠤᠳᠠᠯ ᠪᠣᠯ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ ᠦᠨ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ ᠤᠯᠬᠤ ᠶᠠᠪᠤᠳᠠᠯ ᠶᠤᠮ᠃ <br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠭᠡᠳᠡᠭ ᠨᠢ ᠲᠣᠳᠣᠷᠬᠠᠢ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠵᠤᠷᠪᠤᠰ ᠲᠤ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠵᠤ ᠪᠣᠯᠬᠤ ᠪᠠᠶ᠋ᠢᠷᠢᠰᠢᠯ ᠤᠳ ᠤᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠲᠠᠷᠬᠠᠯᠲᠠ ᠶᠤᠮ᠄ <br />
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<small><math>\frac{-\hbar}{2m}\frac{\partial^2\Psi(x,t)}{\partial x^2} + V(x,t)\Psi(x,t) = i\hbar\frac{\partial \Psi(x,t)}{\partial t}</math></small><br />
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ᠻᠢᠨᠧᠲᠢᠭ<sup>kinetic</sup> ᠧᠨᠧᠷᠭᠢ<sup>energy</sup><br />
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ᠫᠣᠲ᠋ᠧᠨ᠍ᠼᠢᠶᠠᠯ<sup>potencial</sup> ᠧᠨᠧᠷᠭᠢ<br />
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ᠨᠡᠢᠢᠲᠡ ᠧᠨᠧᠷᠭᠢ<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠲᠦ ᠰᠢᠲᠦᠭᠰᠡᠨ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠠᠭᠤᠷᠢ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠨᠢᠭᠡ ᠪᠣᠯ ᠳᠡᠭᠡᠷᠡ ᠳᠤᠷᠠᠳᠤᠭᠰᠠᠨ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡᠬᠦ ᠲᠣᠮᠢᠶᠠ ᠶᠢ ᠠᠰᠢᠭᠯᠠᠭᠰᠠᠨ ᠢᠶ᠋ᠠᠷ ᠤᠰᠤ ᠲᠥᠷᠦᠭᠴᠢ ᠶᠢᠨ ᠮᠣᠯᠧᠻᠤᠯ ᠳᠠᠬᠢ ᠬᠣᠶᠠᠷ ᠠᠲ᠋ᠣᠮ ᠨᠢ ᠻᠣᠸᠠᠯᠧᠨ᠍ᠲ<sup>covalent</sup> ᠬᠣᠯᠪᠣᠭᠠ ᠭᠡᠳᠡᠭ ᠢ ᠡᠭᠦᠰᠬᠡᠨ ᠧᠯᠧᠻᠲ᠋ᠷᠣᠨ ᠨᠤᠭᠤᠳ ᠢᠢᠠᠨ ᠬᠤᠪᠢᠶᠠᠯᠴᠠᠵᠤ ᠪᠠᠶ᠋ᠢᠳᠠᠭ ᠢ ᠪᠠᠲᠤᠯᠠᠭᠰᠠᠨ᠃<ref>Josiah Wu. Real Life Applications of Complex Numbers. 2020 https://issuu.com/harrowhongkong/docs/final_scientific_harrovian_issue_vi-i/s/11488755</ref><br />
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== ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠢᠢᠨ ᠣᠨᠤᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠶᠢ ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ᠂ ᠢᠯᠠᠩᠭᠤᠶᠠ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠨ ᠲᠣᠭᠲᠠᠪᠤᠷᠢᠲᠠᠢ ᠪᠠᠶᠢᠳᠠᠯ ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠭᠡᠨ ᠳᠤ ᠬᠡᠷᠡᠭᠯᠡᠳᠡᠭ᠃ <br />
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ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠳᠤ 《ᠰᠢᠰᠲ᠋ᠧᠮ》 ᠭᠡᠳᠡᠭ ᠦᠭᠡ ᠶᠢ ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ ᠪᠥᠭᠡᠳ ᠡᠨᠡ ᠨᠢ ᠵᠠᠪᠠᠯ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠢ ᠬᠡᠯᠡᠳᠡᠭ ᠦᠭᠡᠢ᠃ <br />
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ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠡᠭᠦᠨ ᠢ ᠬᠥᠷᠥᠩᠭᠡ ᠶᠢᠨ ᠵᠠᠬᠠ ᠵᠡᠭᠡᠯᠢ ᠳ᠋ᠡᠬᠢ ᠥᠭᠡᠷᠡᠴᠢᠯᠡᠯᠲᠡ᠂ ᠬᠢᠮᠢ ᠶᠢᠨ ᠤᠷᠪᠠᠯ᠂ ᠦᠢᠯᠡ ᠶᠠᠪᠤᠴᠠ ᠶᠢ ᠰᠤᠳᠤᠯᠬᠤ ᠳᠤ ᠬᠡᠷᠡᠭᠯᠡᠵᠦ ᠪᠣᠯᠤᠨᠠ᠃ <br />
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ᠲᠡᠭᠦᠨᠴᠢᠯᠡᠨ ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠷᠣᠪᠣᠲ ᠤᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠳᠤ ᠥᠷᠭᠡᠨ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ ᠲᠤᠯᠠ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠠᠨᠠᠯᠢᠽ ᠴᠤ ᠪᠠᠰᠠ ᠲᠡᠭᠦᠨ ᠳᠦ ᠬᠡᠷᠡᠭᠯᠡᠭᠳᠡᠨᠡ ᠭᠡᠰᠡᠨ ᠦᠭᠡ᠃<ref>Ujjvala Y. Gawarguru, Mitali K. Tibdewal, Rajashri A. Naphade, Rahul M. Jethwani. The Review of Introduction & Application of Complex Number in Engineering. 2nd National Conference Recent Innovations in Science and Engineering (NC-RISE 17). Volume: 5 Issue: 9. pp55 – 57. ISSN: 2321-8169. https://ijritcc.org/download/conferences/NC-RISE_17/Track_6_(ASH)/1506931102_02-10-2017.pdf</ref><br />
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ᠬᠢᠨᠠᠯᠲᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠳᠤ ᠵᠠᠷᠢᠮ ᠳᠤ ᠪᠠᠨ ᠯᠠᠫ᠊ᠯᠠᠰ ᠤᠨ<sup>Laplace</sup> ᠬᠤᠪᠢᠷᠠᠭᠠᠯᠲᠠ ᠶᠢ ᠠᠰᠢᠭᠯᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠳ ᠢ<sup>system</sup> ᠴᠠᠭ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠮᠤᠵᠢ ᠠᠴᠠ ᠳᠠᠪᠲᠠᠮᠵᠢ ᠶᠢᠨ ᠮᠤᠵᠢ ᠳᠤ ᠰᠢᠯᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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ᠳᠠᠷᠠᠭᠠ ᠨᠢ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠨ ᠲᠡᠭ ᠪᠠ ᠲᠤᠶ᠋ᠢᠯ ᠤᠳ ᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠰᠢᠨᠵᠢᠯᠡᠨᠡ᠃<br />
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ᠢᠵᠠᠭᠤᠷ ᠤᠨ ᠮᠥᠷ ᠦᠨ ᠠᠷᠭᠠ ᠪᠠᠷ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠳᠦ ᠲᠡᠭ ᠪᠠ ᠲᠤᠶ᠋ᠢᠯ ᠤᠳ ᠵᠡᠭᠦᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠠᠶ᠋ᠢᠨᠠ ᠤᠤ᠂ ᠡᠰᠡᠪᠡᠯ ᠪᠠᠷᠠᠭᠤᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠠᠶ᠋ᠢᠨᠠ ᠤᠤ ᠭᠡᠳᠡᠭ ᠨᠢ ᠴᠢᠬᠤᠯᠠ᠃ <br />
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ᠥᠭᠡᠷᠡ ᠪᠡᠷ ᠬᠡᠯᠡᠪᠡᠯ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠨᠢ ᠲᠡᠭ ᠡᠴᠡ ᠶᠡᠬᠡ ᠪᠠᠢᠢᠨᠠ ᠤᠤ ᠡᠰᠡᠪᠡᠯ ᠪᠠᠭᠠ ᠪᠠᠶ᠋ᠢᠨᠠ ᠤᠤ ᠭᠡᠳᠡᠭ ᠨᠢ ᠴᠢᠬᠤᠯᠠ ᠶᠤᠮ᠃ ᠬᠡᠷᠪᠡ ᠴᠠᠭ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠬᠤᠪᠢᠷᠠᠯᠲᠠ ᠦᠭᠡᠢ ᠱᠤᠭᠤᠮᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠨᠢ᠄<br />
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- ᠪᠠᠷᠠᠭᠤᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠲᠤᠶ᠋ᠢᠯ ᠲᠠᠢ ᠪᠣᠯ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠂<br />
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- ᠵᠡᠭᠦᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠦᠬᠦ ᠲᠤᠶ᠋ᠢᠯ ᠨᠢ ᠪᠠᠶ᠋ᠢᠪᠠᠯ ᠲᠣᠭᠲᠠᠪᠤᠷᠢᠲᠠᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠂<br />
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- ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠲᠤᠶ᠋ᠢᠯ ᠲᠠᠢ ᠪᠣᠯ ᠶᠠᠯᠢᠭᠦᠢ ᠲᠣᠭᠲᠠᠪᠤᠷᠢᠲᠠᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠃<br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==</div>
Angka
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-10-10T09:51:25Z
<p>Angka: Edit</p>
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'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ<sup>complex</sup> ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠯᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃ <br />
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|style = "height: 225px;"|ᠺᠣᠮᠫᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠢᠢ ᠲᠥᠯᠦᠭᠡᠯᠡᠵᠦ ᠪᠤᠶ ᠠᠷᠭᠠᠨ᠋ᠳ᠋ ᠤᠨ<sup>Argand</sup> ᠳᠢᠶᠠᠭ᠋ᠷᠠᠮ ᠳᠡᠭᠡᠷ ᠡ᠋ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠸᠧᠺᠲ᠋ᠣᠷ<sup>vector</sup> ᠡᠭᠦᠰᠬᠡᠵᠦ ᠪᠤᠶ ᠬᠣᠣᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ <span style="writing-mode: horizontal-tb;"><i>(a, b)</i></span> ᠳᠦᠷᠰᠦᠯᠡᠨ ᠦᠵᠡᠭᠦᠯᠵᠦ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
ᠡᠭᠦᠨ ᠳ᠋ᠦ: <br><span style="writing-mode: horizontal-tb;"><i>Re</i></span> ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ᠂ <br><span style="writing-mode: horizontal-tb;"><i>Im</i></span> ᠢᠨᠦ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠣᠯᠬᠤ ᠠᠭᠠᠳ <br><span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠢᠨᠦ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠳ᠋ᠦ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠬᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠮᠥᠨ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
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<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
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|style = "height: 220px;"|ᠬᠣᠶᠠᠷ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃ <br />
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= ᠬᠠᠰᠤᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠳᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
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<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math><br />
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= ᠦᠷᠡᠵᠢᠬᠦ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃<ref>''Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И.'' Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — <abbr>М.</abbr>: Просвещение, 1998. — 288 с. — <nowiki>ISBN 5-09-008036-4</nowiki>.</ref> <br />
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<math>(a + bi) (c + di) = </math> <math>= ac + bci - bd + adi = </math> <math>= (ac - bd) + (bc + ad)i </math> <br />
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|style = "height: 225px;"|<span style="writing-mode: horizontal-tb;"><i>2 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭ ᠠ᠋ ᠢᠢ (ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠬᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>3 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭᠠᠨ ᠳᠤ (ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃<br />
ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠢ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠣᠷᠤᠶ ᠲᠠᠢ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠵᠤ᠂ <span style="writing-mode: horizontal-tb;"><i>√5</i></span> ᠢᠢᠠᠷ ᠰᠤᠩᠭᠠᠨ ᠠ᠋᠃ ᠡᠨᠡ ᠢᠨᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠾᠢᠫᠣᠲ᠋ᠧᠨᠦᠽ ᠤᠨ<sup>hypotenuse</sup> ᠤᠷᠲᠤ ᠪᠣᠯᠠᠢ᠃ <br />
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= ᠬᠤᠪᠢᠶᠠᠬᠤ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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<math>\frac{a+bi}{c+di}=</math> <br />
<math>=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=</math> <br />
<math>=\frac{ac+bd}{c^2+d^2}+\left(\frac{bc-ad}{c^2+d^2}\right)i.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ<sup>quadrate</sup> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨᠡ᠋᠃<ref>Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01</ref><br />
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<math>(x+yi)^2=x^2-y^2 + 2xyi.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> <span style="writing-mode: horizontal-tb;"><i>(b ≠ 0)</i></span> ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ <br />
<math> \pm (\gamma + \delta i)</math> ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄<br />
<math>\gamma = \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠠ <math>\delta = (\sgn b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠣᠯᠤᠮᠤᠢ᠃<br />
<span style="writing-mode: horizontal-tb;"><i>sgn b</i></span> ᠭᠡᠭᠴᠢ ᠢᠨᠦ <math>\sgn b = \begin{cases} \ \ 1, & b > 0 \\ \ \ 0, & b = 0 \\ -1, & b < 0 \end{cases}</math> ᠭᠡᠵᠦ ᠲᠠᠢᠢᠯᠠᠭᠳᠠᠨᠠ᠃ <br />
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ᠡᠭᠦᠨ ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ ᠢᠢ ᠲᠤᠬᠠᠢᠢᠲᠠ ᠳ᠋ᠤ <math> \pm (\gamma + \delta i)</math> ᠲᠣᠭᠠ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠦ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠲᠣᠭᠠ ᠢᠢ ᠭᠠᠷᠭᠠᠵᠤ ᠢᠷᠡᠬᠦ ᠶᠣᠰᠤᠲᠠᠢ᠃ <br />
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=ᠬᠡᠷᠡᠭᠯᠡᠭᠡ=<br />
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== ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ==<br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠳᠠᠪᠲᠠᠯᠲᠠ ᠪᠠᠷ ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠠᠢᠢᠭᠤᠯᠬᠤ᠃<br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠯ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠳ᠋ᠤ ᠡᠭᠦᠰᠬᠡᠳᠡᠭ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ<sup>fractal</sup> ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠵᠡᠰᠱᠢᠶ ᠡ᠋ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡ ᠬᠦ ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ <span style="writing-mode: horizontal-tb;"><i>z = 0</i></span> ᠡᠴᠡ ᠡᠬᠢᠯᠡᠭᠡᠳ ᠢᠲ᠋ᠧᠷᠠᠼ ᠬᠢᠬᠦ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;"><i>f<sub>c</sub>(z)=z<sup>2</sup>+c</i></span> ᠹᠦᠨ᠍ᠻᠼ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠢᠢᠬᠤ ᠨᠥᠬᠦᠴᠡᠯ ᠪᠦᠬᠦᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠤ ᠣᠯᠠᠨᠯᠢᠬ ᠪᠠᠢᠢᠳᠠᠭ᠃ <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ ᠪᠠᠢᠢᠭᠠᠯᠢ ᠢᠢᠨ ᠭᠠᠢᠢᠬᠠᠮᠰᠢᠭᠲᠤ ᠳᠦᠷᠰᠦ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠦᠨ ᠰᠠᠢᠢᠬᠠᠨ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠶᠡᠷᠦ ᠪᠤᠰᠤ ᠢᠢᠨ ᠭᠣᠶᠤᠮᠰᠠᠭ ᠬᠣᠣᠰᠯᠠᠯ ᠢ ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠳᠡᠭ᠃<ref>ᠷᠡᠨᠴᠡᠨ ᠦ ᠡᠩᠬᠡᠪᠠᠲᠤ᠃ '''ᠫᠢᠲ᠋ᠾᠠᠭᠣᠷ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌'''᠃ ᠮᠣᠩᠭᠤᠯ ᠤᠯᠤᠯᠰ ᠊ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠪᠠ ᠲᠣᠭᠠᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠢᠢᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ https://imdt.ac.mn/c/1013874?content=1150891&fbclid=IwAR1HVqeXwT-h7dijj9EeIUzDvFustH99GAr9LHYQIU61XIByuBHCkSm10So 2020</ref> <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠵᠠᠭᠠᠭ ᠢ ᠳᠠᠪᠠᠬᠤ ᠳᠤ ᠪᠠᠨ ᠵᠦᠯᠢᠶ ᠠ᠋ ᠢᠢᠨ<sup>Julia</sup> ᠹᠷᠠᠻᠲᠠᠯ ᠤᠳ ᠦᠷᠭᠡᠯᠵᠢ ᠬᠣᠯᠪᠤᠯᠲᠠ ᠪᠠᠨ ᠠᠯᠳᠠᠵᠤ᠂ ᠹᠠᠲᠣᠤ ᠢᠢᠨ<sup>Fatou</sup> ᠲᠣᠭᠤᠰᠤ ᠪᠣᠯᠤᠨ ᠬᠦᠪᠢᠷᠠᠳᠠᠭ᠃<ref>ᠨᠡᠭᠡᠭᠡᠯᠲᠡᠲᠡᠢ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠳ᠃ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠳᠠᠯ᠃ ᠪᠢᠳᠡᠨ ᠦ ᠡᠷᠭᠢᠨ ᠲᠣᠭᠣᠷᠢᠨ ᠳ᠋ᠠᠬᠢ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠬᠡᠷᠬᠢᠨ ᠠᠵᠢᠯᠯᠠᠳᠠᠭ ᠪᠤᠢ᠃ ᠹᠷᠠᠻᠲᠯᠠ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠶᠢᠨ ᠹᠷᠠᠻᠲᠯᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢ᠌ᠭ᠌᠃ https://ultrait.ru/mn/smartphones/otkrytie-fraktalov-beskonechnost-fraktalov-kak-ustroen-mir.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/10/02</ref><br />
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|style = "height: 225px;"|ᠳᠦᠷᠰᠦ ᠠᠮᠢᠯᠠᠭᠤᠯᠤᠨ ᠵᠦᠯᠢᠶ ᠠ᠋ ᠢᠢᠨ <sup>Julia</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠦᠵᠡᠭᠦᠯᠦᠭᠰᠡᠨ ᠢᠨᠦ᠃<br />
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== ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ==<br />
ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌<sup>quantum mechanics</sup> ᠠᠲ᠋ᠣᠮ ᠠᠴᠠ ᠵᠢᠵᠢᠭ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ᠂ ᠶᠠᠭᠤᠨ ᠤ ᠡᠮᠦᠨᠡ ᠪᠣᠽᠣᠨ<sup>boson</sup> (ᠵᠢᠱᠢᠶᠡ ᠨᠢ ᠹᠣᠲ᠋ᠣᠨ<sup>photon</sup>ᠪᠠ ᠹᠧᠷᠮᠢᠣᠨ ᠤ<sup>fermion</sup> (ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠨᠧᠦᠲ᠋ᠷᠣᠨ) ᠬᠥᠳᠡᠯᠭᠡᠭᠡᠨ᠂ ᠬᠠᠷᠢᠯᠴᠠᠨ ᠦᠢᠯᠡᠴᠢᠯᠡᠯ ᠢ ᠰᠤᠳᠤᠯᠳᠠᠭ᠃ <br />
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ᠴᠢᠩᠭᠢᠬᠦ ᠳᠦ ᠪᠡᠨ ᠲᠡᠳᠡᠭᠡᠷ ᠦᠨ ᠦᠢᠯᠡ ᠬᠥᠳᠡᠯᠦᠯ ᠦᠨ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭ᠌ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠯᠲᠠ ᠶᠢ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠤᠳᠬᠠ ᠪᠠᠷ ᠭᠠᠷᠭᠠᠳᠠᠭ᠃ <br />
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ᠴᠤᠬᠤᠮ ᠳᠠᠭᠠᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠦᠨᠳᠦᠰᠦ ᠰᠠᠭᠤᠷᠢ ᠪᠣᠯᠳᠠᠭ᠃ <br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠲᠤ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ <sup>Schrödinger</sup> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠠᠴᠢ ᠬᠣᠯᠪᠣᠭᠳᠠᠯ ᠨᠢ ᠰᠣᠩᠭᠣᠳᠠᠭ ᠹᠢᠽᠢᡘ ᠳ᠋ᠡᠬᠢ ᠨᠧᠧᠲ᠋ᠣᠨ᠍ ᠤ ᠬᠣᠶᠠᠳᠤᠭᠠᠷ ᠬᠠᠤᠯᠢ ᠶᠢᠨ ᠠᠳᠠᠯᠢ ᠶᠤᠮ᠃ <br />
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ᠠᠯᠢ ᠠᠯᠢ ᠨᠢ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠷᠢᠯᠠᠯ᠂ ᠢᠮᠫᠦ᠋ᠯᠰ ᠢ <sup>impuls</sup> ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭᠴᠢᠯᠠᠨ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠳᠠᠭ᠃ <br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠨᠢ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ<sup>function</sup> ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠬᠦ ᠳᠦ ᠲᠣᠬᠢᠷᠠᠮᠵᠢ ᠲᠠᠢ ᠤᠴᠢᠷ ᠠᠴᠠ ᠹᠢᠽᠢᠻ ᠦᠨ ᠡᠨᠡ ᠰᠠᠯᠪᠤᠷᠢ ᠳᠤ ᠵᠠᠶ᠋ᠢᠯᠠᠰᠢ ᠦᠭᠡᠢ ᠱᠠᠭᠠᠷᠳᠠᠯᠭᠠ ᠲᠠᠢ ᠶᠤᠮ᠃ <br />
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ᠴᠠᠭᠠᠰᠢᠯᠠᠪᠠᠯ᠂ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠢᠭᠤᠳ ᠨᠥᠯᠥᠭᠡ ᠪᠡᠷ ᠬᠢᠮᠢ ᠶᠢᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠬᠥᠭᠵᠢᠯ ᠡᠷᠴᠢᠮᠵᠢᠭᠰᠡᠨ᠃ <br />
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1927 ᠣᠨ ᠳᠤ ᠸᠠᠯᠲ᠋ᠧᠷ ᠾᠠᠢᠢᠲ᠋ᠯᠧᠷ<sup>Walter Heitler</sup> ᠂ ᠹᠷᠢᠼ ᠯᠣᠨ᠍ᠳᠣᠨ<sup>Fritz London</sup> ᠨᠠᠷ ᠸᠠᠯᠧᠨ᠍ᠲ ᠤᠨ ᠬᠣᠯᠪᠣᠭᠠᠰᠤ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠲᠣᠮᠢᠶᠠᠯᠠᠭᠰᠠᠨ᠃ <br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠨᠢᠭᠡ ᠭᠣᠣᠯ ᠠᠰᠠᠭᠤᠳᠠᠯ ᠪᠣᠯ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ ᠦᠨ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ ᠤᠯᠬᠤ ᠶᠠᠪᠤᠳᠠᠯ ᠶᠤᠮ᠃ <br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠭᠡᠳᠡᠭ ᠨᠢ ᠲᠣᠳᠣᠷᠬᠠᠢ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠵᠤᠷᠪᠤᠰ ᠲᠤ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠵᠤ ᠪᠣᠯᠬᠤ ᠪᠠᠶ᠋ᠢᠷᠢᠰᠢᠯ ᠤᠳ ᠤᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠲᠠᠷᠬᠠᠯᠲᠠ ᠶᠤᠮ᠄ <br />
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<small><math>\frac{-\hbar}{2m}\frac{\partial^2\Psi(x,t)}{\partial x^2} + V(x,t)\Psi(x,t) = i\hbar\frac{\partial \Psi(x,t)}{\partial t}</math></small><br />
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ᠻᠢᠨᠧᠲᠢᠭ<sup>kinetic</sup> ᠧᠨᠧᠷᠭᠢ<sup>energy</sup><br />
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ᠫᠣᠲ᠋ᠧᠨ᠍ᠼᠢᠶᠠᠯ<sup>potencial</sup> ᠧᠨᠧᠷᠭᠢ<br />
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ᠨᠡᠢᠢᠲᠡ ᠧᠨᠧᠷᠭᠢ<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠲᠦ ᠰᠢᠲᠦᠭᠰᠡᠨ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠠᠭᠤᠷᠢ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠨᠢᠭᠡ ᠪᠣᠯ ᠳᠡᠭᠡᠷᠡ ᠳᠤᠷᠠᠳᠤᠭᠰᠠᠨ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡᠬᠦ ᠲᠣᠮᠢᠶᠠ ᠶᠢ ᠠᠰᠢᠭᠯᠠᠭᠰᠠᠨ ᠢᠶ᠋ᠠᠷ ᠤᠰᠤ ᠲᠥᠷᠦᠭᠴᠢ ᠶᠢᠨ ᠮᠣᠯᠧᠻᠤᠯ ᠳᠠᠬᠢ ᠬᠣᠶᠠᠷ ᠠᠲ᠋ᠣᠮ ᠨᠢ ᠻᠣᠸᠠᠯᠧᠨ᠍ᠲ<sup>covalent</sup> ᠬᠣᠯᠪᠣᠭᠠ ᠭᠡᠳᠡᠭ ᠢ ᠡᠭᠦᠰᠬᠡᠨ ᠧᠯᠧᠻᠲ᠋ᠷᠣᠨ ᠨᠤᠭᠤᠳ ᠢᠢᠠᠨ ᠬᠤᠪᠢᠶᠠᠯᠴᠠᠵᠤ ᠪᠠᠶ᠋ᠢᠳᠠᠭ ᠢ ᠪᠠᠲᠤᠯᠠᠭᠰᠠᠨ᠃<ref>Josiah Wu. Real Life Applications of Complex Numbers. 2020 https://issuu.com/harrowhongkong/docs/final_scientific_harrovian_issue_vi-i/s/11488755</ref><br />
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== ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠢᠢᠨ ᠣᠨᠤᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠶᠢ ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ᠂ ᠢᠯᠠᠩᠭᠤᠶᠠ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠨ ᠲᠣᠭᠲᠠᠪᠤᠷᠢᠲᠠᠢ ᠪᠠᠶᠢᠳᠠᠯ ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠭᠡᠨ ᠳᠤ ᠬᠡᠷᠡᠭᠯᠡᠳᠡᠭ᠃ <br />
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ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠳᠤ 《ᠰᠢᠰᠲ᠋ᠧᠮ》 ᠭᠡᠳᠡᠭ ᠦᠭᠡ ᠶᠢ ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ ᠪᠥᠭᠡᠳ ᠡᠨᠡ ᠨᠢ ᠵᠠᠪᠠᠯ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠢ ᠬᠡᠯᠡᠳᠡᠭ ᠦᠭᠡᠢ᠃ <br />
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ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠡᠭᠦᠨ ᠢ ᠬᠥᠷᠥᠩᠭᠡ ᠶᠢᠨ ᠵᠠᠬᠠ ᠵᠡᠭᠡᠯᠢ ᠳ᠋ᠡᠬᠢ ᠥᠭᠡᠷᠡᠴᠢᠯᠡᠯᠲᠡ᠂ ᠬᠢᠮᠢ ᠶᠢᠨ ᠤᠷᠪᠠᠯ᠂ ᠦᠢᠯᠡ ᠶᠠᠪᠤᠴᠠ ᠶᠢ ᠰᠤᠳᠤᠯᠬᠤ ᠳᠤ ᠬᠡᠷᠡᠭᠯᠡᠵᠦ ᠪᠣᠯᠤᠨᠠ᠃ <br />
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ᠲᠡᠭᠦᠨᠴᠢᠯᠡᠨ ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠷᠣᠪᠣᠲ ᠤᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠳᠤ ᠥᠷᠭᠡᠨ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ ᠲᠤᠯᠠ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠠᠨᠠᠯᠢᠽ ᠴᠤ ᠪᠠᠰᠠ ᠲᠡᠭᠦᠨ ᠳᠦ ᠬᠡᠷᠡᠭᠯᠡᠭᠳᠡᠨᠡ ᠭᠡᠰᠡᠨ ᠦᠭᠡ᠃<ref>Ujjvala Y. Gawarguru, Mitali K. Tibdewal, Rajashri A. Naphade, Rahul M. Jethwani. The Review of Introduction & Application of Complex Number in Engineering. 2nd National Conference Recent Innovations in Science and Engineering (NC-RISE 17). Volume: 5 Issue: 9. pp55 – 57. ISSN: 2321-8169. https://ijritcc.org/download/conferences/NC-RISE_17/Track_6_(ASH)/1506931102_02-10-2017.pdf</ref><br />
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ᠬᠢᠨᠠᠯᠲᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠳᠤ ᠵᠠᠷᠢᠮ ᠳᠤ ᠪᠠᠨ ᠯᠠᠫ᠊ᠯᠠᠰ ᠤᠨ<sup>Laplace</sup> ᠬᠤᠪᠢᠷᠠᠭᠠᠯᠲᠠ ᠶᠢ ᠠᠰᠢᠭᠯᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠳ ᠢ<sup>system</sup> ᠴᠠᠭ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠮᠤᠵᠢ ᠠᠴᠠ ᠳᠠᠪᠲᠠᠮᠵᠢ ᠶᠢᠨ ᠮᠤᠵᠢ ᠳᠤ ᠰᠢᠯᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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ᠳᠠᠷᠠᠭᠠ ᠨᠢ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠨ ᠲᠡᠭ ᠪᠠ ᠲᠤᠶ᠋ᠢᠯ ᠤᠳ ᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠰᠢᠨᠵᠢᠯᠡᠨᠡ᠃<br />
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ᠢᠵᠠᠭᠤᠷ ᠤᠨ ᠮᠥᠷ ᠦᠨ ᠠᠷᠭᠠ ᠪᠠᠷ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠳᠦ ᠲᠡᠭ ᠪᠠ ᠲᠤᠶ᠋ᠢᠯ ᠤᠳ ᠵᠡᠭᠦᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠠᠶ᠋ᠢᠨᠠ ᠤᠤ᠂ ᠡᠰᠡᠪᠡᠯ ᠪᠠᠷᠠᠭᠤᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠠᠶ᠋ᠢᠨᠠ ᠤᠤ ᠭᠡᠳᠡᠭ ᠨᠢ ᠴᠢᠬᠤᠯᠠ᠃ <br />
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ᠥᠭᠡᠷᠡ ᠪᠡᠷ ᠬᠡᠯᠡᠪᠡᠯ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠨᠢ ᠲᠡᠭ ᠡᠴᠡ ᠶᠡᠬᠡ ᠪᠠᠢᠢᠨᠠ ᠤᠤ ᠡᠰᠡᠪᠡᠯ ᠪᠠᠭᠠ ᠪᠠᠶ᠋ᠢᠨᠠ ᠤᠤ ᠭᠡᠳᠡᠭ ᠨᠢ ᠴᠢᠬᠤᠯᠠ ᠶᠤᠮ᠃ ᠬᠡᠷᠪᠡ ᠴᠠᠭ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠬᠤᠪᠢᠷᠠᠯᠲᠠ ᠦᠭᠡᠢ ᠱᠤᠭᠤᠮᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠨᠢ᠄<br />
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- ᠪᠠᠷᠠᠭᠤᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠲᠤᠶ᠋ᠢᠯ ᠲᠠᠢ ᠪᠣᠯ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠂<br />
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- ᠵᠡᠭᠦᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠦᠬᠦ ᠲᠤᠶ᠋ᠢᠯ ᠨᠢ ᠪᠠᠶ᠋ᠢᠪᠠᠯ ᠲᠣᠭᠲᠠᠪᠤᠷᠢᠲᠠᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠂<br />
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- ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠲᠤᠶ᠋ᠢᠯ ᠲᠠᠢ ᠪᠣᠯ ᠶᠠᠯᠢᠭᠦᠢ ᠲᠣᠭᠲᠠᠪᠤᠷᠢᠲᠠᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠃<br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==</div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%BA%E1%A0%A3%E1%A0%AE%E1%A0%AB%E1%A0%8A%E1%A0%AF%E1%A0%A7%E1%A0%BA%E1%A0%B0_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%A0&diff=2750
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-10-08T11:13:23Z
<p>Angka: Edit</p>
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'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ<sup>complex</sup> ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠯᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃ <br />
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|style = "height: 225px;"|ᠺᠣᠮᠫᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠢᠢ ᠲᠥᠯᠦᠭᠡᠯᠡᠵᠦ ᠪᠤᠶ ᠠᠷᠭᠠᠨ᠋ᠳ᠋ ᠤᠨ<sup>Argand</sup> ᠳᠢᠶᠠᠭ᠋ᠷᠠᠮ ᠳᠡᠭᠡᠷ ᠡ᠋ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠸᠧᠺᠲ᠋ᠣᠷ<sup>vector</sup> ᠡᠭᠦᠰᠬᠡᠵᠦ ᠪᠤᠶ ᠬᠣᠣᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ <span style="writing-mode: horizontal-tb;"><i>(a, b)</i></span> ᠳᠦᠷᠰᠦᠯᠡᠨ ᠦᠵᠡᠭᠦᠯᠵᠦ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
ᠡᠭᠦᠨ ᠳ᠋ᠦ: <br><span style="writing-mode: horizontal-tb;"><i>Re</i></span> ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ᠂ <br><span style="writing-mode: horizontal-tb;"><i>Im</i></span> ᠢᠨᠦ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠣᠯᠬᠤ ᠠᠭᠠᠳ <br><span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠢᠨᠦ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠳ᠋ᠦ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠬᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠮᠥᠨ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
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<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
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{| class="wikitable"<br />
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|style = "height: 220px;"|ᠬᠣᠶᠠᠷ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃ <br />
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= ᠬᠠᠰᠤᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠳᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
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<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math><br />
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= ᠦᠷᠡᠵᠢᠬᠦ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃<ref>''Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И.'' Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — <abbr>М.</abbr>: Просвещение, 1998. — 288 с. — <nowiki>ISBN 5-09-008036-4</nowiki>.</ref> <br />
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<math>(a + bi) (c + di) = </math> <math>= ac + bci - bd + adi = </math> <math>= (ac - bd) + (bc + ad)i </math> <br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|<span style="writing-mode: horizontal-tb;"><i>2 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭ ᠠ᠋ ᠢᠢ (ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠬᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>3 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭᠠᠨ ᠳᠤ (ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃<br />
ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠢ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠣᠷᠤᠶ ᠲᠠᠢ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠵᠤ᠂ <span style="writing-mode: horizontal-tb;"><i>√5</i></span> ᠢᠢᠠᠷ ᠰᠤᠩᠭᠠᠨ ᠠ᠋᠃ ᠡᠨᠡ ᠢᠨᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠾᠢᠫᠣᠲ᠋ᠧᠨᠦᠽ ᠤᠨ<sup>hypotenuse</sup> ᠤᠷᠲᠤ ᠪᠣᠯᠠᠢ᠃ <br />
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= ᠬᠤᠪᠢᠶᠠᠬᠤ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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<math>\frac{a+bi}{c+di}=</math> <br />
<math>=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=</math> <br />
<math>=\frac{ac+bd}{c^2+d^2}+\left(\frac{bc-ad}{c^2+d^2}\right)i.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ<sup>quadrate</sup> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨᠡ᠋᠃<ref>Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01</ref><br />
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<math>(x+yi)^2=x^2-y^2 + 2xyi.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> <span style="writing-mode: horizontal-tb;"><i>(b ≠ 0)</i></span> ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ <br />
<math> \pm (\gamma + \delta i)</math> ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄<br />
<math>\gamma = \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠠ <math>\delta = (\sgn b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠣᠯᠤᠮᠤᠢ᠃<br />
<span style="writing-mode: horizontal-tb;"><i>sgn b</i></span> ᠭᠡᠭᠴᠢ ᠢᠨᠦ <math>\sgn b = \begin{cases} \ \ 1, & b > 0 \\ \ \ 0, & b = 0 \\ -1, & b < 0 \end{cases}</math> ᠭᠡᠵᠦ ᠲᠠᠢᠢᠯᠠᠭᠳᠠᠨᠠ᠃ <br />
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ᠡᠭᠦᠨ ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ ᠢᠢ ᠲᠤᠬᠠᠢᠢᠲᠠ ᠳ᠋ᠤ <math> \pm (\gamma + \delta i)</math> ᠲᠣᠭᠠ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠦ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠲᠣᠭᠠ ᠢᠢ ᠭᠠᠷᠭᠠᠵᠤ ᠢᠷᠡᠬᠦ ᠶᠣᠰᠤᠲᠠᠢ᠃ <br />
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=ᠬᠡᠷᠡᠭᠯᠡᠭᠡ=<br />
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== ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ==<br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠳᠠᠪᠲᠠᠯᠲᠠ ᠪᠠᠷ ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠠᠢᠢᠭᠤᠯᠬᠤ᠃<br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠯ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠳ᠋ᠤ ᠡᠭᠦᠰᠬᠡᠳᠡᠭ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ<sup>fractal</sup> ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠵᠡᠰᠱᠢᠶ ᠡ᠋ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡ ᠬᠦ ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ <span style="writing-mode: horizontal-tb;"><i>z = 0</i></span> ᠡᠴᠡ ᠡᠬᠢᠯᠡᠭᠡᠳ ᠢᠲ᠋ᠧᠷᠠᠼ ᠬᠢᠬᠦ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;"><i>f<sub>c</sub>(z)=z<sup>2</sup>+c</i></span> ᠹᠦᠨ᠍ᠻᠼ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠢᠢᠬᠤ ᠨᠥᠬᠦᠴᠡᠯ ᠪᠦᠬᠦᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠤ ᠣᠯᠠᠨᠯᠢᠬ ᠪᠠᠢᠢᠳᠠᠭ᠃ <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ ᠪᠠᠢᠢᠭᠠᠯᠢ ᠢᠢᠨ ᠭᠠᠢᠢᠬᠠᠮᠰᠢᠭᠲᠤ ᠳᠦᠷᠰᠦ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠦᠨ ᠰᠠᠢᠢᠬᠠᠨ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠶᠡᠷᠦ ᠪᠤᠰᠤ ᠢᠢᠨ ᠭᠣᠶᠤᠮᠰᠠᠭ ᠬᠣᠣᠰᠯᠠᠯ ᠢ ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠳᠡᠭ᠃<ref>ᠷᠡᠨᠴᠡᠨ ᠦ ᠡᠩᠬᠡᠪᠠᠲᠤ᠃ '''ᠫᠢᠲ᠋ᠾᠠᠭᠣᠷ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌'''᠃ ᠮᠣᠩᠭᠤᠯ ᠤᠯᠤᠯᠰ ᠊ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠪᠠ ᠲᠣᠭᠠᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠢᠢᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ https://imdt.ac.mn/c/1013874?content=1150891&fbclid=IwAR1HVqeXwT-h7dijj9EeIUzDvFustH99GAr9LHYQIU61XIByuBHCkSm10So 2020</ref> <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠵᠠᠭᠠᠭ ᠢ ᠳᠠᠪᠠᠬᠤ ᠳᠤ ᠪᠠᠨ ᠵᠦᠯᠢᠶ ᠠ᠋ ᠢᠢᠨ<sup>Julia</sup> ᠹᠷᠠᠻᠲᠠᠯ ᠤᠳ ᠦᠷᠭᠡᠯᠵᠢ ᠬᠣᠯᠪᠤᠯᠲᠠ ᠪᠠᠨ ᠠᠯᠳᠠᠵᠤ᠂ ᠹᠠᠲᠣᠤ ᠢᠢᠨ<sup>Fatou</sup> ᠲᠣᠭᠤᠰᠤ ᠪᠣᠯᠤᠨ ᠬᠦᠪᠢᠷᠠᠳᠠᠭ᠃<ref>ᠨᠡᠭᠡᠭᠡᠯᠲᠡᠲᠡᠢ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠳ᠃ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠳᠠᠯ᠃ ᠪᠢᠳᠡᠨ ᠦ ᠡᠷᠭᠢᠨ ᠲᠣᠭᠣᠷᠢᠨ ᠳ᠋ᠠᠬᠢ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠬᠡᠷᠬᠢᠨ ᠠᠵᠢᠯᠯᠠᠳᠠᠭ ᠪᠤᠢ᠃ ᠹᠷᠠᠻᠲᠯᠠ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠶᠢᠨ ᠹᠷᠠᠻᠲᠯᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢ᠌ᠭ᠌᠃ https://ultrait.ru/mn/smartphones/otkrytie-fraktalov-beskonechnost-fraktalov-kak-ustroen-mir.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/10/02</ref><br />
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== ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ==<br />
ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌<sup>quantum mechanics</sup> ᠠᠲ᠋ᠣᠮ ᠠᠴᠠ ᠵᠢᠵᠢᠭ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ᠂ ᠶᠠᠭᠤᠨ ᠤ ᠡᠮᠦᠨᠡ ᠪᠣᠽᠣᠨ<sup>boson</sup> (ᠵᠢᠱᠢᠶᠡ ᠨᠢ ᠹᠣᠲ᠋ᠣᠨ<sup>photon</sup>ᠪᠠ ᠹᠧᠷᠮᠢᠣᠨ ᠤ<sup>fermion</sup> (ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠨᠧᠦᠲ᠋ᠷᠣᠨ) ᠬᠥᠳᠡᠯᠭᠡᠭᠡᠨ᠂ ᠬᠠᠷᠢᠯᠴᠠᠨ ᠦᠢᠯᠡᠴᠢᠯᠡᠯ ᠢ ᠰᠤᠳᠤᠯᠳᠠᠭ᠃ <br />
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ᠴᠢᠩᠭᠢᠬᠦ ᠳᠦ ᠪᠡᠨ ᠲᠡᠳᠡᠭᠡᠷ ᠦᠨ ᠦᠢᠯᠡ ᠬᠥᠳᠡᠯᠦᠯ ᠦᠨ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭ᠌ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠯᠲᠠ ᠶᠢ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠤᠳᠬᠠ ᠪᠠᠷ ᠭᠠᠷᠭᠠᠳᠠᠭ᠃ <br />
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ᠴᠤᠬᠤᠮ ᠳᠠᠭᠠᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠦᠨᠳᠦᠰᠦ ᠰᠠᠭᠤᠷᠢ ᠪᠣᠯᠳᠠᠭ᠃ <br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠲᠤ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ <sup>Schrödinger</sup> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠠᠴᠢ ᠬᠣᠯᠪᠣᠭᠳᠠᠯ ᠨᠢ ᠰᠣᠩᠭᠣᠳᠠᠭ ᠹᠢᠽᠢᡘ ᠳ᠋ᠡᠬᠢ ᠨᠧᠧᠲ᠋ᠣᠨ᠍ ᠤ ᠬᠣᠶᠠᠳᠤᠭᠠᠷ ᠬᠠᠤᠯᠢ ᠶᠢᠨ ᠠᠳᠠᠯᠢ ᠶᠤᠮ᠃ <br />
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ᠠᠯᠢ ᠠᠯᠢ ᠨᠢ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠷᠢᠯᠠᠯ᠂ ᠢᠮᠫᠦ᠋ᠯᠰ ᠢ <sup>impuls</sup> ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭᠴᠢᠯᠠᠨ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠳᠠᠭ᠃ <br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠨᠢ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ<sup>function</sup> ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠬᠦ ᠳᠦ ᠲᠣᠬᠢᠷᠠᠮᠵᠢ ᠲᠠᠢ ᠤᠴᠢᠷ ᠠᠴᠠ ᠹᠢᠽᠢᠻ ᠦᠨ ᠡᠨᠡ ᠰᠠᠯᠪᠤᠷᠢ ᠳᠤ ᠵᠠᠶ᠋ᠢᠯᠠᠰᠢ ᠦᠭᠡᠢ ᠱᠠᠭᠠᠷᠳᠠᠯᠭᠠ ᠲᠠᠢ ᠶᠤᠮ᠃ <br />
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ᠴᠠᠭᠠᠰᠢᠯᠠᠪᠠᠯ᠂ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠢᠭᠤᠳ ᠨᠥᠯᠥᠭᠡ ᠪᠡᠷ ᠬᠢᠮᠢ ᠶᠢᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠬᠥᠭᠵᠢᠯ ᠡᠷᠴᠢᠮᠵᠢᠭᠰᠡᠨ᠃ <br />
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1927 ᠣᠨ ᠳᠤ ᠸᠠᠯᠲ᠋ᠧᠷ ᠾᠠᠢᠢᠲ᠋ᠯᠧᠷ<sup>Walter Heitler</sup> ᠂ ᠹᠷᠢᠼ ᠯᠣᠨ᠍ᠳᠣᠨ<sup>Fritz London</sup> ᠨᠠᠷ ᠸᠠᠯᠧᠨ᠍ᠲ ᠤᠨ ᠬᠣᠯᠪᠣᠭᠠᠰᠤ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠲᠣᠮᠢᠶᠠᠯᠠᠭᠰᠠᠨ᠃ <br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠨᠢᠭᠡ ᠭᠣᠣᠯ ᠠᠰᠠᠭᠤᠳᠠᠯ ᠪᠣᠯ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ ᠦᠨ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ ᠤᠯᠬᠤ ᠶᠠᠪᠤᠳᠠᠯ ᠶᠤᠮ᠃ <br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠭᠡᠳᠡᠭ ᠨᠢ ᠲᠣᠳᠣᠷᠬᠠᠢ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠵᠤᠷᠪᠤᠰ ᠲᠤ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠵᠤ ᠪᠣᠯᠬᠤ ᠪᠠᠶ᠋ᠢᠷᠢᠰᠢᠯ ᠤᠳ ᠤᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠲᠠᠷᠬᠠᠯᠲᠠ ᠶᠤᠮ᠄ <br />
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<small><math>\frac{-\hbar}{2m}\frac{\partial^2\Psi(x,t)}{\partial x^2} + V(x,t)\Psi(x,t) = i\hbar\frac{\partial \Psi(x,t)}{\partial t}</math></small><br />
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ᠻᠢᠨᠧᠲᠢᠭ<sup>kinetic</sup> ᠧᠨᠧᠷᠭᠢ<sup>energy</sup><br />
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ᠫᠣᠲ᠋ᠧᠨ᠍ᠼᠢᠶᠠᠯ<sup>potencial</sup> ᠧᠨᠧᠷᠭᠢ<br />
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ᠨᠡᠢᠢᠲᠡ ᠧᠨᠧᠷᠭᠢ<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠲᠦ ᠰᠢᠲᠦᠭᠰᠡᠨ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠠᠭᠤᠷᠢ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠨᠢᠭᠡ ᠪᠣᠯ ᠳᠡᠭᠡᠷᠡ ᠳᠤᠷᠠᠳᠤᠭᠰᠠᠨ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡᠬᠦ ᠲᠣᠮᠢᠶᠠ ᠶᠢ ᠠᠰᠢᠭᠯᠠᠭᠰᠠᠨ ᠢᠶ᠋ᠠᠷ ᠤᠰᠤ ᠲᠥᠷᠦᠭᠴᠢ ᠶᠢᠨ ᠮᠣᠯᠧᠻᠤᠯ ᠳᠠᠬᠢ ᠬᠣᠶᠠᠷ ᠠᠲ᠋ᠣᠮ ᠨᠢ ᠻᠣᠸᠠᠯᠧᠨ᠍ᠲ<sup>covalent</sup> ᠬᠣᠯᠪᠣᠭᠠ ᠭᠡᠳᠡᠭ ᠢ ᠡᠭᠦᠰᠬᠡᠨ ᠧᠯᠧᠻᠲ᠋ᠷᠣᠨ ᠨᠤᠭᠤᠳ ᠢᠢᠠᠨ ᠬᠤᠪᠢᠶᠠᠯᠴᠠᠵᠤ ᠪᠠᠶ᠋ᠢᠳᠠᠭ ᠢ ᠪᠠᠲᠤᠯᠠᠭᠰᠠᠨ᠃<ref>Josiah Wu. Real Life Applications of Complex Numbers. 2020 https://issuu.com/harrowhongkong/docs/final_scientific_harrovian_issue_vi-i/s/11488755</ref><br />
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== ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠢᠢᠨ ᠣᠨᠤᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠶᠢ ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ᠂ ᠢᠯᠠᠩᠭᠤᠶᠠ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠨ ᠲᠣᠭᠲᠠᠪᠤᠷᠢᠲᠠᠢ ᠪᠠᠶᠢᠳᠠᠯ ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠭᠡᠨ ᠳᠤ ᠬᠡᠷᠡᠭᠯᠡᠳᠡᠭ᠃ <br />
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ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠳᠤ 《ᠰᠢᠰᠲ᠋ᠧᠮ》 ᠭᠡᠳᠡᠭ ᠦᠭᠡ ᠶᠢ ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ ᠪᠥᠭᠡᠳ ᠡᠨᠡ ᠨᠢ ᠵᠠᠪᠠᠯ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠢ ᠬᠡᠯᠡᠳᠡᠭ ᠦᠭᠡᠢ᠃ <br />
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ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠡᠭᠦᠨ ᠢ ᠬᠥᠷᠥᠩᠭᠡ ᠶᠢᠨ ᠵᠠᠬᠠ ᠵᠡᠭᠡᠯᠢ ᠳ᠋ᠡᠬᠢ ᠥᠭᠡᠷᠡᠴᠢᠯᠡᠯᠲᠡ᠂ ᠬᠢᠮᠢ ᠶᠢᠨ ᠤᠷᠪᠠᠯ᠂ ᠦᠢᠯᠡ ᠶᠠᠪᠤᠴᠠ ᠶᠢ ᠰᠤᠳᠤᠯᠬᠤ ᠳᠤ ᠬᠡᠷᠡᠭᠯᠡᠵᠦ ᠪᠣᠯᠤᠨᠠ᠃ <br />
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ᠲᠡᠭᠦᠨᠴᠢᠯᠡᠨ ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠷᠣᠪᠣᠲ ᠤᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠳᠤ ᠥᠷᠭᠡᠨ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ ᠲᠤᠯᠠ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠠᠨᠠᠯᠢᠽ ᠴᠤ ᠪᠠᠰᠠ ᠲᠡᠭᠦᠨ ᠳᠦ ᠬᠡᠷᠡᠭᠯᠡᠭᠳᠡᠨᠡ ᠭᠡᠰᠡᠨ ᠦᠭᠡ᠃<ref>Ujjvala Y. Gawarguru, Mitali K. Tibdewal, Rajashri A. Naphade, Rahul M. Jethwani. The Review of Introduction & Application of Complex Number in Engineering. 2nd National Conference Recent Innovations in Science and Engineering (NC-RISE 17). Volume: 5 Issue: 9. pp55 – 57. ISSN: 2321-8169. https://ijritcc.org/download/conferences/NC-RISE_17/Track_6_(ASH)/1506931102_02-10-2017.pdf</ref><br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==</div>
Angka
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-10-06T14:20:11Z
<p>Angka: Edit</p>
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'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ<sup>complex</sup> ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠯᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃ <br />
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|style = "height: 225px;"|ᠺᠣᠮᠫᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠢᠢ ᠲᠥᠯᠦᠭᠡᠯᠡᠵᠦ ᠪᠤᠶ ᠠᠷᠭᠠᠨ᠋ᠳ᠋ ᠤᠨ<sup>Argand</sup> ᠳᠢᠶᠠᠭ᠋ᠷᠠᠮ ᠳᠡᠭᠡᠷ ᠡ᠋ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠸᠧᠺᠲ᠋ᠣᠷ<sup>vector</sup> ᠡᠭᠦᠰᠬᠡᠵᠦ ᠪᠤᠶ ᠬᠣᠣᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ <span style="writing-mode: horizontal-tb;"><i>(a, b)</i></span> ᠳᠦᠷᠰᠦᠯᠡᠨ ᠦᠵᠡᠭᠦᠯᠵᠦ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
ᠡᠭᠦᠨ ᠳ᠋ᠦ: <br><span style="writing-mode: horizontal-tb;"><i>Re</i></span> ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ᠂ <br><span style="writing-mode: horizontal-tb;"><i>Im</i></span> ᠢᠨᠦ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠣᠯᠬᠤ ᠠᠭᠠᠳ <br><span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠢᠨᠦ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠳ᠋ᠦ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠬᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠮᠥᠨ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
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<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
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|style = "height: 220px;"|ᠬᠣᠶᠠᠷ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃ <br />
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= ᠬᠠᠰᠤᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠳᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
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<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math><br />
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= ᠦᠷᠡᠵᠢᠬᠦ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃<ref>''Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И.'' Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — <abbr>М.</abbr>: Просвещение, 1998. — 288 с. — <nowiki>ISBN 5-09-008036-4</nowiki>.</ref> <br />
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<math>(a + bi) (c + di) = </math> <math>= ac + bci - bd + adi = </math> <math>= (ac - bd) + (bc + ad)i </math> <br />
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|style = "height: 225px;"|<span style="writing-mode: horizontal-tb;"><i>2 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭ ᠠ᠋ ᠢᠢ (ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠬᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>3 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭᠠᠨ ᠳᠤ (ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃<br />
ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠢ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠣᠷᠤᠶ ᠲᠠᠢ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠵᠤ᠂ <span style="writing-mode: horizontal-tb;"><i>√5</i></span> ᠢᠢᠠᠷ ᠰᠤᠩᠭᠠᠨ ᠠ᠋᠃ ᠡᠨᠡ ᠢᠨᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠾᠢᠫᠣᠲ᠋ᠧᠨᠦᠽ ᠤᠨ<sup>hypotenuse</sup> ᠤᠷᠲᠤ ᠪᠣᠯᠠᠢ᠃ <br />
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= ᠬᠤᠪᠢᠶᠠᠬᠤ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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<math>\frac{a+bi}{c+di}=</math> <br />
<math>=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=</math> <br />
<math>=\frac{ac+bd}{c^2+d^2}+\left(\frac{bc-ad}{c^2+d^2}\right)i.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ<sup>quadrate</sup> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨᠡ᠋᠃<ref>Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01</ref><br />
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<math>(x+yi)^2=x^2-y^2 + 2xyi.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> <span style="writing-mode: horizontal-tb;"><i>(b ≠ 0)</i></span> ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ <br />
<math> \pm (\gamma + \delta i)</math> ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄<br />
<math>\gamma = \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠠ <math>\delta = (\sgn b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠣᠯᠤᠮᠤᠢ᠃<br />
<span style="writing-mode: horizontal-tb;"><i>sgn b</i></span> ᠭᠡᠭᠴᠢ ᠢᠨᠦ <math>\sgn b = \begin{cases} \ \ 1, & b > 0 \\ \ \ 0, & b = 0 \\ -1, & b < 0 \end{cases}</math> ᠭᠡᠵᠦ ᠲᠠᠢᠢᠯᠠᠭᠳᠠᠨᠠ᠃ <br />
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ᠡᠭᠦᠨ ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ ᠢᠢ ᠲᠤᠬᠠᠢᠢᠲᠠ ᠳ᠋ᠤ <math> \pm (\gamma + \delta i)</math> ᠲᠣᠭᠠ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠦ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠲᠣᠭᠠ ᠢᠢ ᠭᠠᠷᠭᠠᠵᠤ ᠢᠷᠡᠬᠦ ᠶᠣᠰᠤᠲᠠᠢ᠃ <br />
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=ᠬᠡᠷᠡᠭᠯᠡᠭᠡ=<br />
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== ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ==<br />
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|style = "height: 225px;"|ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠳᠠᠪᠲᠠᠯᠲᠠ ᠪᠠᠷ ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠠᠢᠢᠭᠤᠯᠬᠤ᠃<br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠯ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠳ᠋ᠤ ᠡᠭᠦᠰᠬᠡᠳᠡᠭ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ<sup>fractal</sup> ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠵᠡᠰᠱᠢᠶ ᠡ᠋ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡ ᠬᠦ ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ <span style="writing-mode: horizontal-tb;"><i>z = 0</i></span> ᠡᠴᠡ ᠡᠬᠢᠯᠡᠭᠡᠳ ᠢᠲ᠋ᠧᠷᠠᠼ ᠬᠢᠬᠦ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;"><i>f<sub>c</sub>(z)=z<sup>2</sup>+c</i></span> ᠹᠦᠨ᠍ᠻᠼ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠢᠢᠬᠤ ᠨᠥᠬᠦᠴᠡᠯ ᠪᠦᠬᠦᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠤ ᠣᠯᠠᠨᠯᠢᠬ ᠪᠠᠢᠢᠳᠠᠭ᠃ <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ ᠪᠠᠢᠢᠭᠠᠯᠢ ᠢᠢᠨ ᠭᠠᠢᠢᠬᠠᠮᠰᠢᠭᠲᠤ ᠳᠦᠷᠰᠦ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠦᠨ ᠰᠠᠢᠢᠬᠠᠨ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠶᠡᠷᠦ ᠪᠤᠰᠤ ᠢᠢᠨ ᠭᠣᠶᠤᠮᠰᠠᠭ ᠬᠣᠣᠰᠯᠠᠯ ᠢ ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠳᠡᠭ᠃<ref>ᠷᠡᠨᠴᠡᠨ ᠦ ᠡᠩᠬᠡᠪᠠᠲᠤ᠃ '''ᠫᠢᠲ᠋ᠾᠠᠭᠣᠷ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌'''᠃ ᠮᠣᠩᠭᠤᠯ ᠤᠯᠤᠯᠰ ᠊ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠪᠠ ᠲᠣᠭᠠᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠢᠢᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ https://imdt.ac.mn/c/1013874?content=1150891&fbclid=IwAR1HVqeXwT-h7dijj9EeIUzDvFustH99GAr9LHYQIU61XIByuBHCkSm10So 2020</ref> <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠵᠠᠭᠠᠭ ᠢ ᠳᠠᠪᠠᠬᠤ ᠳᠤ ᠪᠠᠨ ᠵᠦᠯᠢᠶ ᠠ᠋ ᠢᠢᠨ<sup>Julia</sup> ᠹᠷᠠᠻᠲᠠᠯ ᠤᠳ ᠦᠷᠭᠡᠯᠵᠢ ᠬᠣᠯᠪᠤᠯᠲᠠ ᠪᠠᠨ ᠠᠯᠳᠠᠵᠤ᠂ ᠹᠠᠲᠣᠤ ᠢᠢᠨ<sup>Fatou</sup> ᠲᠣᠭᠤᠰᠤ ᠪᠣᠯᠤᠨ ᠬᠦᠪᠢᠷᠠᠳᠠᠭ᠃<ref>ᠨᠡᠭᠡᠭᠡᠯᠲᠡᠲᠡᠢ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠳ᠃ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠳᠠᠯ᠃ ᠪᠢᠳᠡᠨ ᠦ ᠡᠷᠭᠢᠨ ᠲᠣᠭᠣᠷᠢᠨ ᠳ᠋ᠠᠬᠢ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠬᠡᠷᠬᠢᠨ ᠠᠵᠢᠯᠯᠠᠳᠠᠭ ᠪᠤᠢ᠃ ᠹᠷᠠᠻᠲᠯᠠ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠶᠢᠨ ᠹᠷᠠᠻᠲᠯᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢ᠌ᠭ᠌᠃ https://ultrait.ru/mn/smartphones/otkrytie-fraktalov-beskonechnost-fraktalov-kak-ustroen-mir.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/10/02</ref><br />
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|style = "height: 225px;"|ᠳᠦᠷᠰᠦ ᠠᠮᠢᠯᠠᠭᠤᠯᠤᠨ ᠵᠦᠯᠢᠶ ᠠ᠋ ᠢᠢᠨ <sup>Julia</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠦᠵᠡᠭᠦᠯᠦᠭᠰᠡᠨ ᠢᠨᠦ᠃<br />
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== ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ==<br />
ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌<sup>quantum mechanics</sup> ᠠᠲ᠋ᠣᠮ ᠠᠴᠠ ᠵᠢᠵᠢᠭ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ᠂ ᠶᠠᠭᠤᠨ ᠤ ᠡᠮᠦᠨᠡ ᠪᠣᠽᠣᠨ<sup>boson</sup> (ᠵᠢᠱᠢᠶᠡ ᠨᠢ ᠹᠣᠲ᠋ᠣᠨ<sup>photon</sup>ᠪᠠ ᠹᠧᠷᠮᠢᠣᠨ ᠤ<sup>fermion</sup> (ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠨᠧᠦᠲ᠋ᠷᠣᠨ) ᠬᠥᠳᠡᠯᠭᠡᠭᠡᠨ᠂ ᠬᠠᠷᠢᠯᠴᠠᠨ ᠦᠢᠯᠡᠴᠢᠯᠡᠯ ᠢ ᠰᠤᠳᠤᠯᠳᠠᠭ᠃ <br />
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ᠴᠢᠩᠭᠢᠬᠦ ᠳᠦ ᠪᠡᠨ ᠲᠡᠳᠡᠭᠡᠷ ᠦᠨ ᠦᠢᠯᠡ ᠬᠥᠳᠡᠯᠦᠯ ᠦᠨ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭ᠌ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠯᠲᠠ ᠶᠢ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠤᠳᠬᠠ ᠪᠠᠷ ᠭᠠᠷᠭᠠᠳᠠᠭ᠃ <br />
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ᠴᠤᠬᠤᠮ ᠳᠠᠭᠠᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠦᠨᠳᠦᠰᠦ ᠰᠠᠭᠤᠷᠢ ᠪᠣᠯᠳᠠᠭ᠃ <br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠲᠤ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ <sup>Schrödinger</sup> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠠᠴᠢ ᠬᠣᠯᠪᠣᠭᠳᠠᠯ ᠨᠢ ᠰᠣᠩᠭᠣᠳᠠᠭ ᠹᠢᠽᠢᡘ ᠳ᠋ᠡᠬᠢ ᠨᠧᠧᠲ᠋ᠣᠨ᠍ ᠤ ᠬᠣᠶᠠᠳᠤᠭᠠᠷ ᠬᠠᠤᠯᠢ ᠶᠢᠨ ᠠᠳᠠᠯᠢ ᠶᠤᠮ᠃ <br />
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ᠠᠯᠢ ᠠᠯᠢ ᠨᠢ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠷᠢᠯᠠᠯ᠂ ᠢᠮᠫᠦ᠋ᠯᠰ ᠢ <sup>impuls</sup> ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭᠴᠢᠯᠠᠨ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠳᠠᠭ᠃ <br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠨᠢ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ<sup>function</sup> ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠬᠦ ᠳᠦ ᠲᠣᠬᠢᠷᠠᠮᠵᠢ ᠲᠠᠢ ᠤᠴᠢᠷ ᠠᠴᠠ ᠹᠢᠽᠢᠻ ᠦᠨ ᠡᠨᠡ ᠰᠠᠯᠪᠤᠷᠢ ᠳᠤ ᠵᠠᠶ᠋ᠢᠯᠠᠰᠢ ᠦᠭᠡᠢ ᠱᠠᠭᠠᠷᠳᠠᠯᠭᠠ ᠲᠠᠢ ᠶᠤᠮ᠃ <br />
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ᠴᠠᠭᠠᠰᠢᠯᠠᠪᠠᠯ᠂ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠢᠭᠤᠳ ᠨᠥᠯᠥᠭᠡ ᠪᠡᠷ ᠬᠢᠮᠢ ᠶᠢᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠬᠥᠭᠵᠢᠯ ᠡᠷᠴᠢᠮᠵᠢᠭᠰᠡᠨ᠃ <br />
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1927 ᠣᠨ ᠳᠤ ᠸᠠᠯᠲ᠋ᠧᠷ ᠾᠠᠢᠢᠲ᠋ᠯᠧᠷ<sup>Walter Heitler</sup> ᠂ ᠹᠷᠢᠼ ᠯᠣᠨ᠍ᠳᠣᠨ<sup>Fritz London</sup> ᠨᠠᠷ ᠸᠠᠯᠧᠨ᠍ᠲ ᠤᠨ ᠬᠣᠯᠪᠣᠭᠠᠰᠤ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠲᠣᠮᠢᠶᠠᠯᠠᠭᠰᠠᠨ᠃ <br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠨᠢᠭᠡ ᠭᠣᠣᠯ ᠠᠰᠠᠭᠤᠳᠠᠯ ᠪᠣᠯ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ ᠦᠨ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ ᠤᠯᠬᠤ ᠶᠠᠪᠤᠳᠠᠯ ᠶᠤᠮ᠃ <br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠭᠡᠳᠡᠭ ᠨᠢ ᠲᠣᠳᠣᠷᠬᠠᠢ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠵᠤᠷᠪᠤᠰ ᠲᠤ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠵᠤ ᠪᠣᠯᠬᠤ ᠪᠠᠶ᠋ᠢᠷᠢᠰᠢᠯ ᠤᠳ ᠤᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠲᠠᠷᠬᠠᠯᠲᠠ ᠶᠤᠮ᠄ <br />
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<small><math>\frac{-\hbar}{2m}\frac{\partial^2\Psi(x,t)}{\partial x^2} + V(x,t)\Psi(x,t) = i\hbar\frac{\partial \Psi(x,t)}{\partial t}</math></small><br />
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ᠻᠢᠨᠧᠲᠢᠭ<sup>kinetic</sup> ᠧᠨᠧᠷᠭᠢ<sup>energy</sup><br />
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ᠫᠣᠲ᠋ᠧᠨ᠍ᠼᠢᠶᠠᠯ<sup>potencial</sup> ᠧᠨᠧᠷᠭᠢ<br />
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ᠨᠡᠢᠢᠲᠡ ᠧᠨᠧᠷᠭᠢ<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠲᠦ ᠰᠢᠲᠦᠭᠰᠡᠨ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠠᠭᠤᠷᠢ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠨᠢᠭᠡ ᠪᠣᠯ ᠳᠡᠭᠡᠷᠡ ᠳᠤᠷᠠᠳᠤᠭᠰᠠᠨ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡᠬᠦ ᠲᠣᠮᠢᠶᠠ ᠶᠢ ᠠᠰᠢᠭᠯᠠᠭᠰᠠᠨ ᠢᠶ᠋ᠠᠷ ᠤᠰᠤ ᠲᠥᠷᠦᠭᠴᠢ ᠶᠢᠨ ᠮᠣᠯᠧᠻᠤᠯ ᠳᠠᠬᠢ ᠬᠣᠶᠠᠷ ᠠᠲ᠋ᠣᠮ ᠨᠢ ᠻᠣᠸᠠᠯᠧᠨ᠍ᠲ<sup>covalent</sup> ᠬᠣᠯᠪᠣᠭᠠ ᠭᠡᠳᠡᠭ ᠢ ᠡᠭᠦᠰᠬᠡᠨ ᠧᠯᠧᠻᠲ᠋ᠷᠣᠨ ᠨᠤᠭᠤᠳ ᠢᠢᠠᠨ ᠬᠤᠪᠢᠶᠠᠯᠴᠠᠵᠤ ᠪᠠᠶ᠋ᠢᠳᠠᠭ ᠢ ᠪᠠᠲᠤᠯᠠᠭᠰᠠᠨ᠃<ref>Josiah Wu. Real Life Applications of Complex Numbers. 2020 https://issuu.com/harrowhongkong/docs/final_scientific_harrovian_issue_vi-i/s/11488755</ref><br />
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== ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠢᠢᠨ ᠣᠨᠤᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠶᠢ ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ᠂ ᠢᠯᠠᠩᠭᠤᠶᠠ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠨ ᠲᠣᠭᠲᠠᠪᠤᠷᠢᠲᠠᠢ ᠪᠠᠶᠢᠳᠠᠯ ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠭᠡᠨ ᠳᠤ ᠬᠡᠷᠡᠭᠯᠡᠳᠡᠭ᠃ <br />
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ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠳᠤ 《ᠰᠢᠰᠲ᠋ᠧᠮ》 ᠭᠡᠳᠡᠭ ᠦᠭᠡ ᠶᠢ ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ ᠪᠥᠭᠡᠳ ᠡᠨᠡ ᠨᠢ ᠵᠠᠪᠠᠯ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠢ ᠬᠡᠯᠡᠳᠡᠭ ᠦᠭᠡᠢ᠃ <br />
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ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠡᠭᠦᠨ ᠢ ᠬᠥᠷᠥᠩᠭᠡ ᠶᠢᠨ ᠵᠠᠬᠠ ᠵᠡᠭᠡᠯᠢ ᠳ᠋ᠡᠬᠢ ᠥᠭᠡᠷᠡᠴᠢᠯᠡᠯᠲᠡ᠂ ᠬᠢᠮᠢ ᠶᠢᠨ ᠤᠷᠪᠠᠯ᠂ ᠦᠢᠯᠡ ᠶᠠᠪᠤᠴᠠ ᠶᠢ ᠰᠤᠳᠤᠯᠬᠤ ᠳᠤ ᠬᠡᠷᠡᠭᠯᠡᠵᠦ ᠪᠣᠯᠤᠨᠠ᠃ <br />
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ᠲᠡᠭᠦᠨᠴᠢᠯᠡᠨ ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠷᠣᠪᠣᠲ ᠤᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠳᠤ ᠥᠷᠭᠡᠨ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ ᠲᠤᠯᠠ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠠᠨᠠᠯᠢᠽ ᠴᠤ ᠪᠠᠰᠠ ᠲᠡᠭᠦᠨ ᠳᠦ ᠬᠡᠷᠡᠭᠯᠡᠭᠳᠡᠨᠡ ᠭᠡᠰᠡᠨ ᠦᠭᠡ᠃ <br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==</div>
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-10-06T04:07:02Z
<p>Angka: Edit</p>
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'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ<sup>complex</sup> ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠯᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃ <br />
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|style = "height: 225px;"|ᠺᠣᠮᠫᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠢᠢ ᠲᠥᠯᠦᠭᠡᠯᠡᠵᠦ ᠪᠤᠶ ᠠᠷᠭᠠᠨ᠋ᠳ᠋ ᠤᠨ<sup>Argand</sup> ᠳᠢᠶᠠᠭ᠋ᠷᠠᠮ ᠳᠡᠭᠡᠷ ᠡ᠋ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠸᠧᠺᠲ᠋ᠣᠷ<sup>vector</sup> ᠡᠭᠦᠰᠬᠡᠵᠦ ᠪᠤᠶ ᠬᠣᠣᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ <span style="writing-mode: horizontal-tb;"><i>(a, b)</i></span> ᠳᠦᠷᠰᠦᠯᠡᠨ ᠦᠵᠡᠭᠦᠯᠵᠦ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
ᠡᠭᠦᠨ ᠳ᠋ᠦ: <br><span style="writing-mode: horizontal-tb;"><i>Re</i></span> ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ᠂ <br><span style="writing-mode: horizontal-tb;"><i>Im</i></span> ᠢᠨᠦ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠣᠯᠬᠤ ᠠᠭᠠᠳ <br><span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠢᠨᠦ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠳ᠋ᠦ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠬᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠮᠥᠨ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
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<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
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|style = "height: 220px;"|ᠬᠣᠶᠠᠷ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃ <br />
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= ᠬᠠᠰᠤᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠳᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
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<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math><br />
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= ᠦᠷᠡᠵᠢᠬᠦ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃<ref>''Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И.'' Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — <abbr>М.</abbr>: Просвещение, 1998. — 288 с. — <nowiki>ISBN 5-09-008036-4</nowiki>.</ref> <br />
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<math>(a + bi) (c + di) = </math> <math>= ac + bci - bd + adi = </math> <math>= (ac - bd) + (bc + ad)i </math> <br />
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|style = "height: 225px;"|<span style="writing-mode: horizontal-tb;"><i>2 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭ ᠠ᠋ ᠢᠢ (ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠬᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>3 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭᠠᠨ ᠳᠤ (ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃<br />
ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠢ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠣᠷᠤᠶ ᠲᠠᠢ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠵᠤ᠂ <span style="writing-mode: horizontal-tb;"><i>√5</i></span> ᠢᠢᠠᠷ ᠰᠤᠩᠭᠠᠨ ᠠ᠋᠃ ᠡᠨᠡ ᠢᠨᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠾᠢᠫᠣᠲ᠋ᠧᠨᠦᠽ ᠤᠨ<sup>hypotenuse</sup> ᠤᠷᠲᠤ ᠪᠣᠯᠠᠢ᠃ <br />
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= ᠬᠤᠪᠢᠶᠠᠬᠤ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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<math>\frac{a+bi}{c+di}=</math> <br />
<math>=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=</math> <br />
<math>=\frac{ac+bd}{c^2+d^2}+\left(\frac{bc-ad}{c^2+d^2}\right)i.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ<sup>quadrate</sup> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨᠡ᠋᠃<ref>Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01</ref><br />
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<math>(x+yi)^2=x^2-y^2 + 2xyi.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> <span style="writing-mode: horizontal-tb;"><i>(b ≠ 0)</i></span> ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ <br />
<math> \pm (\gamma + \delta i)</math> ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄<br />
<math>\gamma = \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠠ <math>\delta = (\sgn b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠣᠯᠤᠮᠤᠢ᠃<br />
<span style="writing-mode: horizontal-tb;"><i>sgn b</i></span> ᠭᠡᠭᠴᠢ ᠢᠨᠦ <math>\sgn b = \begin{cases} \ \ 1, & b > 0 \\ \ \ 0, & b = 0 \\ -1, & b < 0 \end{cases}</math> ᠭᠡᠵᠦ ᠲᠠᠢᠢᠯᠠᠭᠳᠠᠨᠠ᠃ <br />
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ᠡᠭᠦᠨ ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ ᠢᠢ ᠲᠤᠬᠠᠢᠢᠲᠠ ᠳ᠋ᠤ <math> \pm (\gamma + \delta i)</math> ᠲᠣᠭᠠ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠦ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠲᠣᠭᠠ ᠢᠢ ᠭᠠᠷᠭᠠᠵᠤ ᠢᠷᠡᠬᠦ ᠶᠣᠰᠤᠲᠠᠢ᠃ <br />
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=ᠬᠡᠷᠡᠭᠯᠡᠭᠡ=<br />
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== ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ==<br />
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|style = "height: 225px;"|ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠳᠠᠪᠲᠠᠯᠲᠠ ᠪᠠᠷ ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠠᠢᠢᠭᠤᠯᠬᠤ᠃<br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠯ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠳ᠋ᠤ ᠡᠭᠦᠰᠬᠡᠳᠡᠭ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ<sup>fractal</sup> ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠵᠡᠰᠱᠢᠶ ᠡ᠋ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡ ᠬᠦ ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ <span style="writing-mode: horizontal-tb;"><i>z = 0</i></span> ᠡᠴᠡ ᠡᠬᠢᠯᠡᠭᠡᠳ ᠢᠲ᠋ᠧᠷᠠᠼ ᠬᠢᠬᠦ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;"><i>f<sub>c</sub>(z)=z<sup>2</sup>+c</i></span> ᠹᠦᠨ᠍ᠻᠼ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠢᠢᠬᠤ ᠨᠥᠬᠦᠴᠡᠯ ᠪᠦᠬᠦᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠤ ᠣᠯᠠᠨᠯᠢᠬ ᠪᠠᠢᠢᠳᠠᠭ᠃ <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ ᠪᠠᠢᠢᠭᠠᠯᠢ ᠢᠢᠨ ᠭᠠᠢᠢᠬᠠᠮᠰᠢᠭᠲᠤ ᠳᠦᠷᠰᠦ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠦᠨ ᠰᠠᠢᠢᠬᠠᠨ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠶᠡᠷᠦ ᠪᠤᠰᠤ ᠢᠢᠨ ᠭᠣᠶᠤᠮᠰᠠᠭ ᠬᠣᠣᠰᠯᠠᠯ ᠢ ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠳᠡᠭ᠃<ref>ᠷᠡᠨᠴᠡᠨ ᠦ ᠡᠩᠬᠡᠪᠠᠲᠤ᠃ '''ᠫᠢᠲ᠋ᠾᠠᠭᠣᠷ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌'''᠃ ᠮᠣᠩᠭᠤᠯ ᠤᠯᠤᠯᠰ ᠊ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠪᠠ ᠲᠣᠭᠠᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠢᠢᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ https://imdt.ac.mn/c/1013874?content=1150891&fbclid=IwAR1HVqeXwT-h7dijj9EeIUzDvFustH99GAr9LHYQIU61XIByuBHCkSm10So 2020</ref> <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠵᠠᠭᠠᠭ ᠢ ᠳᠠᠪᠠᠬᠤ ᠳᠤ ᠪᠠᠨ ᠵᠦᠯᠢᠶ ᠠ᠋ ᠢᠢᠨ<sup>Julia</sup> ᠹᠷᠠᠻᠲᠠᠯ ᠤᠳ ᠦᠷᠭᠡᠯᠵᠢ ᠬᠣᠯᠪᠤᠯᠲᠠ ᠪᠠᠨ ᠠᠯᠳᠠᠵᠤ᠂ ᠹᠠᠲᠣᠤ ᠢᠢᠨ<sup>Fatou</sup> ᠲᠣᠭᠤᠰᠤ ᠪᠣᠯᠤᠨ ᠬᠦᠪᠢᠷᠠᠳᠠᠭ᠃<ref>ᠨᠡᠭᠡᠭᠡᠯᠲᠡᠲᠡᠢ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠳ᠃ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠳᠠᠯ᠃ ᠪᠢᠳᠡᠨ ᠦ ᠡᠷᠭᠢᠨ ᠲᠣᠭᠣᠷᠢᠨ ᠳ᠋ᠠᠬᠢ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠬᠡᠷᠬᠢᠨ ᠠᠵᠢᠯᠯᠠᠳᠠᠭ ᠪᠤᠢ᠃ ᠹᠷᠠᠻᠲᠯᠠ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠶᠢᠨ ᠹᠷᠠᠻᠲᠯᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢ᠌ᠭ᠌᠃ https://ultrait.ru/mn/smartphones/otkrytie-fraktalov-beskonechnost-fraktalov-kak-ustroen-mir.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/10/02</ref><br />
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|style = "height: 225px;"|ᠳᠦᠷᠰᠦ ᠠᠮᠢᠯᠠᠭᠤᠯᠤᠨ ᠵᠦᠯᠢᠶ ᠠ᠋ ᠢᠢᠨ <sup>Julia</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠦᠵᠡᠭᠦᠯᠦᠭᠰᠡᠨ ᠢᠨᠦ᠃<br />
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== ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ==<br />
ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌<sup>quantum mechanics</sup> ᠠᠲ᠋ᠣᠮ ᠠᠴᠠ ᠵᠢᠵᠢᠭ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ᠂ ᠶᠠᠭᠤᠨ ᠤ ᠡᠮᠦᠨᠡ ᠪᠣᠽᠣᠨ<sup>boson</sup> (ᠵᠢᠱᠢᠶᠡ ᠨᠢ ᠹᠣᠲ᠋ᠣᠨ<sup>photon</sup>ᠪᠠ ᠹᠧᠷᠮᠢᠣᠨ ᠤ<sup>fermion</sup> (ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠨᠧᠦᠲ᠋ᠷᠣᠨ) ᠬᠥᠳᠡᠯᠭᠡᠭᠡᠨ᠂ ᠬᠠᠷᠢᠯᠴᠠᠨ ᠦᠢᠯᠡᠴᠢᠯᠡᠯ ᠢ ᠰᠤᠳᠤᠯᠳᠠᠭ᠃ <br />
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ᠴᠢᠩᠭᠢᠬᠦ ᠳᠦ ᠪᠡᠨ ᠲᠡᠳᠡᠭᠡᠷ ᠦᠨ ᠦᠢᠯᠡ ᠬᠥᠳᠡᠯᠦᠯ ᠦᠨ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭ᠌ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠯᠲᠠ ᠶᠢ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠤᠳᠬᠠ ᠪᠠᠷ ᠭᠠᠷᠭᠠᠳᠠᠭ᠃ <br />
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ᠴᠤᠬᠤᠮ ᠳᠠᠭᠠᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠦᠨᠳᠦᠰᠦ ᠰᠠᠭᠤᠷᠢ ᠪᠣᠯᠳᠠᠭ᠃ <br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠲᠤ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ <sup>Schrödinger</sup> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠠᠴᠢ ᠬᠣᠯᠪᠣᠭᠳᠠᠯ ᠨᠢ ᠰᠣᠩᠭᠣᠳᠠᠭ ᠹᠢᠽᠢᡘ ᠳ᠋ᠡᠬᠢ ᠨᠧᠧᠲ᠋ᠣᠨ᠍ ᠤ ᠬᠣᠶᠠᠳᠤᠭᠠᠷ ᠬᠠᠤᠯᠢ ᠶᠢᠨ ᠠᠳᠠᠯᠢ ᠶᠤᠮ᠃ <br />
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ᠠᠯᠢ ᠠᠯᠢ ᠨᠢ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠷᠢᠯᠠᠯ᠂ ᠢᠮᠫᠦ᠋ᠯᠰ ᠢ <sup>impuls</sup> ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭᠴᠢᠯᠠᠨ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠳᠠᠭ᠃ <br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠨᠢ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ<sup>function</sup> ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠬᠦ ᠳᠦ ᠲᠣᠬᠢᠷᠠᠮᠵᠢ ᠲᠠᠢ ᠤᠴᠢᠷ ᠠᠴᠠ ᠹᠢᠽᠢᠻ ᠦᠨ ᠡᠨᠡ ᠰᠠᠯᠪᠤᠷᠢ ᠳᠤ ᠵᠠᠶ᠋ᠢᠯᠠᠰᠢ ᠦᠭᠡᠢ ᠱᠠᠭᠠᠷᠳᠠᠯᠭᠠ ᠲᠠᠢ ᠶᠤᠮ᠃ <br />
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ᠴᠠᠭᠠᠰᠢᠯᠠᠪᠠᠯ᠂ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠢᠭᠤᠳ ᠨᠥᠯᠥᠭᠡ ᠪᠡᠷ ᠬᠢᠮᠢ ᠶᠢᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠬᠥᠭᠵᠢᠯ ᠡᠷᠴᠢᠮᠵᠢᠭᠰᠡᠨ᠃ <br />
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1927 ᠣᠨ ᠳᠤ ᠸᠠᠯᠲ᠋ᠧᠷ ᠾᠠᠢᠢᠲ᠋ᠯᠧᠷ<sup>Walter Heitler</sup> ᠂ ᠹᠷᠢᠼ ᠯᠣᠨ᠍ᠳᠣᠨ<sup>Fritz London</sup> ᠨᠠᠷ ᠸᠠᠯᠧᠨ᠍ᠲ ᠤᠨ ᠬᠣᠯᠪᠣᠭᠠᠰᠤ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠲᠣᠮᠢᠶᠠᠯᠠᠭᠰᠠᠨ᠃ <br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠨᠢᠭᠡ ᠭᠣᠣᠯ ᠠᠰᠠᠭᠤᠳᠠᠯ ᠪᠣᠯ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ ᠦᠨ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ ᠤᠯᠬᠤ ᠶᠠᠪᠤᠳᠠᠯ ᠶᠤᠮ᠃ <br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠭᠡᠳᠡᠭ ᠨᠢ ᠲᠣᠳᠣᠷᠬᠠᠢ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠵᠤᠷᠪᠤᠰ ᠲᠤ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠵᠤ ᠪᠣᠯᠬᠤ ᠪᠠᠶ᠋ᠢᠷᠢᠰᠢᠯ ᠤᠳ ᠤᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠲᠠᠷᠬᠠᠯᠲᠠ ᠶᠤᠮ᠄ <br />
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<small><math>\frac{-\hbar}{2m}\frac{\partial^2\Psi(x,t)}{\partial x^2} + V(x,t)\Psi(x,t) = i\hbar\frac{\partial \Psi(x,t)}{\partial t}</math></small><br />
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ᠻᠢᠨᠧᠲᠢᠭ<sup>kinetic</sup> ᠧᠨᠧᠷᠭᠢ<sup>energy</sup><br />
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ᠫᠣᠲ᠋ᠧᠨ᠍ᠼᠢᠶᠠᠯ<sup>potencial</sup> ᠧᠨᠧᠷᠭᠢ<br />
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ᠨᠡᠢᠢᠲᠡ ᠧᠨᠧᠷᠭᠢ<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠲᠦ ᠰᠢᠲᠦᠭᠰᠡᠨ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠠᠭᠤᠷᠢ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠨᠢᠭᠡ ᠪᠣᠯ ᠳᠡᠭᠡᠷᠡ ᠳᠤᠷᠠᠳᠤᠭᠰᠠᠨ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡᠬᠦ ᠲᠣᠮᠢᠶᠠ ᠶᠢ ᠠᠰᠢᠭᠯᠠᠭᠰᠠᠨ ᠢᠶ᠋ᠠᠷ ᠤᠰᠤ ᠲᠥᠷᠦᠭᠴᠢ ᠶᠢᠨ ᠮᠣᠯᠧᠻᠤᠯ ᠳᠠᠬᠢ ᠬᠣᠶᠠᠷ ᠠᠲ᠋ᠣᠮ ᠨᠢ ᠻᠣᠸᠠᠯᠧᠨ᠍ᠲ<sup>covalent</sup> ᠬᠣᠯᠪᠣᠭᠠ ᠭᠡᠳᠡᠭ ᠢ ᠡᠭᠦᠰᠬᠡᠨ ᠧᠯᠧᠻᠲ᠋ᠷᠣᠨ ᠨᠤᠭᠤᠳ ᠢᠢᠠᠨ ᠬᠤᠪᠢᠶᠠᠯᠴᠠᠵᠤ ᠪᠠᠶ᠋ᠢᠳᠠᠭ ᠢ ᠪᠠᠲᠤᠯᠠᠭᠰᠠᠨ᠃<ref>Josiah Wu. Real Life Applications of Complex Numbers. 2020 https://issuu.com/harrowhongkong/docs/final_scientific_harrovian_issue_vi-i/s/11488755</ref><br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==</div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%BA%E1%A0%A3%E1%A0%AE%E1%A0%AB%E1%A0%8A%E1%A0%AF%E1%A0%A7%E1%A0%BA%E1%A0%B0_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%A0&diff=2747
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-10-05T06:43:38Z
<p>Angka: Edit</p>
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'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ<sup>complex</sup> ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠯᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃ <br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|ᠺᠣᠮᠫᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠢᠢ ᠲᠥᠯᠦᠭᠡᠯᠡᠵᠦ ᠪᠤᠶ ᠠᠷᠭᠠᠨ᠋ᠳ᠋ ᠤᠨ<sup>Argand</sup> ᠳᠢᠶᠠᠭ᠋ᠷᠠᠮ ᠳᠡᠭᠡᠷ ᠡ᠋ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠸᠧᠺᠲ᠋ᠣᠷ<sup>vector</sup> ᠡᠭᠦᠰᠬᠡᠵᠦ ᠪᠤᠶ ᠬᠣᠣᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ <span style="writing-mode: horizontal-tb;"><i>(a, b)</i></span> ᠳᠦᠷᠰᠦᠯᠡᠨ ᠦᠵᠡᠭᠦᠯᠵᠦ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
ᠡᠭᠦᠨ ᠳ᠋ᠦ: <br><span style="writing-mode: horizontal-tb;"><i>Re</i></span> ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ᠂ <br><span style="writing-mode: horizontal-tb;"><i>Im</i></span> ᠢᠨᠦ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠣᠯᠬᠤ ᠠᠭᠠᠳ <br><span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠢᠨᠦ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠳ᠋ᠦ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠬᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠮᠥᠨ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
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<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
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{| class="wikitable"<br />
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|style = "height: 220px;"|ᠬᠣᠶᠠᠷ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃ <br />
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= ᠬᠠᠰᠤᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠳᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
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<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math><br />
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= ᠦᠷᠡᠵᠢᠬᠦ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃<ref>''Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И.'' Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — <abbr>М.</abbr>: Просвещение, 1998. — 288 с. — <nowiki>ISBN 5-09-008036-4</nowiki>.</ref> <br />
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<math>(a + bi) (c + di) = </math> <math>= ac + bci - bd + adi = </math> <math>= (ac - bd) + (bc + ad)i </math> <br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|<span style="writing-mode: horizontal-tb;"><i>2 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭ ᠠ᠋ ᠢᠢ (ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠬᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>3 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭᠠᠨ ᠳᠤ (ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃<br />
ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠢ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠣᠷᠤᠶ ᠲᠠᠢ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠵᠤ᠂ <span style="writing-mode: horizontal-tb;"><i>√5</i></span> ᠢᠢᠠᠷ ᠰᠤᠩᠭᠠᠨ ᠠ᠋᠃ ᠡᠨᠡ ᠢᠨᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠾᠢᠫᠣᠲ᠋ᠧᠨᠦᠽ ᠤᠨ<sup>hypotenuse</sup> ᠤᠷᠲᠤ ᠪᠣᠯᠠᠢ᠃ <br />
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= ᠬᠤᠪᠢᠶᠠᠬᠤ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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<math>\frac{a+bi}{c+di}=</math> <br />
<math>=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=</math> <br />
<math>=\frac{ac+bd}{c^2+d^2}+\left(\frac{bc-ad}{c^2+d^2}\right)i.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ<sup>quadrate</sup> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨᠡ᠋᠃<ref>Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01</ref><br />
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<math>(x+yi)^2=x^2-y^2 + 2xyi.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> <span style="writing-mode: horizontal-tb;"><i>(b ≠ 0)</i></span> ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ <br />
<math> \pm (\gamma + \delta i)</math> ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄<br />
<math>\gamma = \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠠ <math>\delta = (\sgn b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠣᠯᠤᠮᠤᠢ᠃<br />
<span style="writing-mode: horizontal-tb;"><i>sgn b</i></span> ᠭᠡᠭᠴᠢ ᠢᠨᠦ <math>\sgn b = \begin{cases} \ \ 1, & b > 0 \\ \ \ 0, & b = 0 \\ -1, & b < 0 \end{cases}</math> ᠭᠡᠵᠦ ᠲᠠᠢᠢᠯᠠᠭᠳᠠᠨᠠ᠃ <br />
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ᠡᠭᠦᠨ ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ ᠢᠢ ᠲᠤᠬᠠᠢᠢᠲᠠ ᠳ᠋ᠤ <math> \pm (\gamma + \delta i)</math> ᠲᠣᠭᠠ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠦ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠲᠣᠭᠠ ᠢᠢ ᠭᠠᠷᠭᠠᠵᠤ ᠢᠷᠡᠬᠦ ᠶᠣᠰᠤᠲᠠᠢ᠃ <br />
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=ᠬᠡᠷᠡᠭᠯᠡᠭᠡ=<br />
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== ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ==<br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠳᠠᠪᠲᠠᠯᠲᠠ ᠪᠠᠷ ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠠᠢᠢᠭᠤᠯᠬᠤ᠃<br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠯ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠳ᠋ᠤ ᠡᠭᠦᠰᠬᠡᠳᠡᠭ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ<sup>fractal</sup> ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠵᠡᠰᠱᠢᠶ ᠡ᠋ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡ ᠬᠦ ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ <span style="writing-mode: horizontal-tb;"><i>z = 0</i></span> ᠡᠴᠡ ᠡᠬᠢᠯᠡᠭᠡᠳ ᠢᠲ᠋ᠧᠷᠠᠼ ᠬᠢᠬᠦ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;"><i>f<sub>c</sub>(z)=z<sup>2</sup>+c</i></span> ᠹᠦᠨ᠍ᠻᠼ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠢᠢᠬᠤ ᠨᠥᠬᠦᠴᠡᠯ ᠪᠦᠬᠦᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠤ ᠣᠯᠠᠨᠯᠢᠬ ᠪᠠᠢᠢᠳᠠᠭ᠃ <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ ᠪᠠᠢᠢᠭᠠᠯᠢ ᠢᠢᠨ ᠭᠠᠢᠢᠬᠠᠮᠰᠢᠭᠲᠤ ᠳᠦᠷᠰᠦ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠦᠨ ᠰᠠᠢᠢᠬᠠᠨ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠶᠡᠷᠦ ᠪᠤᠰᠤ ᠢᠢᠨ ᠭᠣᠶᠤᠮᠰᠠᠭ ᠬᠣᠣᠰᠯᠠᠯ ᠢ ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠳᠡᠭ᠃<ref>ᠷᠡᠨᠴᠡᠨ ᠦ ᠡᠩᠬᠡᠪᠠᠲᠤ᠃ '''ᠫᠢᠲ᠋ᠾᠠᠭᠣᠷ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌'''᠃ ᠮᠣᠩᠭᠤᠯ ᠤᠯᠤᠯᠰ ᠊ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠪᠠ ᠲᠣᠭᠠᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠢᠢᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ https://imdt.ac.mn/c/1013874?content=1150891&fbclid=IwAR1HVqeXwT-h7dijj9EeIUzDvFustH99GAr9LHYQIU61XIByuBHCkSm10So 2020</ref> <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠵᠠᠭᠠᠭ ᠢ ᠳᠠᠪᠠᠬᠤ ᠳᠤ ᠪᠠᠨ ᠵᠦᠯᠢᠶ ᠠ᠋ ᠢᠢᠨ<sup>Julia</sup> ᠹᠷᠠᠻᠲᠠᠯ ᠤᠳ ᠦᠷᠭᠡᠯᠵᠢ ᠬᠣᠯᠪᠤᠯᠲᠠ ᠪᠠᠨ ᠠᠯᠳᠠᠵᠤ᠂ ᠹᠠᠲᠣᠤ ᠢᠢᠨ<sup>Fatou</sup> ᠲᠣᠭᠤᠰᠤ ᠪᠣᠯᠤᠨ ᠬᠦᠪᠢᠷᠠᠳᠠᠭ᠃<ref>ᠨᠡᠭᠡᠭᠡᠯᠲᠡᠲᠡᠢ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠳ᠃ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠳᠠᠯ᠃ ᠪᠢᠳᠡᠨ ᠦ ᠡᠷᠭᠢᠨ ᠲᠣᠭᠣᠷᠢᠨ ᠳ᠋ᠠᠬᠢ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠬᠡᠷᠬᠢᠨ ᠠᠵᠢᠯᠯᠠᠳᠠᠭ ᠪᠤᠢ᠃ ᠹᠷᠠᠻᠲᠯᠠ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠶᠢᠨ ᠹᠷᠠᠻᠲᠯᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢ᠌ᠭ᠌᠃ https://ultrait.ru/mn/smartphones/otkrytie-fraktalov-beskonechnost-fraktalov-kak-ustroen-mir.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/10/02</ref><br />
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== ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ==<br />
ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌<sup>quantum mechanics</sup> ᠠᠲ᠋ᠣᠮ ᠠᠴᠠ ᠵᠢᠵᠢᠭ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ᠂ ᠶᠠᠭᠤᠨ ᠤ ᠡᠮᠦᠨᠡ ᠪᠣᠽᠣᠨ<sup>boson</sup> (ᠵᠢᠱᠢᠶᠡ ᠨᠢ ᠹᠣᠲ᠋ᠣᠨ<sup>photon</sup>ᠪᠠ ᠹᠧᠷᠮᠢᠣᠨ ᠤ<sup>fermion</sup> (ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠨᠧᠦᠲ᠋ᠷᠣᠨ) ᠬᠥᠳᠡᠯᠭᠡᠭᠡᠨ᠂ ᠬᠠᠷᠢᠯᠴᠠᠨ ᠦᠢᠯᠡᠴᠢᠯᠡᠯ ᠢ ᠰᠤᠳᠤᠯᠳᠠᠭ᠃ <br />
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ᠴᠢᠩᠭᠢᠬᠦ ᠳᠦ ᠪᠡᠨ ᠲᠡᠳᠡᠭᠡᠷ ᠦᠨ ᠦᠢᠯᠡ ᠬᠥᠳᠡᠯᠦᠯ ᠦᠨ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭ᠌ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠯᠲᠠ ᠶᠢ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠤᠳᠬᠠ ᠪᠠᠷ ᠭᠠᠷᠭᠠᠳᠠᠭ᠃ <br />
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ᠴᠤᠬᠤᠮ ᠳᠠᠭᠠᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠦᠨᠳᠦᠰᠦ ᠰᠠᠭᠤᠷᠢ ᠪᠣᠯᠳᠠᠭ᠃ <br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠲᠤ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ <sup>Schrödinger</sup> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠠᠴᠢ ᠬᠣᠯᠪᠣᠭᠳᠠᠯ ᠨᠢ ᠰᠣᠩᠭᠣᠳᠠᠭ ᠹᠢᠽᠢᡘ ᠳ᠋ᠡᠬᠢ ᠨᠧᠧᠲ᠋ᠣᠨ᠍ ᠤ ᠬᠣᠶᠠᠳᠤᠭᠠᠷ ᠬᠠᠤᠯᠢ ᠶᠢᠨ ᠠᠳᠠᠯᠢ ᠶᠤᠮ᠃ <br />
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ᠠᠯᠢ ᠠᠯᠢ ᠨᠢ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠷᠢᠯᠠᠯ᠂ ᠢᠮᠫᠦ᠋ᠯᠰ ᠢ <sup>impuls</sup> ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭᠴᠢᠯᠠᠨ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠳᠠᠭ᠃ <br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠨᠢ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ<sup>function</sup> ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠬᠦ ᠳᠦ ᠲᠣᠬᠢᠷᠠᠮᠵᠢ ᠲᠠᠢ ᠤᠴᠢᠷ ᠠᠴᠠ ᠹᠢᠽᠢᠻ ᠦᠨ ᠡᠨᠡ ᠰᠠᠯᠪᠤᠷᠢ ᠳᠤ ᠵᠠᠶ᠋ᠢᠯᠠᠰᠢ ᠦᠭᠡᠢ ᠱᠠᠭᠠᠷᠳᠠᠯᠭᠠ ᠲᠠᠢ ᠶᠤᠮ᠃ <br />
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ᠴᠠᠭᠠᠰᠢᠯᠠᠪᠠᠯ᠂ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠢᠭᠤᠳ ᠨᠥᠯᠥᠭᠡ ᠪᠡᠷ ᠬᠢᠮᠢ ᠶᠢᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠬᠥᠭᠵᠢᠯ ᠡᠷᠴᠢᠮᠵᠢᠭᠰᠡᠨ᠃ <br />
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1927 ᠣᠨ ᠳᠤ ᠸᠠᠯᠲ᠋ᠧᠷ ᠾᠠᠢᠢᠲ᠋ᠯᠧᠷ<sup>Walter Heitler</sup> ᠂ ᠹᠷᠢᠼ ᠯᠣᠨ᠍ᠳᠣᠨ<sup>Fritz London</sup> ᠨᠠᠷ ᠸᠠᠯᠧᠨ᠍ᠲ ᠤᠨ ᠬᠣᠯᠪᠣᠭᠠᠰᠤ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠲᠣᠮᠢᠶᠠᠯᠠᠭᠰᠠᠨ᠃ <br />
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ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠨᠢᠭᠡ ᠭᠣᠣᠯ ᠠᠰᠠᠭᠤᠳᠠᠯ ᠪᠣᠯ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ ᠦᠨ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ ᠤᠯᠬᠤ ᠶᠠᠪᠤᠳᠠᠯ ᠶᠤᠮ᠃ <br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠭᠡᠳᠡᠭ ᠨᠢ ᠲᠣᠳᠣᠷᠬᠠᠢ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠵᠤᠷᠪᠤᠰ ᠲᠤ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠵᠤ ᠪᠣᠯᠬᠤ ᠪᠠᠶ᠋ᠢᠷᠢᠰᠢᠯ ᠤᠳ ᠤᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠲᠠᠷᠬᠠᠯᠲᠠ ᠶᠤᠮ᠄ <br />
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<small><math>\frac{-\hbar}{2m}\frac{\partial^2\Psi(x,t)}{\partial x^2} + V(x,t)\Psi(x,t) = i\hbar\frac{\partial \Psi(x,t)}{\partial t}</math></small><br />
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ᠻᠢᠨᠧᠲᠢᠭ<sup>kinetic</sup> ᠧᠨᠧᠷᠭᠢ<sup>energy</sup><br />
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ᠫᠥᠲ᠋ᠧᠨ᠍ᠼᠢᠶᠠᠯ<sup>potencial</sup> ᠧᠨᠧᠷᠭᠢ<br />
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ᠨᠡᠢᠢᠲᠡ ᠧᠨᠧᠷᠭᠢ<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠲᠦ ᠰᠢᠲᠦᠭᠰᠡᠨ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠠᠭᠤᠷᠢ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠨᠢᠭᠡ ᠪᠣᠯ ᠳᠡᠭᠡᠷᠡ ᠳᠤᠷᠠᠳᠤᠭᠰᠠᠨ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡᠬᠦ ᠲᠣᠮᠢᠶᠠ ᠶᠢ ᠠᠰᠢᠭᠯᠠᠭᠰᠠᠨ ᠢᠶ᠋ᠠᠷ ᠤᠰᠤ ᠲᠥᠷᠦᠭᠴᠢ ᠶᠢᠨ ᠮᠣᠯᠧᠻᠤᠯ ᠳᠠᠬᠢ ᠬᠣᠶᠠᠷ ᠠᠲ᠋ᠣᠮ ᠨᠢ ᠻᠣᠸᠠᠯᠧᠨ᠍ᠲ<sup>covalent</sup> ᠬᠣᠯᠪᠣᠭᠠ ᠭᠡᠳᠡᠭ ᠢ ᠡᠭᠦᠰᠬᠡᠨ ᠧᠯᠧᠻᠲ᠋ᠷᠣᠨ ᠨᠤᠭᠤᠳ ᠢᠢᠠᠨ ᠬᠤᠪᠢᠶᠠᠯᠴᠠᠵᠤ ᠪᠠᠶ᠋ᠢᠳᠠᠭ ᠢ ᠪᠠᠲᠤᠯᠠᠭᠰᠠᠨ᠃<ref>Josiah Wu. Real Life Applications of Complex Numbers. 2020 https://issuu.com/harrowhongkong/docs/final_scientific_harrovian_issue_vi-i/s/11488755</ref><br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==</div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%BA%E1%A0%A3%E1%A0%AE%E1%A0%AB%E1%A0%8A%E1%A0%AF%E1%A0%A7%E1%A0%BA%E1%A0%B0_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%A0&diff=2746
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-10-05T06:16:32Z
<p>Angka: Edit</p>
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'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ<sup>complex</sup> ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠯᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃ <br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|ᠺᠣᠮᠫᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠢᠢ ᠲᠥᠯᠦᠭᠡᠯᠡᠵᠦ ᠪᠤᠶ ᠠᠷᠭᠠᠨ᠋ᠳ᠋ ᠤᠨ<sup>Argand</sup> ᠳᠢᠶᠠᠭ᠋ᠷᠠᠮ ᠳᠡᠭᠡᠷ ᠡ᠋ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠸᠧᠺᠲ᠋ᠣᠷ<sup>vector</sup> ᠡᠭᠦᠰᠬᠡᠵᠦ ᠪᠤᠶ ᠬᠣᠣᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ <span style="writing-mode: horizontal-tb;"><i>(a, b)</i></span> ᠳᠦᠷᠰᠦᠯᠡᠨ ᠦᠵᠡᠭᠦᠯᠵᠦ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
ᠡᠭᠦᠨ ᠳ᠋ᠦ: <br><span style="writing-mode: horizontal-tb;"><i>Re</i></span> ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ᠂ <br><span style="writing-mode: horizontal-tb;"><i>Im</i></span> ᠢᠨᠦ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠣᠯᠬᠤ ᠠᠭᠠᠳ <br><span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠢᠨᠦ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠳ᠋ᠦ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠬᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠮᠥᠨ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
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<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
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{| class="wikitable"<br />
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|style = "height: 220px;"|ᠬᠣᠶᠠᠷ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃ <br />
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= ᠬᠠᠰᠤᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠳᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
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<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math><br />
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= ᠦᠷᠡᠵᠢᠬᠦ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃<ref>''Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И.'' Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — <abbr>М.</abbr>: Просвещение, 1998. — 288 с. — <nowiki>ISBN 5-09-008036-4</nowiki>.</ref> <br />
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<math>(a + bi) (c + di) = </math> <math>= ac + bci - bd + adi = </math> <math>= (ac - bd) + (bc + ad)i </math> <br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|<span style="writing-mode: horizontal-tb;"><i>2 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭ ᠠ᠋ ᠢᠢ (ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠬᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>3 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭᠠᠨ ᠳᠤ (ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃<br />
ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠢ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠣᠷᠤᠶ ᠲᠠᠢ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠵᠤ᠂ <span style="writing-mode: horizontal-tb;"><i>√5</i></span> ᠢᠢᠠᠷ ᠰᠤᠩᠭᠠᠨ ᠠ᠋᠃ ᠡᠨᠡ ᠢᠨᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠾᠢᠫᠣᠲ᠋ᠧᠨᠦᠽ ᠤᠨ<sup>hypotenuse</sup> ᠤᠷᠲᠤ ᠪᠣᠯᠠᠢ᠃ <br />
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= ᠬᠤᠪᠢᠶᠠᠬᠤ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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<math>\frac{a+bi}{c+di}=</math> <br />
<math>=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=</math> <br />
<math>=\frac{ac+bd}{c^2+d^2}+\left(\frac{bc-ad}{c^2+d^2}\right)i.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ<sup>quadrate</sup> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨᠡ᠋᠃<ref>Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01</ref><br />
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<math>(x+yi)^2=x^2-y^2 + 2xyi.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> <span style="writing-mode: horizontal-tb;"><i>(b ≠ 0)</i></span> ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ <br />
<math> \pm (\gamma + \delta i)</math> ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄<br />
<math>\gamma = \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠠ <math>\delta = (\sgn b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠣᠯᠤᠮᠤᠢ᠃<br />
<span style="writing-mode: horizontal-tb;"><i>sgn b</i></span> ᠭᠡᠭᠴᠢ ᠢᠨᠦ <math>\sgn b = \begin{cases} \ \ 1, & b > 0 \\ \ \ 0, & b = 0 \\ -1, & b < 0 \end{cases}</math> ᠭᠡᠵᠦ ᠲᠠᠢᠢᠯᠠᠭᠳᠠᠨᠠ᠃ <br />
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ᠡᠭᠦᠨ ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ ᠢᠢ ᠲᠤᠬᠠᠢᠢᠲᠠ ᠳ᠋ᠤ <math> \pm (\gamma + \delta i)</math> ᠲᠣᠭᠠ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠦ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠲᠣᠭᠠ ᠢᠢ ᠭᠠᠷᠭᠠᠵᠤ ᠢᠷᠡᠬᠦ ᠶᠣᠰᠤᠲᠠᠢ᠃ <br />
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=ᠬᠡᠷᠡᠭᠯᠡᠭᠡ=<br />
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== ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ==<br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠳᠠᠪᠲᠠᠯᠲᠠ ᠪᠠᠷ ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠠᠢᠢᠭᠤᠯᠬᠤ᠃<br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠯ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠳ᠋ᠤ ᠡᠭᠦᠰᠬᠡᠳᠡᠭ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ<sup>fractal</sup> ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠵᠡᠰᠱᠢᠶ ᠡ᠋ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡ ᠬᠦ ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ <span style="writing-mode: horizontal-tb;"><i>z = 0</i></span> ᠡᠴᠡ ᠡᠬᠢᠯᠡᠭᠡᠳ ᠢᠲ᠋ᠧᠷᠠᠼ ᠬᠢᠬᠦ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;"><i>f<sub>c</sub>(z)=z<sup>2</sup>+c</i></span> ᠹᠦᠨ᠍ᠻᠼ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠢᠢᠬᠤ ᠨᠥᠬᠦᠴᠡᠯ ᠪᠦᠬᠦᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠤ ᠣᠯᠠᠨᠯᠢᠬ ᠪᠠᠢᠢᠳᠠᠭ᠃ <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ ᠪᠠᠢᠢᠭᠠᠯᠢ ᠢᠢᠨ ᠭᠠᠢᠢᠬᠠᠮᠰᠢᠭᠲᠤ ᠳᠦᠷᠰᠦ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠦᠨ ᠰᠠᠢᠢᠬᠠᠨ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠶᠡᠷᠦ ᠪᠤᠰᠤ ᠢᠢᠨ ᠭᠣᠶᠤᠮᠰᠠᠭ ᠬᠣᠣᠰᠯᠠᠯ ᠢ ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠳᠡᠭ᠃<ref>ᠷᠡᠨᠴᠡᠨ ᠦ ᠡᠩᠬᠡᠪᠠᠲᠤ᠃ '''ᠫᠢᠲ᠋ᠾᠠᠭᠣᠷ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌'''᠃ ᠮᠣᠩᠭᠤᠯ ᠤᠯᠤᠯᠰ ᠊ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠪᠠ ᠲᠣᠭᠠᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠢᠢᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ https://imdt.ac.mn/c/1013874?content=1150891&fbclid=IwAR1HVqeXwT-h7dijj9EeIUzDvFustH99GAr9LHYQIU61XIByuBHCkSm10So 2020</ref> <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠵᠠᠭᠠᠭ ᠢ ᠳᠠᠪᠠᠬᠤ ᠳᠤ ᠪᠠᠨ ᠵᠦᠯᠢᠶ ᠠ᠋ ᠢᠢᠨ<sup>Julia</sup> ᠹᠷᠠᠻᠲᠠᠯ ᠤᠳ ᠦᠷᠭᠡᠯᠵᠢ ᠬᠣᠯᠪᠤᠯᠲᠠ ᠪᠠᠨ ᠠᠯᠳᠠᠵᠤ᠂ ᠹᠠᠲᠣᠤ ᠢᠢᠨ<sup>Fatou</sup> ᠲᠣᠭᠤᠰᠤ ᠪᠣᠯᠤᠨ ᠬᠦᠪᠢᠷᠠᠳᠠᠭ᠃<ref>ᠨᠡᠭᠡᠭᠡᠯᠲᠡᠲᠡᠢ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠳ᠃ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠳᠠᠯ᠃ ᠪᠢᠳᠡᠨ ᠦ ᠡᠷᠭᠢᠨ ᠲᠣᠭᠣᠷᠢᠨ ᠳ᠋ᠠᠬᠢ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠬᠡᠷᠬᠢᠨ ᠠᠵᠢᠯᠯᠠᠳᠠᠭ ᠪᠤᠢ᠃ ᠹᠷᠠᠻᠲᠯᠠ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠶᠢᠨ ᠹᠷᠠᠻᠲᠯᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢ᠌ᠭ᠌᠃ https://ultrait.ru/mn/smartphones/otkrytie-fraktalov-beskonechnost-fraktalov-kak-ustroen-mir.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/10/02</ref><br />
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== ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ==<br />
ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌<sup>quantum mechanics</sup> ᠠᠲ᠋ᠣᠮ ᠠᠴᠠ ᠵᠢᠵᠢᠭ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ᠂ ᠶᠠᠭᠤᠨ ᠤ ᠡᠮᠦᠨᠡ ᠪᠣᠽᠣᠨ<sup>boson</sup> (ᠵᠢᠱᠢᠶᠡ ᠨᠢ ᠹᠣᠲ᠋ᠣᠨ<sup>photon</sup>ᠪᠠ ᠹᠧᠷᠮᠢᠣᠨ ᠤ<sup>fermion</sup> (ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠨᠧᠦᠲ᠋ᠷᠣᠨ) ᠬᠥᠳᠡᠯᠭᠡᠭᠡᠨ᠂ ᠬᠠᠷᠢᠯᠴᠠᠨ ᠦᠢᠯᠡᠴᠢᠯᠡᠯ ᠢ ᠰᠤᠳᠤᠯᠳᠠᠭ᠃ ᠴᠢᠩᠭᠢᠬᠦ ᠳᠦ ᠪᠡᠨ ᠲᠡᠳᠡᠭᠡᠷ ᠦᠨ ᠦᠢᠯᠡ ᠬᠥᠳᠡᠯᠦᠯ ᠦᠨ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭ᠌ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠯᠲᠠ ᠶᠢ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠤᠳᠬᠠ ᠪᠠᠷ ᠭᠠᠷᠭᠠᠳᠠᠭ᠃ ᠴᠤᠬᠤᠮ ᠳᠠᠭᠠᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠦᠨᠳᠦᠰᠦ ᠰᠠᠭᠤᠷᠢ ᠪᠣᠯᠳᠠᠭ᠃ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠲᠤ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ <sup>Schrödinger</sup> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠠᠴᠢ ᠬᠣᠯᠪᠣᠭᠳᠠᠯ ᠨᠢ ᠰᠣᠩᠭᠣᠳᠠᠭ ᠹᠢᠽᠢᡘ ᠳ᠋ᠡᠬᠢ ᠨᠧᠧᠲ᠋ᠣᠨ᠍ ᠤ ᠬᠣᠶᠠᠳᠤᠭᠠᠷ ᠬᠠᠤᠯᠢ ᠶᠢᠨ ᠠᠳᠠᠯᠢ ᠶᠤᠮ᠃ ᠠᠯᠢ ᠠᠯᠢ ᠨᠢ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠷᠢᠯᠠᠯ᠂ ᠢᠮᠫᠦ᠋ᠯᠰ ᠢ <sup>impuls</sup> ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭᠴᠢᠯᠠᠨ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠳᠠᠭ᠃ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠨᠢ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ<sup>function</sup> ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠬᠦ ᠳᠦ ᠲᠣᠬᠢᠷᠠᠮᠵᠢ ᠲᠠᠢ ᠤᠴᠢᠷ ᠠᠴᠠ ᠹᠢᠽᠢᠻ ᠦᠨ ᠡᠨᠡ ᠰᠠᠯᠪᠤᠷᠢ ᠳᠤ ᠵᠠᠶ᠋ᠢᠯᠠᠰᠢ ᠦᠭᠡᠢ ᠱᠠᠭᠠᠷᠳᠠᠯᠭᠠ ᠲᠠᠢ ᠶᠤᠮ᠃ ᠴᠠᠭᠠᠰᠢᠯᠠᠪᠠᠯ᠂ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠢᠭᠤᠳ ᠨᠥᠯᠥᠭᠡ ᠪᠡᠷ ᠬᠢᠮᠢ ᠶᠢᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠬᠥᠭᠵᠢᠯ ᠡᠷᠴᠢᠮᠵᠢᠭᠰᠡᠨ᠃ 1927 ᠣᠨ ᠳᠤ ᠸᠠᠯᠲ᠋ᠧᠷ ᠾᠠᠢᠢᠲ᠋ᠯᠧᠷ<sup>Walter Heitler</sup> ᠂ ᠹᠷᠢᠼ ᠯᠣᠨ᠍ᠳᠣᠨ<sup>Fritz London</sup> ᠨᠠᠷ ᠸᠠᠯᠧᠨ᠍ᠲ ᠤᠨ ᠬᠣᠯᠪᠣᠭᠠᠰᠤ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠲᠣᠮᠢᠶᠠᠯᠠᠭᠰᠠᠨ᠃ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠨᠢᠭᠡ ᠭᠣᠣᠯ ᠠᠰᠠᠭᠤᠳᠠᠯ ᠪᠣᠯ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ ᠦᠨ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ ᠤᠯᠬᠤ ᠶᠠᠪᠤᠳᠠᠯ ᠶᠤᠮ᠃ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠭᠡᠳᠡᠭ ᠨᠢ ᠲᠣᠳᠣᠷᠬᠠᠢ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠵᠤᠷᠪᠤᠰ ᠲᠤ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠵᠤ ᠪᠣᠯᠬᠤ ᠪᠠᠶ᠋ᠢᠷᠢᠰᠢᠯ ᠤᠳ ᠤᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠲᠠᠷᠬᠠᠯᠲᠠ ᠶᠤᠮ᠄ <br />
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<math>\frac{-\hbar}{2m}\frac{\partial^2\Psi(x,t)}{\partial x^2} + V(x,t)\Psi(x,t) = i\hbar\frac{\partial \Psi(x,t)}{\partial t}</math> <br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠲᠦ ᠰᠢᠲᠦᠭᠰᠡᠨ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠠᠭᠤᠷᠢ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠨᠢᠭᠡ ᠪᠣᠯ ᠳᠡᠭᠡᠷᠡ ᠳᠤᠷᠠᠳᠤᠭᠰᠠᠨ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡᠬᠦ ᠲᠣᠮᠢᠶᠠ ᠶᠢ ᠠᠰᠢᠭᠯᠠᠭᠰᠠᠨ ᠢᠶ᠋ᠠᠷ ᠤᠰᠤ ᠲᠥᠷᠦᠭᠴᠢ ᠶᠢᠨ ᠮᠣᠯᠧᠻᠤᠯ ᠳᠠᠬᠢ ᠬᠣᠶᠠᠷ ᠠᠲ᠋ᠣᠮ ᠨᠢ ᠻᠣᠸᠠᠯᠧᠨ᠍ᠲ<sup>covalent</sup> ᠬᠣᠯᠪᠣᠭᠠ ᠭᠡᠳᠡᠭ ᠢ ᠡᠭᠦᠰᠬᠡᠨ ᠧᠯᠧᠻᠲ᠋ᠷᠣᠨ ᠨᠤᠭᠤᠳ ᠢᠢᠠᠨ ᠬᠤᠪᠢᠶᠠᠯᠴᠠᠵᠤ ᠪᠠᠶ᠋ᠢᠳᠠᠭ ᠢ ᠪᠠᠲᠤᠯᠠᠭᠰᠠᠨ᠃<ref>Josiah Wu. Real Life Applications of Complex Numbers. 2020<nowiki/>https://issuu.com/harrowhongkong/docs/final_scientific_harrovian_issue_vi-i/s/11488755</ref><br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==</div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%BA%E1%A0%A3%E1%A0%AE%E1%A0%AB%E1%A0%8A%E1%A0%AF%E1%A0%A7%E1%A0%BA%E1%A0%B0_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%A0&diff=2745
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-10-05T05:57:07Z
<p>Angka: Edit</p>
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'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ<sup>complex</sup> ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠯᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃ <br />
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|style = "height: 225px;"|ᠺᠣᠮᠫᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠢᠢ ᠲᠥᠯᠦᠭᠡᠯᠡᠵᠦ ᠪᠤᠶ ᠠᠷᠭᠠᠨ᠋ᠳ᠋ ᠤᠨ<sup>Argand</sup> ᠳᠢᠶᠠᠭ᠋ᠷᠠᠮ ᠳᠡᠭᠡᠷ ᠡ᠋ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠸᠧᠺᠲ᠋ᠣᠷ<sup>vector</sup> ᠡᠭᠦᠰᠬᠡᠵᠦ ᠪᠤᠶ ᠬᠣᠣᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ <span style="writing-mode: horizontal-tb;"><i>(a, b)</i></span> ᠳᠦᠷᠰᠦᠯᠡᠨ ᠦᠵᠡᠭᠦᠯᠵᠦ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
ᠡᠭᠦᠨ ᠳ᠋ᠦ: <br><span style="writing-mode: horizontal-tb;"><i>Re</i></span> ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ᠂ <br><span style="writing-mode: horizontal-tb;"><i>Im</i></span> ᠢᠨᠦ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠣᠯᠬᠤ ᠠᠭᠠᠳ <br><span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠢᠨᠦ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠳ᠋ᠦ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠬᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠮᠥᠨ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
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<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
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{| class="wikitable"<br />
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|style = "height: 220px;"|ᠬᠣᠶᠠᠷ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃ <br />
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= ᠬᠠᠰᠤᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠳᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
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<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math><br />
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= ᠦᠷᠡᠵᠢᠬᠦ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃<ref>''Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И.'' Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — <abbr>М.</abbr>: Просвещение, 1998. — 288 с. — <nowiki>ISBN 5-09-008036-4</nowiki>.</ref> <br />
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<math>(a + bi) (c + di) = </math> <math>= ac + bci - bd + adi = </math> <math>= (ac - bd) + (bc + ad)i </math> <br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|<span style="writing-mode: horizontal-tb;"><i>2 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭ ᠠ᠋ ᠢᠢ (ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠬᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>3 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭᠠᠨ ᠳᠤ (ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃<br />
ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠢ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠣᠷᠤᠶ ᠲᠠᠢ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠵᠤ᠂ <span style="writing-mode: horizontal-tb;"><i>√5</i></span> ᠢᠢᠠᠷ ᠰᠤᠩᠭᠠᠨ ᠠ᠋᠃ ᠡᠨᠡ ᠢᠨᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠾᠢᠫᠣᠲ᠋ᠧᠨᠦᠽ ᠤᠨ<sup>hypotenuse</sup> ᠤᠷᠲᠤ ᠪᠣᠯᠠᠢ᠃ <br />
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= ᠬᠤᠪᠢᠶᠠᠬᠤ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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<math>\frac{a+bi}{c+di}=</math> <br />
<math>=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=</math> <br />
<math>=\frac{ac+bd}{c^2+d^2}+\left(\frac{bc-ad}{c^2+d^2}\right)i.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ<sup>quadrate</sup> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨᠡ᠋᠃<ref>Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01</ref><br />
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<math>(x+yi)^2=x^2-y^2 + 2xyi.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> <span style="writing-mode: horizontal-tb;"><i>(b ≠ 0)</i></span> ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ <br />
<math> \pm (\gamma + \delta i)</math> ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄<br />
<math>\gamma = \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠠ <math>\delta = (\sgn b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠣᠯᠤᠮᠤᠢ᠃<br />
<span style="writing-mode: horizontal-tb;"><i>sgn b</i></span> ᠭᠡᠭᠴᠢ ᠢᠨᠦ <math>\sgn b = \begin{cases} \ \ 1, & b > 0 \\ \ \ 0, & b = 0 \\ -1, & b < 0 \end{cases}</math> ᠭᠡᠵᠦ ᠲᠠᠢᠢᠯᠠᠭᠳᠠᠨᠠ᠃ <br />
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ᠡᠭᠦᠨ ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ ᠢᠢ ᠲᠤᠬᠠᠢᠢᠲᠠ ᠳ᠋ᠤ <math> \pm (\gamma + \delta i)</math> ᠲᠣᠭᠠ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠦ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠲᠣᠭᠠ ᠢᠢ ᠭᠠᠷᠭᠠᠵᠤ ᠢᠷᠡᠬᠦ ᠶᠣᠰᠤᠲᠠᠢ᠃ <br />
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=ᠬᠡᠷᠡᠭᠯᠡᠭᠡ=<br />
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== ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ==<br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠳᠠᠪᠲᠠᠯᠲᠠ ᠪᠠᠷ ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠠᠢᠢᠭᠤᠯᠬᠤ᠃<br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠯ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠳ᠋ᠤ ᠡᠭᠦᠰᠬᠡᠳᠡᠭ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ<sup>fractal</sup> ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠵᠡᠰᠱᠢᠶ ᠡ᠋ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡ ᠬᠦ ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ <span style="writing-mode: horizontal-tb;"><i>z = 0</i></span> ᠡᠴᠡ ᠡᠬᠢᠯᠡᠭᠡᠳ ᠢᠲ᠋ᠧᠷᠠᠼ ᠬᠢᠬᠦ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;"><i>f<sub>c</sub>(z)=z<sup>2</sup>+c</i></span> ᠹᠦᠨ᠍ᠻᠼ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠢᠢᠬᠤ ᠨᠥᠬᠦᠴᠡᠯ ᠪᠦᠬᠦᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠤ ᠣᠯᠠᠨᠯᠢᠬ ᠪᠠᠢᠢᠳᠠᠭ᠃ <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ ᠪᠠᠢᠢᠭᠠᠯᠢ ᠢᠢᠨ ᠭᠠᠢᠢᠬᠠᠮᠰᠢᠭᠲᠤ ᠳᠦᠷᠰᠦ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠦᠨ ᠰᠠᠢᠢᠬᠠᠨ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠶᠡᠷᠦ ᠪᠤᠰᠤ ᠢᠢᠨ ᠭᠣᠶᠤᠮᠰᠠᠭ ᠬᠣᠣᠰᠯᠠᠯ ᠢ ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠳᠡᠭ᠃<ref>ᠷᠡᠨᠴᠡᠨ ᠦ ᠡᠩᠬᠡᠪᠠᠲᠤ᠃ '''ᠫᠢᠲ᠋ᠾᠠᠭᠣᠷ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌'''᠃ ᠮᠣᠩᠭᠤᠯ ᠤᠯᠤᠯᠰ ᠊ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠪᠠ ᠲᠣᠭᠠᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠢᠢᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ https://imdt.ac.mn/c/1013874?content=1150891&fbclid=IwAR1HVqeXwT-h7dijj9EeIUzDvFustH99GAr9LHYQIU61XIByuBHCkSm10So 2020</ref> <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠵᠠᠭᠠᠭ ᠢ ᠳᠠᠪᠠᠬᠤ ᠳᠤ ᠪᠠᠨ ᠵᠦᠯᠢᠶ ᠠ᠋ ᠢᠢᠨ<sup>Julia</sup> ᠹᠷᠠᠻᠲᠠᠯ ᠤᠳ ᠦᠷᠭᠡᠯᠵᠢ ᠬᠣᠯᠪᠤᠯᠲᠠ ᠪᠠᠨ ᠠᠯᠳᠠᠵᠤ᠂ ᠹᠠᠲᠣᠤ ᠢᠢᠨ<sup>Fatou</sup> ᠲᠣᠭᠤᠰᠤ ᠪᠣᠯᠤᠨ ᠬᠦᠪᠢᠷᠠᠳᠠᠭ᠃<ref>ᠨᠡᠭᠡᠭᠡᠯᠲᠡᠲᠡᠢ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠳ᠃ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠳᠠᠯ᠃ ᠪᠢᠳᠡᠨ ᠦ ᠡᠷᠭᠢᠨ ᠲᠣᠭᠣᠷᠢᠨ ᠳ᠋ᠠᠬᠢ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠬᠡᠷᠬᠢᠨ ᠠᠵᠢᠯᠯᠠᠳᠠᠭ ᠪᠤᠢ᠃ ᠹᠷᠠᠻᠲᠯᠠ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠶᠢᠨ ᠹᠷᠠᠻᠲᠯᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢ᠌ᠭ᠌᠃ https://ultrait.ru/mn/smartphones/otkrytie-fraktalov-beskonechnost-fraktalov-kak-ustroen-mir.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/10/02</ref><br />
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== ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ==<br />
ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌<sup>quantum mechanics</sup> ᠠᠲ᠋ᠣᠮ ᠠᠴᠠ ᠵᠢᠵᠢᠭ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ᠂ ᠶᠠᠭᠤᠨ ᠤ ᠡᠮᠦᠨᠡ ᠪᠣᠽᠣᠨ<sup>boson</sup> (ᠵᠢᠱᠢᠶᠡ ᠨᠢ ᠹᠣᠲ᠋ᠣᠨ<sup>photon</sup>ᠪᠠ ᠹᠧᠷᠮᠢᠣᠨ ᠤ<sup>fermion</sup> (ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠨᠧᠦᠲ᠋ᠷᠣᠨ) ᠬᠥᠳᠡᠯᠭᠡᠭᠡᠨ᠂ ᠬᠠᠷᠢᠯᠴᠠᠨ ᠦᠢᠯᠡᠴᠢᠯᠡᠯ ᠢ ᠰᠤᠳᠤᠯᠳᠠᠭ᠃ ᠴᠢᠩᠭᠢᠬᠦ ᠳᠦ ᠪᠡᠨ ᠲᠡᠳᠡᠭᠡᠷ ᠦᠨ ᠦᠢᠯᠡ ᠬᠥᠳᠡᠯᠦᠯ ᠦᠨ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭ᠌ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠯᠲᠠ ᠶᠢ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠤᠳᠬᠠ ᠪᠠᠷ ᠭᠠᠷᠭᠠᠳᠠᠭ᠃ ᠴᠤᠬᠤᠮ ᠳᠠᠭᠠᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠦᠨᠳᠦᠰᠦ ᠰᠠᠭᠤᠷᠢ ᠪᠣᠯᠳᠠᠭ᠃ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠲᠤ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ <sup>Schrödinger</sup> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠠᠴᠢ ᠬᠣᠯᠪᠣᠭᠳᠠᠯ ᠨᠢ ᠰᠣᠩᠭᠣᠳᠠᠭ ᠹᠢᠽᠢᡘ ᠳ᠋ᠡᠬᠢ ᠨᠧᠧᠲ᠋ᠣᠨ᠍ ᠤ ᠬᠣᠶᠠᠳᠤᠭᠠᠷ ᠬᠠᠤᠯᠢ ᠶᠢᠨ ᠠᠳᠠᠯᠢ ᠶᠤᠮ᠃ ᠠᠯᠢ ᠠᠯᠢ ᠨᠢ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠷᠢᠯᠠᠯ᠂ ᠢᠮᠫᠦ᠋ᠯᠰ ᠢ <sup>impuls</sup> ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭᠴᠢᠯᠠᠨ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠳᠠᠭ᠃ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠨᠢ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ<sup>function</sup> ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠬᠦ ᠳᠦ ᠲᠣᠬᠢᠷᠠᠮᠵᠢ ᠲᠠᠢ ᠤᠴᠢᠷ ᠠᠴᠠ ᠹᠢᠽᠢᠻ ᠦᠨ ᠡᠨᠡ ᠰᠠᠯᠪᠤᠷᠢ ᠳᠤ ᠵᠠᠶ᠋ᠢᠯᠠᠰᠢ ᠦᠭᠡᠢ ᠱᠠᠭᠠᠷᠳᠠᠯᠭᠠ ᠲᠠᠢ ᠶᠤᠮ᠃ ᠴᠠᠭᠠᠰᠢᠯᠠᠪᠠᠯ᠂ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠢᠭᠤᠳ ᠨᠥᠯᠥᠭᠡ ᠪᠡᠷ ᠬᠢᠮᠢ ᠶᠢᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠬᠥᠭᠵᠢᠯ ᠡᠷᠴᠢᠮᠵᠢᠭᠰᠡᠨ᠃ 1927 ᠣᠨ ᠳᠤ ᠸᠠᠯᠲ᠋ᠧᠷ ᠾᠠᠢᠢᠲ᠋ᠯᠧᠷ<sup>Walter Heitler</sup> ᠂ ᠹᠷᠢᠼ ᠯᠣᠨ᠍ᠳᠣᠨ<sup>Fritz London</sup> ᠨᠠᠷ ᠸᠠᠯᠧᠨ᠍ᠲ ᠤᠨ ᠬᠣᠯᠪᠣᠭᠠᠰᠤ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠲᠣᠮᠢᠶᠠᠯᠠᠭᠰᠠᠨ᠃ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠨᠢᠭᠡ ᠭᠣᠣᠯ ᠠᠰᠠᠭᠤᠳᠠᠯ ᠪᠣᠯ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ ᠦᠨ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢ ᠤᠯᠬᠤ ᠶᠠᠪᠤᠳᠠᠯ ᠶᠤᠮ᠃ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠭᠡᠳᠡᠭ ᠨᠢ ᠲᠣᠳᠣᠷᠬᠠᠢ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠵᠤᠷᠪᠤᠰ ᠲᠤ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠵᠤ ᠪᠣᠯᠬᠤ ᠪᠠᠶ᠋ᠢᠷᠢᠰᠢᠯ ᠤᠳ ᠤᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠮᠠᠭᠠᠳᠯᠠᠯ ᠤᠨ ᠲᠠᠷᠬᠠᠯᠲᠠ ᠶᠤᠮ. ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠲᠦ ᠰᠢᠲᠦᠭᠰᠡᠨ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠠᠭᠤᠷᠢ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠨᠢᠭᠡ ᠪᠣᠯ ᠳᠡᠭᠡᠷᠡ ᠳᠤᠷᠠᠳᠤᠭᠰᠠᠨ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡᠬᠦ ᠲᠣᠮᠢᠶᠠ ᠶᠢ ᠠᠰᠢᠭᠯᠠᠭᠰᠠᠨ ᠢᠶ᠋ᠠᠷ ᠤᠰᠤ ᠲᠥᠷᠦᠭᠴᠢ ᠶᠢᠨ ᠮᠣᠯᠧᠻᠤᠯ ᠳᠠᠬᠢ ᠬᠣᠶᠠᠷ ᠠᠲ᠋ᠣᠮ ᠨᠢ ᠻᠣᠸᠠᠯᠧᠨ᠍ᠲ<sup>covalent</sup> ᠬᠣᠯᠪᠣᠭᠠ ᠭᠡᠳᠡᠭ ᠢ ᠡᠭᠦᠰᠬᠡᠨ ᠧᠯᠧᠻᠲ᠋ᠷᠣᠨ ᠨᠤᠭᠤᠳ ᠢᠢᠠᠨ ᠬᠤᠪᠢᠶᠠᠯᠴᠠᠵᠤ ᠪᠠᠶ᠋ᠢᠳᠠᠭ ᠢ ᠪᠠᠲᠤᠯᠠᠭᠰᠠᠨ᠃<ref>Josiah Wu. Real Life Applications of Complex Numbers. 2020<nowiki/>https://issuu.com/harrowhongkong/docs/final_scientific_harrovian_issue_vi-i/s/11488755</ref><br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==</div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%BA%E1%A0%A3%E1%A0%AE%E1%A0%AB%E1%A0%8A%E1%A0%AF%E1%A0%A7%E1%A0%BA%E1%A0%B0_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%A0&diff=2744
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-10-03T14:43:26Z
<p>Angka: Images</p>
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'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ<sup>complex</sup> ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠯᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃ <br />
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|style = "height: 225px;"|ᠺᠣᠮᠫᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠢᠢ ᠲᠥᠯᠦᠭᠡᠯᠡᠵᠦ ᠪᠤᠶ ᠠᠷᠭᠠᠨ᠋ᠳ᠋ ᠤᠨ<sup>Argand</sup> ᠳᠢᠶᠠᠭ᠋ᠷᠠᠮ ᠳᠡᠭᠡᠷ ᠡ᠋ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠸᠧᠺᠲ᠋ᠣᠷ<sup>vector</sup> ᠡᠭᠦᠰᠬᠡᠵᠦ ᠪᠤᠶ ᠬᠣᠣᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ <span style="writing-mode: horizontal-tb;"><i>(a, b)</i></span> ᠳᠦᠷᠰᠦᠯᠡᠨ ᠦᠵᠡᠭᠦᠯᠵᠦ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
ᠡᠭᠦᠨ ᠳ᠋ᠦ: <br><span style="writing-mode: horizontal-tb;"><i>Re</i></span> ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ᠂ <br><span style="writing-mode: horizontal-tb;"><i>Im</i></span> ᠢᠨᠦ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠣᠯᠬᠤ ᠠᠭᠠᠳ <br><span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠢᠨᠦ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠳ᠋ᠦ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠬᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠮᠥᠨ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
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<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
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{| class="wikitable"<br />
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|style = "height: 220px;"|ᠬᠣᠶᠠᠷ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃ <br />
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= ᠬᠠᠰᠤᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠳᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
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<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math><br />
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= ᠦᠷᠡᠵᠢᠬᠦ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃<ref>''Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И.'' Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — <abbr>М.</abbr>: Просвещение, 1998. — 288 с. — <nowiki>ISBN 5-09-008036-4</nowiki>.</ref> <br />
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<math>(a + bi) (c + di) = </math> <math>= ac + bci - bd + adi = </math> <math>= (ac - bd) + (bc + ad)i </math> <br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|<span style="writing-mode: horizontal-tb;"><i>2 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭ ᠠ᠋ ᠢᠢ (ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠬᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>3 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭᠠᠨ ᠳᠤ (ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃<br />
ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠢ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠣᠷᠤᠶ ᠲᠠᠢ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠵᠤ᠂ <span style="writing-mode: horizontal-tb;"><i>√5</i></span> ᠢᠢᠠᠷ ᠰᠤᠩᠭᠠᠨ ᠠ᠋᠃ ᠡᠨᠡ ᠢᠨᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠾᠢᠫᠣᠲ᠋ᠧᠨᠦᠽ ᠤᠨ<sup>hypotenuse</sup> ᠤᠷᠲᠤ ᠪᠣᠯᠠᠢ᠃ <br />
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= ᠬᠤᠪᠢᠶᠠᠬᠤ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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<math>\frac{a+bi}{c+di}=</math> <br />
<math>=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=</math> <br />
<math>=\frac{ac+bd}{c^2+d^2}+\left(\frac{bc-ad}{c^2+d^2}\right)i.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ<sup>quadrate</sup> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨᠡ᠋᠃<ref>Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01</ref><br />
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<math>(x+yi)^2=x^2-y^2 + 2xyi.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> <span style="writing-mode: horizontal-tb;"><i>(b ≠ 0)</i></span> ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ <br />
<math> \pm (\gamma + \delta i)</math> ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄<br />
<math>\gamma = \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠠ <math>\delta = (\sgn b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠣᠯᠤᠮᠤᠢ᠃<br />
<span style="writing-mode: horizontal-tb;"><i>sgn b</i></span> ᠭᠡᠭᠴᠢ ᠢᠨᠦ <math>\sgn b = \begin{cases} \ \ 1, & b > 0 \\ \ \ 0, & b = 0 \\ -1, & b < 0 \end{cases}</math> ᠭᠡᠵᠦ ᠲᠠᠢᠢᠯᠠᠭᠳᠠᠨᠠ᠃ <br />
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ᠡᠭᠦᠨ ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ ᠢᠢ ᠲᠤᠬᠠᠢᠢᠲᠠ ᠳ᠋ᠤ <math> \pm (\gamma + \delta i)</math> ᠲᠣᠭᠠ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠦ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠲᠣᠭᠠ ᠢᠢ ᠭᠠᠷᠭᠠᠵᠤ ᠢᠷᠡᠬᠦ ᠶᠣᠰᠤᠲᠠᠢ᠃ <br />
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=ᠬᠡᠷᠡᠭᠯᠡᠭᠡ=<br />
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== ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ==<br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠳᠠᠪᠲᠠᠯᠲᠠ ᠪᠠᠷ ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠠᠢᠢᠭᠤᠯᠬᠤ᠃<br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠯ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠳ᠋ᠤ ᠡᠭᠦᠰᠬᠡᠳᠡᠭ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ<sup>fractal</sup> ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠵᠡᠰᠱᠢᠶ ᠡ᠋ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡ ᠬᠦ ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ <span style="writing-mode: horizontal-tb;"><i>z = 0</i></span> ᠡᠴᠡ ᠡᠬᠢᠯᠡᠭᠡᠳ ᠢᠲ᠋ᠧᠷᠠᠼ ᠬᠢᠬᠦ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;"><i>f<sub>c</sub>(z)=z<sup>2</sup>+c</i></span> ᠹᠦᠨ᠍ᠻᠼ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠢᠢᠬᠤ ᠨᠥᠬᠦᠴᠡᠯ ᠪᠦᠬᠦᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠤ ᠣᠯᠠᠨᠯᠢᠬ ᠪᠠᠢᠢᠳᠠᠭ᠃ <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ ᠪᠠᠢᠢᠭᠠᠯᠢ ᠢᠢᠨ ᠭᠠᠢᠢᠬᠠᠮᠰᠢᠭᠲᠤ ᠳᠦᠷᠰᠦ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠦᠨ ᠰᠠᠢᠢᠬᠠᠨ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠶᠡᠷᠦ ᠪᠤᠰᠤ ᠢᠢᠨ ᠭᠣᠶᠤᠮᠰᠠᠭ ᠬᠣᠣᠰᠯᠠᠯ ᠢ ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠳᠡᠭ᠃<ref>ᠷᠡᠨᠴᠡᠨ ᠦ ᠡᠩᠬᠡᠪᠠᠲᠤ᠃ '''ᠫᠢᠲ᠋ᠾᠠᠭᠣᠷ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌'''᠃ ᠮᠣᠩᠭᠤᠯ ᠤᠯᠤᠯᠰ ᠊ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠪᠠ ᠲᠣᠭᠠᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠢᠢᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ https://imdt.ac.mn/c/1013874?content=1150891&fbclid=IwAR1HVqeXwT-h7dijj9EeIUzDvFustH99GAr9LHYQIU61XIByuBHCkSm10So 2020</ref> <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠵᠠᠭᠠᠭ ᠢ ᠳᠠᠪᠠᠬᠤ ᠳᠤ ᠪᠠᠨ ᠵᠦᠯᠢᠶ ᠠ᠋ ᠢᠢᠨ<sup>Julia</sup> ᠹᠷᠠᠻᠲᠠᠯ ᠤᠳ ᠦᠷᠭᠡᠯᠵᠢ ᠬᠣᠯᠪᠤᠯᠲᠠ ᠪᠠᠨ ᠠᠯᠳᠠᠵᠤ᠂ ᠹᠠᠲᠣᠤ ᠢᠢᠨ<sup>Fatou</sup> ᠲᠣᠭᠤᠰᠤ ᠪᠣᠯᠤᠨ ᠬᠦᠪᠢᠷᠠᠳᠠᠭ᠃<ref>ᠨᠡᠭᠡᠭᠡᠯᠲᠡᠲᠡᠢ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠳ᠃ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠳᠠᠯ᠃ ᠪᠢᠳᠡᠨ ᠦ ᠡᠷᠭᠢᠨ ᠲᠣᠭᠣᠷᠢᠨ ᠳ᠋ᠠᠬᠢ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠬᠡᠷᠬᠢᠨ ᠠᠵᠢᠯᠯᠠᠳᠠᠭ ᠪᠤᠢ᠃ ᠹᠷᠠᠻᠲᠯᠠ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠶᠢᠨ ᠹᠷᠠᠻᠲᠯᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢ᠌ᠭ᠌᠃ https://ultrait.ru/mn/smartphones/otkrytie-fraktalov-beskonechnost-fraktalov-kak-ustroen-mir.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/10/02</ref><br />
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|style = "height: 225px;"|ᠳᠦᠷᠰᠦ ᠠᠮᠢᠯᠠᠭᠤᠯᠤᠨ ᠵᠦᠯᠢᠶ ᠠ᠋ ᠢᠢᠨ <sup>Julia</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠦᠵᠡᠭᠦᠯᠦᠭᠰᠡᠨ ᠢᠨᠦ᠃<br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==</div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%BA%E1%A0%A3%E1%A0%AE%E1%A0%AB%E1%A0%8A%E1%A0%AF%E1%A0%A7%E1%A0%BA%E1%A0%B0_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%A0&diff=2743
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-10-02T15:49:07Z
<p>Angka: Link</p>
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'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ<sup>complex</sup> ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠯᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃ <br />
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|style = "height: 225px;"|ᠺᠣᠮᠫᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠢᠢ ᠲᠥᠯᠦᠭᠡᠯᠡᠵᠦ ᠪᠤᠶ ᠠᠷᠭᠠᠨ᠋ᠳ᠋ ᠤᠨ<sup>Argand</sup> ᠳᠢᠶᠠᠭ᠋ᠷᠠᠮ ᠳᠡᠭᠡᠷ ᠡ᠋ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠸᠧᠺᠲ᠋ᠣᠷ<sup>vector</sup> ᠡᠭᠦᠰᠬᠡᠵᠦ ᠪᠤᠶ ᠬᠣᠣᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ <span style="writing-mode: horizontal-tb;"><i>(a, b)</i></span> ᠳᠦᠷᠰᠦᠯᠡᠨ ᠦᠵᠡᠭᠦᠯᠵᠦ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
ᠡᠭᠦᠨ ᠳ᠋ᠦ: <br><span style="writing-mode: horizontal-tb;"><i>Re</i></span> ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ᠂ <br><span style="writing-mode: horizontal-tb;"><i>Im</i></span> ᠢᠨᠦ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠣᠯᠬᠤ ᠠᠭᠠᠳ <br><span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠢᠨᠦ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠳ᠋ᠦ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠬᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠮᠥᠨ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
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<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
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= ᠬᠠᠰᠤᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠳᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
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<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math><br />
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= ᠦᠷᠡᠵᠢᠬᠦ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃<ref>''Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И.'' Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — <abbr>М.</abbr>: Просвещение, 1998. — 288 с. — <nowiki>ISBN 5-09-008036-4</nowiki>.</ref> <br />
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<math>(a + bi) (c + di) = </math> <math>= ac + bci - bd + adi = </math> <math>= (ac - bd) + (bc + ad)i </math> <br />
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|style = "height: 225px;"|<span style="writing-mode: horizontal-tb;"><i>2 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭ ᠠ᠋ ᠢᠢ (ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠬᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>3 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭᠠᠨ ᠳᠤ (ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃<br />
ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠢ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠣᠷᠤᠶ ᠲᠠᠢ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠵᠤ᠂ <span style="writing-mode: horizontal-tb;"><i>√5</i></span> ᠢᠢᠠᠷ ᠰᠤᠩᠭᠠᠨ ᠠ᠋᠃ ᠡᠨᠡ ᠢᠨᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠾᠢᠫᠣᠲ᠋ᠧᠨᠦᠽ ᠤᠨ<sup>hypotenuse</sup> ᠤᠷᠲᠤ ᠪᠣᠯᠠᠢ᠃ <br />
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= ᠬᠤᠪᠢᠶᠠᠬᠤ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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<math>\frac{a+bi}{c+di}=</math> <br />
<math>=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=</math> <br />
<math>=\frac{ac+bd}{c^2+d^2}+\left(\frac{bc-ad}{c^2+d^2}\right)i.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ<sup>quadrate</sup> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨᠡ᠋᠃<ref>Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01</ref><br />
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<math>(x+yi)^2=x^2-y^2 + 2xyi.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> <span style="writing-mode: horizontal-tb;"><i>(b ≠ 0)</i></span> ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ <br />
<math> \pm (\gamma + \delta i)</math> ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄<br />
<math>\gamma = \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠠ <math>\delta = (\sgn b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠣᠯᠤᠮᠤᠢ᠃<br />
<span style="writing-mode: horizontal-tb;"><i>sgn b</i></span> ᠭᠡᠭᠴᠢ ᠢᠨᠦ <math>\sgn b = \begin{cases} \ \ 1, & b > 0 \\ \ \ 0, & b = 0 \\ -1, & b < 0 \end{cases}</math> ᠭᠡᠵᠦ ᠲᠠᠢᠢᠯᠠᠭᠳᠠᠨᠠ᠃ <br />
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ᠡᠭᠦᠨ ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ ᠢᠢ ᠲᠤᠬᠠᠢᠢᠲᠠ ᠳ᠋ᠤ <math> \pm (\gamma + \delta i)</math> ᠲᠣᠭᠠ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠦ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠲᠣᠭᠠ ᠢᠢ ᠭᠠᠷᠭᠠᠵᠤ ᠢᠷᠡᠬᠦ ᠶᠣᠰᠤᠲᠠᠢ᠃ <br />
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=ᠬᠡᠷᠡᠭᠯᠡᠭᠡ=<br />
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== ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ==<br />
ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠯ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠳ᠋ᠤ ᠡᠭᠦᠰᠬᠡᠳᠡᠭ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ<sup>fractal</sup> ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠵᠡᠰᠱᠢᠶ ᠡ᠋ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡ ᠬᠦ ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ <span style="writing-mode: horizontal-tb;"><i>z = 0</i></span> ᠡᠴᠡ ᠡᠬᠢᠯᠡᠭᠡᠳ ᠢᠲ᠋ᠧᠷᠠᠼ ᠬᠢᠬᠦ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;"><i>f<sub>c</sub>(z)=z<sup>2</sup>+c</i></span> ᠹᠦᠨ᠍ᠻᠼ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠢᠢᠬᠤ ᠨᠥᠬᠦᠴᠡᠯ ᠪᠦᠬᠦᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠤ ᠣᠯᠠᠨᠯᠢᠬ ᠪᠠᠢᠢᠳᠠᠭ᠃ <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ ᠪᠠᠢᠢᠭᠠᠯᠢ ᠢᠢᠨ ᠭᠠᠢᠢᠬᠠᠮᠰᠢᠭᠲᠤ ᠳᠦᠷᠰᠦ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠦᠨ ᠰᠠᠢᠢᠬᠠᠨ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠶᠡᠷᠦ ᠪᠤᠰᠤ ᠢᠢᠨ ᠭᠣᠶᠤᠮᠰᠠᠭ ᠬᠣᠣᠰᠯᠠᠯ ᠢ ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠳᠡᠭ᠃<ref>ᠷᠡᠨᠴᠡᠨ ᠦ ᠡᠩᠬᠡᠪᠠᠲᠤ᠃ '''ᠫᠢᠲ᠋ᠾᠠᠭᠣᠷ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌'''᠃ ᠮᠣᠩᠭᠤᠯ ᠤᠯᠤᠯᠰ ᠊ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠪᠠ ᠲᠣᠭᠠᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠢᠢᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ https://imdt.ac.mn/c/1013874?content=1150891&fbclid=IwAR1HVqeXwT-h7dijj9EeIUzDvFustH99GAr9LHYQIU61XIByuBHCkSm10So 2020</ref> <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠵᠠᠭᠠᠭ ᠢ ᠳᠠᠪᠠᠬᠤ ᠳᠤ ᠪᠠᠨ ᠵᠦᠯᠢᠶ ᠠ᠋ ᠢᠢᠨ<sup>Julia</sup> ᠹᠷᠠᠻᠲᠠᠯ ᠤᠳ ᠦᠷᠭᠡᠯᠵᠢ ᠬᠣᠯᠪᠤᠯᠲᠠ ᠪᠠᠨ ᠠᠯᠳᠠᠵᠤ᠂ ᠹᠠᠲᠣᠤ ᠢᠢᠨ<sup>Fatou</sup> ᠲᠣᠭᠤᠰᠤ ᠪᠣᠯᠤᠨ ᠬᠦᠪᠢᠷᠠᠳᠠᠭ᠃<ref>ᠨᠡᠭᠡᠭᠡᠯᠲᠡᠲᠡᠢ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠳ᠃ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠳᠠᠯ᠃ ᠪᠢᠳᠡᠨ ᠦ ᠡᠷᠭᠢᠨ ᠲᠣᠭᠣᠷᠢᠨ ᠳ᠋ᠠᠬᠢ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠬᠡᠷᠬᠢᠨ ᠠᠵᠢᠯᠯᠠᠳᠠᠭ ᠪᠤᠢ᠃ ᠹᠷᠠᠻᠲᠯᠠ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠶᠢᠨ ᠹᠷᠠᠻᠲᠯᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢ᠌ᠭ᠌᠃ https://ultrait.ru/mn/smartphones/otkrytie-fraktalov-beskonechnost-fraktalov-kak-ustroen-mir.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/10/02</ref><br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==</div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%BA%E1%A0%A3%E1%A0%AE%E1%A0%AB%E1%A0%8A%E1%A0%AF%E1%A0%A7%E1%A0%BA%E1%A0%B0_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%A0&diff=2742
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-10-02T08:50:40Z
<p>Angka: Edit</p>
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'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ<sup>complex</sup> ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠯᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃ <br />
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|style = "height: 225px;"|ᠺᠣᠮᠫᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠢᠢ ᠲᠥᠯᠦᠭᠡᠯᠡᠵᠦ ᠪᠤᠶ ᠠᠷᠭᠠᠨ᠋ᠳ᠋ ᠤᠨ<sup>Argand</sup> ᠳᠢᠶᠠᠭ᠋ᠷᠠᠮ ᠳᠡᠭᠡᠷ ᠡ᠋ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠸᠧᠺᠲ᠋ᠣᠷ<sup>vector</sup> ᠡᠭᠦᠰᠬᠡᠵᠦ ᠪᠤᠶ ᠬᠣᠣᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ <span style="writing-mode: horizontal-tb;"><i>(a, b)</i></span> ᠳᠦᠷᠰᠦᠯᠡᠨ ᠦᠵᠡᠭᠦᠯᠵᠦ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
ᠡᠭᠦᠨ ᠳ᠋ᠦ: <br><span style="writing-mode: horizontal-tb;"><i>Re</i></span> ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ᠂ <br><span style="writing-mode: horizontal-tb;"><i>Im</i></span> ᠢᠨᠦ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠣᠯᠬᠤ ᠠᠭᠠᠳ <br><span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠢᠨᠦ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠳ᠋ᠦ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠬᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠮᠥᠨ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
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<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
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= ᠬᠠᠰᠤᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠳᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
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<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math><br />
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= ᠦᠷᠡᠵᠢᠬᠦ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃<ref>''Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И.'' Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — <abbr>М.</abbr>: Просвещение, 1998. — 288 с. — <nowiki>ISBN 5-09-008036-4</nowiki>.</ref> <br />
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<math>(a + bi) (c + di) = </math> <math>= ac + bci - bd + adi = </math> <math>= (ac - bd) + (bc + ad)i </math> <br />
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|style = "height: 225px;"|<span style="writing-mode: horizontal-tb;"><i>2 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭ ᠠ᠋ ᠢᠢ (ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠬᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>3 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭᠠᠨ ᠳᠤ (ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃<br />
ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠢ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠣᠷᠤᠶ ᠲᠠᠢ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠵᠤ᠂ <span style="writing-mode: horizontal-tb;"><i>√5</i></span> ᠢᠢᠠᠷ ᠰᠤᠩᠭᠠᠨ ᠠ᠋᠃ ᠡᠨᠡ ᠢᠨᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠾᠢᠫᠣᠲ᠋ᠧᠨᠦᠽ ᠤᠨ<sup>hypotenuse</sup> ᠤᠷᠲᠤ ᠪᠣᠯᠠᠢ᠃ <br />
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= ᠬᠤᠪᠢᠶᠠᠬᠤ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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<math>\frac{a+bi}{c+di}=</math> <br />
<math>=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=</math> <br />
<math>=\frac{ac+bd}{c^2+d^2}+\left(\frac{bc-ad}{c^2+d^2}\right)i.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ<sup>quadrate</sup> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨᠡ᠋᠃<ref>Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01</ref><br />
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<math>(x+yi)^2=x^2-y^2 + 2xyi.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> <span style="writing-mode: horizontal-tb;"><i>(b ≠ 0)</i></span> ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ <br />
<math> \pm (\gamma + \delta i)</math> ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄<br />
<math>\gamma = \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠠ <math>\delta = (\sgn b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠣᠯᠤᠮᠤᠢ᠃<br />
<span style="writing-mode: horizontal-tb;"><i>sgn b</i></span> ᠭᠡᠭᠴᠢ ᠢᠨᠦ <math>\sgn b = \begin{cases} \ \ 1, & b > 0 \\ \ \ 0, & b = 0 \\ -1, & b < 0 \end{cases}</math> ᠭᠡᠵᠦ ᠲᠠᠢᠢᠯᠠᠭᠳᠠᠨᠠ᠃ <br />
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ᠡᠭᠦᠨ ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ ᠢᠢ ᠲᠤᠬᠠᠢᠢᠲᠠ ᠳ᠋ᠤ <math> \pm (\gamma + \delta i)</math> ᠲᠣᠭᠠ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠦ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠲᠣᠭᠠ ᠢᠢ ᠭᠠᠷᠭᠠᠵᠤ ᠢᠷᠡᠬᠦ ᠶᠣᠰᠤᠲᠠᠢ᠃ <br />
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=ᠬᠡᠷᠡᠭᠯᠡᠭᠡ=<br />
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== ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ==<br />
ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠯ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠳ᠋ᠤ ᠡᠭᠦᠰᠬᠡᠳᠡᠭ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ<sup>fractal</sup> ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠵᠡᠰᠱᠢᠶ ᠡ᠋ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡ ᠬᠦ ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ <span style="writing-mode: horizontal-tb;"><i>z = 0</i></span> ᠡᠴᠡ ᠡᠬᠢᠯᠡᠭᠡᠳ ᠢᠲ᠋ᠧᠷᠠᠼ ᠬᠢᠬᠦ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;"><i>f<sub>c</sub>(z)=z<sup>2</sup>+c</i></span> ᠹᠦᠨ᠍ᠻᠼ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠢᠢᠬᠤ ᠨᠥᠬᠦᠴᠡᠯ ᠪᠦᠬᠦᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠤ ᠣᠯᠠᠨᠯᠢᠬ ᠪᠠᠢᠢᠳᠠᠭ᠃ <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ ᠪᠠᠢᠢᠭᠠᠯᠢ ᠢᠢᠨ ᠭᠠᠢᠢᠬᠠᠮᠰᠢᠭᠲᠤ ᠳᠦᠷᠰᠦ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠦᠨ ᠰᠠᠢᠢᠬᠠᠨ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠶᠡᠷᠦ ᠪᠤᠰᠤ ᠢᠢᠨ ᠭᠣᠶᠤᠮᠰᠠᠭ ᠬᠣᠣᠰᠯᠠᠯ ᠢ ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠳᠡᠭ᠃<ref>ᠷᠡᠨᠴᠡᠨ ᠦ ᠡᠩᠬᠡᠪᠠᠲᠤ᠃ '''ᠫᠢᠲ᠋ᠾᠠᠭᠣᠷ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌'''᠃ ᠮᠣᠩᠭᠤᠯ ᠤᠯᠤᠯᠰ ᠊ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠪᠠ ᠲᠣᠭᠠᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠢᠢᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ https://imdt.ac.mn/c/1013874?content=1150891&fbclid=IwAR1HVqeXwT-h7dijj9EeIUzDvFustH99GAr9LHYQIU61XIByuBHCkSm10So 2020</ref> <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠵᠠᠭᠠᠭ ᠢ ᠳᠠᠪᠠᠬᠤ ᠳᠤ ᠪᠠᠨ ᠵᠦᠯᠢᠶ ᠠ᠋ ᠢᠢᠨ<sup>Julia</sup> ᠹᠷᠠᠻᠲᠠᠯ ᠤᠳ ᠦᠷᠭᠡᠯᠵᠢ ᠬᠣᠯᠪᠤᠯᠲᠠ ᠪᠠᠨ ᠠᠯᠳᠠᠵᠤ᠂ ᠹᠠᠲᠣᠤ ᠢᠢᠨ<sup>Fatou</sup> ᠲᠣᠭᠤᠰᠤ ᠪᠣᠯᠤᠨ ᠬᠦᠪᠢᠷᠠᠳᠠᠭ᠃<br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==</div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%BA%E1%A0%A3%E1%A0%AE%E1%A0%AB%E1%A0%8A%E1%A0%AF%E1%A0%A7%E1%A0%BA%E1%A0%B0_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%A0&diff=2741
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-10-02T08:43:00Z
<p>Angka: Edit</p>
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'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ<sup>complex</sup> ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠯᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃ <br />
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|style = "height: 225px;"|ᠺᠣᠮᠫᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠢᠢ ᠲᠥᠯᠦᠭᠡᠯᠡᠵᠦ ᠪᠤᠶ ᠠᠷᠭᠠᠨ᠋ᠳ᠋ ᠤᠨ<sup>Argand</sup> ᠳᠢᠶᠠᠭ᠋ᠷᠠᠮ ᠳᠡᠭᠡᠷ ᠡ᠋ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠸᠧᠺᠲ᠋ᠣᠷ<sup>vector</sup> ᠡᠭᠦᠰᠬᠡᠵᠦ ᠪᠤᠶ ᠬᠣᠣᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ <span style="writing-mode: horizontal-tb;"><i>(a, b)</i></span> ᠳᠦᠷᠰᠦᠯᠡᠨ ᠦᠵᠡᠭᠦᠯᠵᠦ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
ᠡᠭᠦᠨ ᠳ᠋ᠦ: <br><span style="writing-mode: horizontal-tb;"><i>Re</i></span> ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ᠂ <br><span style="writing-mode: horizontal-tb;"><i>Im</i></span> ᠢᠨᠦ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠣᠯᠬᠤ ᠠᠭᠠᠳ <br><span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠢᠨᠦ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠳ᠋ᠦ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠬᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠮᠥᠨ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
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<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
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{| class="wikitable"<br />
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|style = "height: 220px;"|ᠬᠣᠶᠠᠷ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃ <br />
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= ᠬᠠᠰᠤᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠳᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
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<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math><br />
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= ᠦᠷᠡᠵᠢᠬᠦ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃<ref>''Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И.'' Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — <abbr>М.</abbr>: Просвещение, 1998. — 288 с. — <nowiki>ISBN 5-09-008036-4</nowiki>.</ref> <br />
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<math>(a + bi) (c + di) = </math> <math>= ac + bci - bd + adi = </math> <math>= (ac - bd) + (bc + ad)i </math> <br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|<span style="writing-mode: horizontal-tb;"><i>2 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭ ᠠ᠋ ᠢᠢ (ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠬᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>3 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭᠠᠨ ᠳᠤ (ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃<br />
ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠢ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠣᠷᠤᠶ ᠲᠠᠢ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠵᠤ᠂ <span style="writing-mode: horizontal-tb;"><i>√5</i></span> ᠢᠢᠠᠷ ᠰᠤᠩᠭᠠᠨ ᠠ᠋᠃ ᠡᠨᠡ ᠢᠨᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠾᠢᠫᠣᠲ᠋ᠧᠨᠦᠽ ᠤᠨ<sup>hypotenuse</sup> ᠤᠷᠲᠤ ᠪᠣᠯᠠᠢ᠃ <br />
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= ᠬᠤᠪᠢᠶᠠᠬᠤ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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<math>\frac{a+bi}{c+di}=</math> <br />
<math>=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=</math> <br />
<math>=\frac{ac+bd}{c^2+d^2}+\left(\frac{bc-ad}{c^2+d^2}\right)i.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ<sup>quadrate</sup> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨᠡ᠋᠃<ref>Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01</ref><br />
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<math>(x+yi)^2=x^2-y^2 + 2xyi.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> <span style="writing-mode: horizontal-tb;"><i>(b ≠ 0)</i></span> ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ <br />
<math> \pm (\gamma + \delta i)</math> ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄<br />
<math>\gamma = \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠠ <math>\delta = (\sgn b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠣᠯᠤᠮᠤᠢ᠃<br />
<span style="writing-mode: horizontal-tb;"><i>sgn b</i></span> ᠭᠡᠭᠴᠢ ᠢᠨᠦ <math>\sgn b = \begin{cases} \ \ 1, & b > 0 \\ \ \ 0, & b = 0 \\ -1, & b < 0 \end{cases}</math> ᠭᠡᠵᠦ ᠲᠠᠢᠢᠯᠠᠭᠳᠠᠨᠠ᠃ <br />
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ᠡᠭᠦᠨ ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ ᠢᠢ ᠲᠤᠬᠠᠢᠢᠲᠠ ᠳ᠋ᠤ <math> \pm (\gamma + \delta i)</math> ᠲᠣᠭᠠ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠦ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠲᠣᠭᠠ ᠢᠢ ᠭᠠᠷᠭᠠᠵᠤ ᠢᠷᠡᠬᠦ ᠶᠣᠰᠤᠲᠠᠢ᠃ <br />
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=ᠬᠡᠷᠡᠭᠯᠡᠭᠡ=<br />
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== ᠹᠷᠠᠻᠲᠠᠯ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ==<br />
ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠯ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠳ᠋ᠤ ᠡᠭᠦᠰᠬᠡᠳᠡᠭ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ<sup>fractal</sup> ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠵᠡᠰᠱᠢᠶ ᠡ᠋ ᠶᠤᠮ᠃ <br />
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ᠡᠨᠡ ᠬᠦ ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ z = 0 ᠡᠴᠡ ᠡᠬᠢᠯᠡᠭᠡᠳ ᠢᠲ᠋ᠧᠷᠠᠼ ᠬᠢᠬᠦ ᠳ᠋ᠦ <math>f_c(z)=z^2+c</math> ᠹᠦᠨ᠍ᠻᠼ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠢᠢᠬᠤ ᠨᠥᠬᠦᠴᠡᠯ ᠪᠦᠬᠦᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠤ ᠣᠯᠠᠨᠯᠢᠬ ᠪᠠᠢᠢᠳᠠᠭ᠃ <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ<sup>Mandelbrot</sup> ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ ᠪᠠᠢᠢᠭᠠᠯᠢ ᠢᠢᠨ ᠭᠠᠢᠢᠬᠠᠮᠰᠢᠭᠲᠤ ᠳᠦᠷᠰᠦ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠦᠨ ᠰᠠᠢᠢᠬᠠᠨ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠶᠡᠷᠦ ᠪᠤᠰᠤ ᠢᠢᠨ ᠭᠣᠶᠤᠮᠰᠠᠭ ᠬᠣᠣᠰᠯᠠᠯ ᠢ ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠳᠡᠭ᠃<ref>ᠷᠡᠨᠴᠡᠨ ᠦ ᠡᠩᠬᠡᠪᠠᠲᠤ᠃ '''ᠫᠢᠲ᠋ᠾᠠᠭᠣᠷ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌'''᠃ ᠮᠣᠩᠭᠤᠯ ᠤᠯᠤᠯᠰ ᠊ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠪᠠ ᠲᠣᠭᠠᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠢᠢᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ https://imdt.ac.mn/c/1013874?content=1150891&fbclid=IwAR1HVqeXwT-h7dijj9EeIUzDvFustH99GAr9LHYQIU61XIByuBHCkSm10So 2020</ref> <br />
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ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ ᠹᠷᠠᠻᠲᠠᠯ ᠤᠨ ᠵᠠᠭᠠᠭ ᠢ ᠳᠠᠪᠠᠬᠤ ᠳᠤ ᠪᠠᠨ ᠵᠦᠯᠢᠶ ᠠ᠋ ᠢᠢᠨ<sup>Julia</sup> ᠹᠷᠠᠻᠲᠠᠯ ᠤᠳ ᠦᠷᠭᠡᠯᠵᠢ ᠬᠣᠯᠪᠤᠯᠲᠠ ᠪᠠᠨ ᠠᠯᠳᠠᠵᠤ᠂ ᠹᠠᠲᠣᠤ ᠢᠢᠨ<sup>Fatou</sup> ᠲᠣᠭᠤᠰᠤ ᠪᠣᠯᠤᠨ ᠬᠦᠪᠢᠷᠠᠳᠠᠭ᠃<br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==</div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%BA%E1%A0%A3%E1%A0%AE%E1%A0%AB%E1%A0%8A%E1%A0%AF%E1%A0%A7%E1%A0%BA%E1%A0%B0_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%A0&diff=2740
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-09-10T14:51:26Z
<p>Angka: Images</p>
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'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ<sup>complex</sup> ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠯᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃ <br />
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|style = "height: 225px;"|ᠺᠣᠮᠫᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠢᠢ ᠲᠥᠯᠦᠭᠡᠯᠡᠵᠦ ᠪᠤᠶ ᠠᠷᠭᠠᠨ᠋ᠳ᠋ ᠤᠨ<sup>Argand</sup> ᠳᠢᠶᠠᠭ᠋ᠷᠠᠮ ᠳᠡᠭᠡᠷ ᠡ᠋ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠸᠧᠺᠲ᠋ᠣᠷ<sup>vector</sup> ᠡᠭᠦᠰᠬᠡᠵᠦ ᠪᠤᠶ ᠬᠣᠣᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ <span style="writing-mode: horizontal-tb;"><i>(a, b)</i></span> ᠳᠦᠷᠰᠦᠯᠡᠨ ᠦᠵᠡᠭᠦᠯᠵᠦ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
ᠡᠭᠦᠨ ᠳ᠋ᠦ: <br><span style="writing-mode: horizontal-tb;"><i>Re</i></span> ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ᠂ <br><span style="writing-mode: horizontal-tb;"><i>Im</i></span> ᠢᠨᠦ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠣᠯᠬᠤ ᠠᠭᠠᠳ <br><span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠢᠨᠦ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠳ᠋ᠦ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠬᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠮᠥᠨ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
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<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
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{| class="wikitable"<br />
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|style = "height: 220px;"|ᠬᠣᠶᠠᠷ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃ <br />
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= ᠬᠠᠰᠤᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠳᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
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<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math><br />
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= ᠦᠷᠡᠵᠢᠬᠦ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃<ref>''Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И.'' Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — <abbr>М.</abbr>: Просвещение, 1998. — 288 с. — <nowiki>ISBN 5-09-008036-4</nowiki>.</ref> <br />
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<math>(a + bi) (c + di) = </math> <math>= ac + bci - bd + adi = </math> <math>= (ac - bd) + (bc + ad)i </math> <br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|<span style="writing-mode: horizontal-tb;"><i>2 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭ ᠠ᠋ ᠢᠢ (ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠬᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>3 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭᠠᠨ ᠳᠤ (ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃<br />
ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠢ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠣᠷᠤᠶ ᠲᠠᠢ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠵᠤ᠂ <span style="writing-mode: horizontal-tb;"><i>√5</i></span> ᠢᠢᠠᠷ ᠰᠤᠩᠭᠠᠨ ᠠ᠋᠃ ᠡᠨᠡ ᠢᠨᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠾᠢᠫᠣᠲ᠋ᠧᠨᠦᠽ ᠤᠨ<sup>hypotenuse</sup> ᠤᠷᠲᠤ ᠪᠣᠯᠠᠢ᠃ <br />
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= ᠬᠤᠪᠢᠶᠠᠬᠤ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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<math>\frac{a+bi}{c+di}=</math> <br />
<math>=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=</math> <br />
<math>=\frac{ac+bd}{c^2+d^2}+\left(\frac{bc-ad}{c^2+d^2}\right)i.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ<sup>quadrate</sup> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨᠡ᠋᠃<ref>Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01</ref><br />
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<math>(x+yi)^2=x^2-y^2 + 2xyi.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> <span style="writing-mode: horizontal-tb;"><i>(b ≠ 0)</i></span> ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ <br />
<math> \pm (\gamma + \delta i)</math> ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄<br />
<math>\gamma = \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠠ <math>\delta = (\sgn b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠣᠯᠤᠮᠤᠢ᠃<br />
<span style="writing-mode: horizontal-tb;"><i>sgn b</i></span> ᠭᠡᠭᠴᠢ ᠢᠨᠦ <math>\sgn b = \begin{cases} \ \ 1, & b > 0 \\ \ \ 0, & b = 0 \\ -1, & b < 0 \end{cases}</math> ᠭᠡᠵᠦ ᠲᠠᠢᠢᠯᠠᠭᠳᠠᠨᠠ᠃ <br />
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ᠡᠭᠦᠨ ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ ᠢᠢ ᠲᠤᠬᠠᠢᠢᠲᠠ ᠳ᠋ᠤ <math> \pm (\gamma + \delta i)</math> ᠲᠣᠭᠠ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠦ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠲᠣᠭᠠ ᠢᠢ ᠭᠠᠷᠭᠠᠵᠤ ᠢᠷᠡᠬᠦ ᠶᠣᠰᠤᠲᠠᠢ᠃ <br />
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==ᠬᠡᠷᠡᠭᠯᠡᠭᠡ==<br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==</div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%BA%E1%A0%A3%E1%A0%AE%E1%A0%AB%E1%A0%8A%E1%A0%AF%E1%A0%A7%E1%A0%BA%E1%A0%B0_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%A0&diff=2739
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-09-10T14:35:27Z
<p>Angka: Images</p>
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'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ<sup>complex</sup> ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠯᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃ <br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|ᠺᠣᠮᠫᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠢᠢ ᠲᠥᠯᠦᠭᠡᠯᠡᠵᠦ ᠪᠤᠶ ᠠᠷᠭᠠᠨ᠋ᠳ᠋ ᠤᠨ<sup>Argand</sup> ᠳᠢᠶᠠᠭ᠋ᠷᠠᠮ ᠳᠡᠭᠡᠷ ᠡ᠋ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠸᠧᠺᠲ᠋ᠣᠷ<sup>vector</sup> ᠡᠭᠦᠰᠬᠡᠵᠦ ᠪᠤᠶ ᠬᠣᠣᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ <span style="writing-mode: horizontal-tb;"><i>(a, b)</i></span> ᠳᠦᠷᠰᠦᠯᠡᠨ ᠦᠵᠡᠭᠦᠯᠵᠦ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
ᠡᠭᠦᠨ ᠳ᠋ᠦ: <br><span style="writing-mode: horizontal-tb;"><i>Re</i></span> ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ᠂ <br><span style="writing-mode: horizontal-tb;"><i>Im</i></span> ᠢᠨᠦ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠣᠯᠬᠤ ᠠᠭᠠᠳ <br><span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠢᠨᠦ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠳ᠋ᠦ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠬᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠮᠥᠨ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
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<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
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= ᠬᠠᠰᠤᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠳᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
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<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math><br />
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= ᠦᠷᠡᠵᠢᠬᠦ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃<ref>''Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И.'' Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — <abbr>М.</abbr>: Просвещение, 1998. — 288 с. — <nowiki>ISBN 5-09-008036-4</nowiki>.</ref> <br />
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<math>(a + bi) (c + di) = </math> <math>= ac + bci - bd + adi = </math> <math>= (ac - bd) + (bc + ad)i </math> <br />
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|style = "height: 225px;"|<span style="writing-mode: horizontal-tb;"><i>2 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭ ᠠ᠋ ᠢᠢ (ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠬᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>3 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭᠠᠨ ᠳᠤ (ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃<br />
ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠢ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠣᠷᠤᠶ ᠲᠠᠢ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠵᠤ᠂ <span style="writing-mode: horizontal-tb;"><i>√5</i></span> ᠢᠢᠠᠷ ᠰᠤᠩᠭᠠᠨ ᠠ᠋᠃ ᠡᠨᠡ ᠢᠨᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠾᠢᠫᠣᠲ᠋ᠧᠨᠦᠽ ᠤᠨ<sup>hypotenuse</sup> ᠤᠷᠲᠤ ᠪᠣᠯᠠᠢ᠃ <br />
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= ᠬᠤᠪᠢᠶᠠᠬᠤ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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<math>\frac{a+bi}{c+di}=</math> <br />
<math>=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=</math> <br />
<math>=\frac{ac+bd}{c^2+d^2}+\left(\frac{bc-ad}{c^2+d^2}\right)i.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ<sup>quadrate</sup> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨᠡ᠋᠃<ref>Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01</ref><br />
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<math>(x+yi)^2=x^2-y^2 + 2xyi.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> <span style="writing-mode: horizontal-tb;"><i>(b ≠ 0)</i></span> ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ <br />
<math> \pm (\gamma + \delta i)</math> ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄<br />
<math>\gamma = \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠠ <math>\delta = (\sgn b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠣᠯᠤᠮᠤᠢ᠃<br />
<span style="writing-mode: horizontal-tb;"><i>sgn b</i></span> ᠭᠡᠭᠴᠢ ᠢᠨᠦ <math>\sgn b = \begin{cases} \ \ 1, & b > 0 \\ \ \ 0, & b = 0 \\ -1, & b < 0 \end{cases}</math> ᠭᠡᠵᠦ ᠲᠠᠢᠢᠯᠠᠭᠳᠠᠨᠠ᠃ <br />
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ᠡᠭᠦᠨ ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ ᠢᠢ ᠲᠤᠬᠠᠢᠢᠲᠠ ᠳ᠋ᠤ <math> \pm (\gamma + \delta i)</math> ᠲᠣᠭᠠ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠦ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠲᠣᠭᠠ ᠢᠢ ᠭᠠᠷᠭᠠᠵᠤ ᠢᠷᠡᠬᠦ ᠶᠣᠰᠤᠲᠠᠢ᠃ <br />
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==ᠬᠡᠷᠡᠭᠯᠡᠭᠡ==<br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==</div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%BA%E1%A0%A3%E1%A0%AE%E1%A0%AB%E1%A0%8A%E1%A0%AF%E1%A0%A7%E1%A0%BA%E1%A0%B0_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%A0&diff=2738
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-09-10T12:07:21Z
<p>Angka: Layout</p>
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'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ<sup>complex</sup> ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠯᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃ <br />
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|style = "height: 225px;"|ᠺᠣᠮᠫᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠢᠢ ᠲᠥᠯᠦᠭᠡᠯᠡᠵᠦ ᠪᠤᠶ ᠠᠷᠭᠠᠨ᠋ᠳ᠋ ᠤᠨ<sup>Argand</sup> ᠳᠢᠶᠠᠭ᠋ᠷᠠᠮ ᠳᠡᠭᠡᠷ ᠡ᠋ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠸᠧᠺᠲ᠋ᠣᠷ<sup>vector</sup> ᠡᠭᠦᠰᠬᠡᠵᠦ ᠪᠤᠶ ᠬᠣᠣᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ <span style="writing-mode: horizontal-tb;"><i>(a, b)</i></span> ᠳᠦᠷᠰᠦᠯᠡᠨ ᠦᠵᠡᠭᠦᠯᠵᠦ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
ᠡᠭᠦᠨ ᠳ᠋ᠦ: <br><span style="writing-mode: horizontal-tb;"><i>Re</i></span> ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ᠂ <br><span style="writing-mode: horizontal-tb;"><i>Im</i></span> ᠢᠨᠦ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠣᠯᠬᠤ ᠠᠭᠠᠳ <br><span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠢᠨᠦ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠳ᠋ᠦ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠬᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠮᠥᠨ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
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<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
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= ᠬᠠᠰᠤᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠳᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
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<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math><br />
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= ᠦᠷᠡᠵᠢᠬᠦ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃<ref>''Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И.'' Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — <abbr>М.</abbr>: Просвещение, 1998. — 288 с. — <nowiki>ISBN 5-09-008036-4</nowiki>.</ref> <br />
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<math>(a + bi) (c + di) = </math> <math>= ac + bci - bd + adi = </math> <math>= (ac - bd) + (bc + ad)i </math> <br />
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= ᠬᠤᠪᠢᠶᠠᠬᠤ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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<math>\frac{a+bi}{c+di}=</math> <br />
<math>=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=</math> <br />
<math>=\frac{ac+bd}{c^2+d^2}+\left(\frac{bc-ad}{c^2+d^2}\right)i.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ<sup>quadrate</sup> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨᠡ᠋᠃<ref>Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01</ref><br />
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<math>(x+yi)^2=x^2-y^2 + 2xyi.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> <span style="writing-mode: horizontal-tb;"><i>(b ≠ 0)</i></span> ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ <br />
<math> \pm (\gamma + \delta i)</math> ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄<br />
<math>\gamma = \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠠ <math>\delta = (\sgn b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠣᠯᠤᠮᠤᠢ᠃<br />
<span style="writing-mode: horizontal-tb;"><i>sgn b</i></span> ᠭᠡᠭᠴᠢ ᠢᠨᠦ <math>\sgn b = \begin{cases} \ \ 1, & b > 0 \\ \ \ 0, & b = 0 \\ -1, & b < 0 \end{cases}</math> ᠭᠡᠵᠦ ᠲᠠᠢᠢᠯᠠᠭᠳᠠᠨᠠ᠃ <br />
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ᠡᠭᠦᠨ ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ ᠢᠢ ᠲᠤᠬᠠᠢᠢᠲᠠ ᠳ᠋ᠤ <math> \pm (\gamma + \delta i)</math> ᠲᠣᠭᠠ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠦ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠲᠣᠭᠠ ᠢᠢ ᠭᠠᠷᠭᠠᠵᠤ ᠢᠷᠡᠬᠦ ᠶᠣᠰᠤᠲᠠᠢ᠃ <br />
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==ᠬᠡᠷᠡᠭᠯᠡᠭᠡ==<br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==</div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%BA%E1%A0%A3%E1%A0%AE%E1%A0%AB%E1%A0%8A%E1%A0%AF%E1%A0%A7%E1%A0%BA%E1%A0%B0_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%A0&diff=2737
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-09-10T11:50:23Z
<p>Angka: Images</p>
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<div><div style="max-height: 80vh; writing-mode: vertical-lr; font-family: 'Noto Sans Mongolian'"><br />
'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠯᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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__TOC__<br />
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== ᠳᠦᠷᠡᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠳ᠋ᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠳ᠋ᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃ <br />
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{| class="wikitable"<br />
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|style = "height: 225px;"|ᠺᠣᠮᠫᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠢᠢ ᠲᠥᠯᠦᠭᠡᠯᠡᠵᠦ ᠪᠤᠶ ᠠᠷᠭᠠᠨ᠋ᠳ᠋ ᠤᠨ ᠳᠢᠶᠠᠭ᠋ᠷᠠᠮ ᠳᠡᠭᠡᠷ ᠡ᠋ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠸᠧᠺᠲ᠋ᠣᠷ ᠡᠭᠦᠰᠬᠡᠵᠦ ᠪᠤᠶ ᠬᠣᠣᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ <span style="writing-mode: horizontal-tb;"><i>(a, b)</i></span> ᠳᠦᠷᠰᠦᠯᠡᠨ ᠦᠵᠡᠭᠦᠯᠵᠦ ᠪᠣᠯᠤᠨ ᠠ᠋᠃ <br />
ᠡᠭᠦᠨ ᠳ᠋ᠦ: <br><span style="writing-mode: horizontal-tb;"><i>Re</i></span> ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ᠂ <br><span style="writing-mode: horizontal-tb;"><i>Im</i></span> ᠢᠨᠦ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠣᠯᠬᠤ ᠠᠭᠠᠳ <br><span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠢᠨᠦ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠳ᠋ᠦ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠬᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠮᠥᠨ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
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<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
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= ᠬᠠᠰᠤᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠳᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
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<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math><br />
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= ᠦᠷᠡᠵᠢᠬᠦ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃<ref>''Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И.'' Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — <abbr>М.</abbr>: Просвещение, 1998. — 288 с. — <nowiki>ISBN 5-09-008036-4</nowiki>.</ref> <br />
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<math>(a + bi) (c + di) = </math> <math>= ac + bci - bd + adi = </math> <math>= (ac - bd) + (bc + ad)i </math> <br />
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= ᠬᠤᠪᠢᠶᠠᠬᠤ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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<math>\frac{a+bi}{c+di}=</math> <br />
<math>=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=</math> <br />
<math>=\frac{ac+bd}{c^2+d^2}+\left(\frac{bc-ad}{c^2+d^2}\right)i.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨᠡ᠋᠃<ref>Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01</ref><br />
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<math>(x+yi)^2=x^2-y^2 + 2xyi.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> <span style="writing-mode: horizontal-tb;"><i>(with b ≠ 0)</i></span> ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ <br />
<math> \pm (\gamma + \delta i)</math> ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄<br />
<math>\gamma = \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠠ <math>\delta = (\sgn b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠣᠯᠤᠮᠤᠢ᠃<br />
<span style="writing-mode: horizontal-tb;"><i>sgn b</i></span> ᠭᠡᠭᠴᠢ ᠢᠨᠦ <math>\sgn b = \begin{cases} \ \ 1, & b > 0 \\ \ \ 0, & b = 0 \\ -1, & b < 0 \end{cases}</math> ᠭᠡᠵᠦ ᠲᠠᠢᠢᠯᠠᠭᠳᠠᠨᠠ᠃ <br />
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ᠡᠭᠦᠨ ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ ᠢᠢ ᠲᠤᠬᠠᠢᠢᠲᠠ ᠳ᠋ᠤ <math> \pm (\gamma + \delta i)</math> ᠲᠣᠭᠠ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠦ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠲᠣᠭᠠ ᠢᠢ ᠭᠠᠷᠭᠠᠵᠤ ᠢᠷᠡᠬᠦ ᠶᠣᠰᠤᠲᠠᠢ᠃ <br />
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==ᠬᠡᠷᠡᠭᠯᠡᠭᠡ==<br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==</div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%BA%E1%A0%A3%E1%A0%AE%E1%A0%AB%E1%A0%8A%E1%A0%AF%E1%A0%A7%E1%A0%BA%E1%A0%B0_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%A0&diff=2729
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-09-01T03:55:54Z
<p>Angka: Spelling</p>
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<div><div style="max-height: 80vh; writing-mode: vertical-lr; font-family: 'Noto Sans Mongolian'"><br />
'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠯᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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__TOC__<br />
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== ᠳᠦᠷᠡᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠳ᠋ᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠳ᠋ᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
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<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
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= ᠬᠠᠰᠤᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠳᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
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<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math><br />
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= ᠦᠷᠡᠵᠢᠬᠦ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃<ref>''Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И.'' Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — <abbr>М.</abbr>: Просвещение, 1998. — 288 с. — <nowiki>ISBN 5-09-008036-4</nowiki>.</ref> <br />
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<math>(a + bi) (c + di) = </math> <math>= ac + bci - bd + adi = </math> <math>= (ac - bd) + (bc + ad)i </math> <br />
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= ᠬᠤᠪᠢᠶᠠᠬᠤ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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<math>\frac{a+bi}{c+di}=</math> <br />
<math>=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=</math> <br />
<math>=\frac{ac+bd}{c^2+d^2}+\left(\frac{bc-ad}{c^2+d^2}\right)i.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨᠡ᠋᠃<ref>Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01</ref><br />
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<math>(x+yi)^2=x^2-y^2 + 2xyi.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> <span style="writing-mode: horizontal-tb;"><i>(with b ≠ 0)</i></span> ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ <br />
<math> \pm (\gamma + \delta i)</math> ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄<br />
<math>\gamma = \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠠ <math>\delta = (\sgn b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠣᠯᠤᠮᠤᠢ᠃<br />
<span style="writing-mode: horizontal-tb;"><i>sgn b</i></span> ᠭᠡᠭᠴᠢ ᠢᠨᠦ <math>\sgn b = \begin{cases} \ \ 1, & b > 0 \\ \ \ 0, & b = 0 \\ -1, & b < 0 \end{cases}</math> ᠭᠡᠵᠦ ᠲᠠᠢᠢᠯᠠᠭᠳᠠᠨᠠ᠃ <br />
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ᠡᠭᠦᠨ ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ ᠢᠢ ᠲᠤᠬᠠᠢᠢᠲᠠ ᠳ᠋ᠤ <math> \pm (\gamma + \delta i)</math> ᠲᠣᠭᠠ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠦ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠲᠣᠭᠠ ᠢᠢ ᠭᠠᠷᠭᠠᠵᠤ ᠢᠷᠡᠬᠦ ᠶᠣᠰᠤᠲᠠᠢ᠃ <br />
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==ᠬᠡᠷᠡᠭᠯᠡᠭᠡ==<br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==</div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%BA%E1%A0%A3%E1%A0%AE%E1%A0%AB%E1%A0%8A%E1%A0%AF%E1%A0%A7%E1%A0%BA%E1%A0%B0_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%A0&diff=2728
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-09-01T03:54:26Z
<p>Angka: Edit</p>
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<div><div style="max-height: 80vh; writing-mode: vertical-lr; font-family: 'Noto Sans Mongolian'"><br />
'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠯᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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__TOC__<br />
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== ᠳᠦᠷᠡᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠳ᠋ᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠳ᠋ᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
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<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
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= ᠬᠠᠰᠤᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠳᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
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<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math><br />
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= ᠦᠷᠡᠵᠢᠬᠦ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃<ref>''Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И.'' Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — <abbr>М.</abbr>: Просвещение, 1998. — 288 с. — <nowiki>ISBN 5-09-008036-4</nowiki>.</ref> <br />
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<math>(a + bi) (c + di) = </math> <math>= ac + bci - bd + adi = </math> <math>= (ac - bd) + (bc + ad)i </math> <br />
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= ᠬᠤᠪᠢᠶᠠᠬᠤ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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<math>\frac{a+bi}{c+di}=</math> <br />
<math>=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=</math> <br />
<math>=\frac{ac+bd}{c^2+d^2}+\left(\frac{bc-ad}{c^2+d^2}\right)i.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨᠡ᠋᠃<ref>Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01</ref><br />
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<math>(x+yi)^2=x^2-y^2 + 2xyi.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> <span style="writing-mode: horizontal-tb;"><i>(with b ≠ 0)</i></span> ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ <br />
<math> \pm (\gamma + \delta i)</math> ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄<br />
<math>\gamma = \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠠ <math>\delta = (\sgn b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}}</math> ᠪᠣᠯᠤᠮᠤᠢ᠃<br />
<span style="writing-mode: horizontal-tb;"><i>sgn B</i></span> ᠭᠡᠭᠴᠢ ᠢᠨᠦ <math>\sgn B = \begin{cases} \ \ 1, & B > 0 \\ \ \ 0, & B = 0 \\ -1, & B < 0 \end{cases}</math> ᠭᠡᠵᠦ ᠲᠠᠢᠢᠯᠠᠭᠳᠠᠨᠠ᠃ <br />
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ᠡᠭᠦᠨ ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ ᠢᠢ ᠲᠤᠬᠠᠢᠢᠲᠠ ᠳ᠋ᠤ <math> \pm (\gamma + \delta i)</math> ᠲᠣᠭᠠ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠦ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠲᠣᠭᠠ ᠢᠢ ᠭᠠᠷᠭᠠᠵᠤ ᠢᠷᠡᠬᠦ ᠶᠣᠰᠤᠲᠠᠢ᠃ <br />
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==ᠬᠡᠷᠡᠭᠯᠡᠭᠡ==<br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==</div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%BA%E1%A0%A3%E1%A0%AE%E1%A0%AB%E1%A0%8A%E1%A0%AF%E1%A0%A7%E1%A0%BA%E1%A0%B0_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%A0&diff=2727
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-09-01T01:00:49Z
<p>Angka: Edit</p>
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'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠯᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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__TOC__<br />
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== ᠳᠦᠷᠡᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠳ᠋ᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠳ᠋ᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
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<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
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= ᠬᠠᠰᠤᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠳᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
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<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math><br />
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= ᠦᠷᠡᠵᠢᠬᠦ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃<ref>''Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И.'' Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — <abbr>М.</abbr>: Просвещение, 1998. — 288 с. — <nowiki>ISBN 5-09-008036-4</nowiki>.</ref> <br />
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<math>(a + bi) (c + di) = </math> <math>= ac + bci - bd + adi = </math> <math>= (ac - bd) + (bc + ad)i </math> <br />
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= ᠬᠤᠪᠢᠶᠠᠬᠤ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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<math>\frac{a+bi}{c+di}=</math> <br />
<math>=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=</math> <br />
<math>=\frac{ac+bd}{c^2+d^2}+\left(\frac{bc-ad}{c^2+d^2}\right)i.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ==<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨᠡ᠋᠃<ref>Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01</ref><br />
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<math>(x+yi)^2=x^2-y^2 + 2xyi.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ==<br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==</div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%BA%E1%A0%A3%E1%A0%AE%E1%A0%AB%E1%A0%8A%E1%A0%AF%E1%A0%A7%E1%A0%BA%E1%A0%B0_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%A0&diff=2726
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-08-18T09:49:24Z
<p>Angka: Edit</p>
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<div><div style="max-height: 80vh; writing-mode: vertical-lr; font-family: 'Noto Sans Mongolian'"><br />
'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠯᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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__TOC__<br />
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== ᠳᠦᠷᠡᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠳ᠋ᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠳ᠋ᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
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<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
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= ᠬᠠᠰᠤᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠳᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
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<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math><br />
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= ᠦᠷᠡᠵᠢᠬᠦ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃<ref>''Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И.'' Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — <abbr>М.</abbr>: Просвещение, 1998. — 288 с. — <nowiki>ISBN 5-09-008036-4</nowiki>.</ref> <br />
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<math>(a + bi) (c + di) = </math> <math>= ac + bci - bd + adi = </math> <math>= (ac - bd) + (bc + ad)i </math> <br />
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= ᠬᠤᠪᠢᠶᠠᠬᠤ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
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<math>\frac{a+bi}{c+di}=</math> <br />
<math>=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=</math> <br />
<math>=\frac{ac+bd}{c^2+d^2}+\left(\frac{bc-ad}{c^2+d^2}\right)i.</math><br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ==<br />
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==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ==<br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==</div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%BA%E1%A0%A3%E1%A0%AE%E1%A0%AB%E1%A0%8A%E1%A0%AF%E1%A0%A7%E1%A0%BA%E1%A0%B0_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%A0&diff=2725
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-08-18T09:37:36Z
<p>Angka: Edit</p>
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<div><div style="max-height: 80vh; writing-mode: vertical-lr; font-family: 'Noto Sans Mongolian'"><br />
'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠯᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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__TOC__<br />
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== ᠳᠦᠷᠡᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠳ᠋ᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠳ᠋ᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
<br />
<br />
<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
<br />
<br />
<br />
<br />
= ᠬᠠᠰᠤᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠳᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
<br />
<br />
<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math><br />
<br />
<br />
= ᠦᠷᠡᠵᠢᠬᠦ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃<ref>''Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И.'' Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — <abbr>М.</abbr>: Просвещение, 1998. — 288 с. — <nowiki>ISBN 5-09-008036-4</nowiki>.</ref> <br />
<br />
<br />
<math>(a + bi) (c + di) = </math> <math>= ac + bci - bd + adi = </math> <math>= (ac - bd) + (bc + ad)i </math> <br />
<br />
<br />
= ᠬᠤᠪᠢᠶᠠᠬᠤ =<br />
ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ <br />
<br />
<br />
<math>\frac{a+bi}{c+di}=</math> <br />
<math>=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=</math> <br />
<math>=\frac{ac+bd}{c^2+d^2}+\left(\frac{bc-ad}{c^2+d^2}\right)i.</math></div>
Angka
https://wikibilig.mn/index.php?title=%D0%9D%D2%AF%D2%AF%D1%80_%D1%85%D1%83%D1%83%D0%B4%D0%B0%D1%81&diff=2724
Нүүр хуудас
2021-08-17T12:53:30Z
<p>Angka: Expanded the list</p>
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<div><span style="writing-mode: vertical-lr; ">᠀᠋</span><br />
[[File:HomePage.JPG|center]]<br />
<div style="text-align:center"><br />
Encyclopædia Mongolica Grandis : Большая монгольская энциклопедия : <br><br />
༄།།ཆེན་པོ་ཧོར་གྱི་ཤེས་བྱ་ཀུན་བཏུས་ཚིག་མཛོད་ཆེན་མོ་བཞུགས་སོ།། : Ұлы Моңғол үлкен энциклопедиясы.<br />
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type=create<br />
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break=no<br />
buttonlabel=Шинэ нийтлэл үүсгэх<br />
default=(ᠨᠡᠢᠲᠡᠯᠡᠯ ᠊ᠦᠨ ᠨᠡᠷ ᠊ᠡ᠋)<br />
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<div style="text-align:center"><br />
[https://wikibilig.mn/index.php?title=%E1%A0%B0%E1%A0%A2%E1%A0%A8_%E2%80%8D%E1%A0%A1%E1%A0%8B_%E1%A0%A8%E1%A0%A1%E1%A0%A2%E1%A0%A2%E1%A0%B2%E1%A0%A1%E1%A0%AF%E1%A0%A1%E1%A0%AF&veaction=edit <span style="color:Blue">'''Шинэ нийтлэл бичих'''] [https://www.youtube.com/watch?v=naD5wFtDDEQ&t=1s <span style="color:Blue">'''Шинэ нийтлэл бичих заавар''']<br />
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{{Mongolian}}<big>ᠨᠡᠢᠢᠲᠡᠯᠡᠯ ᠦᠳ</big><br><br><br />
{{Mongolian}}<br />
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[[ᠤᠰᠤᠨ ᠊ᠤ ᠬᠠᠨ ᠭᠦᠷᠪᠡᠯ]]<br><br />
[[ᠮᠦᠯᠲ᠋ᠢᠯᠢᠩ ᠤᠲᠠᠰᠤᠨ ᠤ ᠭᠠᠷ|ᠤᠲᠠᠰᠤᠨ ᠤ ᠮᠦᠯᠲ᠋ᠢᠯᠢᠩ 0 ᠭᠠᠷ]]<br><br />
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[[ᠨᠡᠭᠦᠷᠡᠰᠦ ᠬᠡᠶᠢᠵᠢᠭᠦᠯᠬᠦ ᠲᠧᠭᠨᠣᠯᠣᠭᠢ]]<br><br />
[[ᠨᠢᠩᠵᠢᠪᠠᠳᠭᠠᠷ᠂ ᠰᠠᠩᠵᠢᠮᠢᠲᠤᠪ ᠊ᠤᠨ]]<br><br />
[[ᠨᠢᠮᠪᠡᠭᠡ ᠶ᠋ᠢᠨ ᠬᠦᠴᠢᠯ ᠦᠨ ᠮᠥᠴᠡᠯᠭᠡ]]<br><br />
[[ᠨᠤᠭᠤ]]<br><br />
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[[Ньютоны хөдөлгөөний хуулиуд|ᠨᠧᠸᠲᠣᠨ ᠊ᠤ ᠬᠥᠳᠡᠯᠭᠡᠭᠡᠨ ᠊ᠦ ᠬᠠᠤᠯᠢ ᠨᠤᠭᠤᠳ]]<br><br />
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[[ᠪᠠᠷᠰᠪᠣᠯᠤᠳ᠂ ᠷᠢᠨᠴᠡᠨ ᠊ᠦ]]<br><br />
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[[ᠬᠣᠶᠠᠷᠲᠤ ᠶᠢᠨ ᠠᠯᠭᠧᠪᠷᠠ|ᠪᠦᠦᠯᠢ ᠢᠢᠨ ᠠᠯᠭᠧᠪᠷᠠ]]<br><br />
[[ᠪᠯᠠᠻᠫᠢᠨ᠍ᠻ]]<br><br />
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[[ᠬᠠᠪᠤᠩ]]<br><br />
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[[Хархорины усан цахилгаан станц|ᠬᠠᠷᠠᠬᠣᠷᠢᠨ ᠣ ᠣᠰᠣᠨ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠰᠲ᠋ᠠᠨ᠍ᠴᠢ]]<br><br />
[[Хархорины усжуулах байгууламж|ᠬᠠᠷᠠᠬᠣᠷᠢᠨ ᠣ ᠣᠰᠣᠵᠢᠭᠣᠯᠬᠣ ᠪᠠᠢᠢᠭᠤᠯᠤᠮᠵᠢ]]<br><br />
[[Харанхуй матери|ᠬᠠᠷᠠᠩᠭᠤᠶ ᠮᠠᠲ᠌ᠧᠷᠢ]]<br><br />
[[ᠬᠡᠦᠬᠡᠳ ᠊ᠦᠨ ᠡᠷᠬᠡ ᠊ᠢᠢᠨ ᠲᠤᠬᠠᠢ ᠻᠣᠨᠸᠧᠨᠼ]]<br><br />
[[Хэлмэгдэгсдийн хөшөө|ᠬᠡᠯᠮᠡᠭᠳᠡᠭᠰᠡᠳ ᠊ᠦᠨ ᠬᠥᠰᠢᠶ ᠊ᠡ᠋]]<br><br />
[[ᠬᠢᠮᠢ ᠊ᠢᠢᠨ ᠬᠥᠳᠡᠯᠭᠡᠭᠦᠷᠲᠦ ᠫᠤᠤᠵᠢᠩ]]<br><br />
[[ᠬᠢᠶᠠᠭᠲᠤ ᠢᠢᠨ ᠭᠡᠷ ᠡ᠋ ᠋(1915)]]<br><br />
[[ᠬᠣᠭᠣᠰᠤᠨ ᠴᠢᠨᠠᠷ]]<br><br />
[[Хоёртын тооллын систем|ᠬᠣᠶᠠᠷᠲᠣ ᠊ᠢᠢᠨ ᠲᠣᠭᠠᠯᠠᠯ ᠊ᠤᠨ ᠰᠢᠰᠲ᠋ᠡᠮ]]<br><br />
[[ᠬᠤᠪᠢᠰᠤᠯ ᠊ᠦᠨ ᠣᠨᠣᠯ]]<br><br />
[[Хутагтын дөрвөн үнэн|ᠬᠣᠲᠣᠬᠲᠣ ᠊ᠢᠢᠨ ᠳᠥᠷᠪᠡᠨ ᠦᠨᠡᠨ]]<br><br />
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[[ᠬᠤᠷᠢᠴᠠᠯ]]<br><br />
[[ᠬᠦᠴᠢᠯ]]<br><br />
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[[Монголын газрын гадаргын геологийн түүх|ᠮᠣᠩᠭᠣᠯ ᠊ᠤᠨ ᠭᠠᠵᠠᠷ ᠊ᠤᠨ ᠭᠠᠳᠠᠷᠭᠤ ᠊ᠢᠢᠨ ᠭᠧᠣᠯᠣᠭᠢ ᠊ᠢᠢᠨ ᠲᠡᠦᠬᠡ]]<br><br />
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[[Монголын түүхийн цаг тоолол|ᠮᠣᠩᠭᠣᠯ ᠊ᠤᠨ ᠲᠡᠦᠬᠡ ᠊ᠢᠢᠨ ᠴᠠᠭ ᠲᠣᠭᠠᠯᠠᠯ]]<br><br />
[[ᠮᠣᠩᠭᠤᠯ ᠊ᠤᠨ ᠴᠠᠭᠳᠠᠭ ᠊ᠠ᠋]]<br><br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-08-17T12:49:52Z
<p>Angka: Edit</p>
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<div><div style="max-height: 80vh; writing-mode: vertical-lr; font-family: 'Noto Sans Mongolian'"><br />
'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠯᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠡᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠳ᠋ᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠳ᠋ᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃<ref name=":0">ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17</ref> <br />
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<math>(a + bi) + (c + di) = </math> <math>= (a + c) + (b + d)i </math><br />
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= ᠬᠠᠰᠠᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠳᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃<ref name=":0" /> <br />
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<math>(a + bi) - (c + di) = </math> <math>= (a - c) + (b - d)i </math></div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%BA%E1%A0%A3%E1%A0%AE%E1%A0%AB%E1%A0%8A%E1%A0%AF%E1%A0%A7%E1%A0%BA%E1%A0%B0_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%A0&diff=2722
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-08-17T12:38:20Z
<p>Angka: Edit</p>
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<div><div style="max-height: 80vh; writing-mode: vertical-lr; font-family: 'Noto Sans Mongolian'"><br />
'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠯᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠡᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠳ᠋ᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠳ᠋ᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃<br />
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== ᠦᠢᠯᠡᠳᠦᠯ ==<br />
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= ᠨᠡᠮᠡᠬᠦ =<br />
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃ <br />
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= ᠬᠠᠰᠠᠬᠤ =<br />
ᠨᠢᠭᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠳᠡ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃</div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%AC%E1%A0%A0%E1%A0%AD%E1%A0%A4%E1%A0%B7%E1%A0%AE%E1%A0%A0%E1%A0%AD_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%8A%E1%A0%A0%E1%A0%8B&diff=2721
ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠊ᠠ᠋
2021-08-03T11:12:02Z
<p>Angka: Layout</p>
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<div><div style="max-height: 80vh; writing-mode: vertical-lr; font-family: 'Noto Sans Mongolian'"><br />
ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠢᠨᠦ ᠲᠡᠭ <span style="writing-mode: horizontal-tb;"><i>(0)</i></span> ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ [[ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ]] ᠢᠢ '''ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋''' ᠬᠠᠮᠠᠮᠦᠢ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠺᠸᠠᠲᠷᠠᠲ<sup>quadrate</sup> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠬᠠᠰᠤᠬᠤ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠬᠠᠷᠢᠭᠤ ᠭᠠᠷᠤᠨ ᠠ᠋᠃<ref name=":0">https://www.livescience.com/42748-imaginary-numbers.html Live Science. Elaine J. Hom. What Are Imaginary Numbers? 2014/01/22. </ref> <br />
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ᠲᠡᠭᠡᠬᠦ ᠪᠡᠷ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠠᠨᠤ ᠬᠠᠰᠤᠬᠤ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠶ ᠲᠣᠭᠠᠨ ᠤ ᠬᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠮᠥᠨ᠃ <br />
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ᠲᠡᠭ <span style="writing-mode: horizontal-tb;"><i>(0)</i></span> ᠲᠣᠭ ᠠ᠋ ᠪᠣᠳᠠᠲᠤ ᠪᠤᠶᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠠᠯᠢ ᠨᠢ ᠴᠤ ᠪᠠᠢᠢᠵᠤ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
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__TOC__<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
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{| class="wikitable"<br />
|+<br />
|[[File:Rotations_on_the_complex_plane.svg|thumb]]<br />
|style = "height: 150px;"|ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭᠠ ᠪᠦᠬᠦᠢ ᠲᠣᠭᠠᠴᠠᠭᠠᠯᠠᠯ ᠤᠨ ᠳᠦᠷᠰᠦᠯᠡᠯ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃<br />
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ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠳᠦᠷᠰᠦᠯᠡᠯ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;"><i>–1</i></span> ᠢᠢᠡᠷ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ᠢ ᠲᠣᠭᠤᠷᠢᠭᠰᠠᠨ <span style="writing-mode: horizontal-tb;"><i>180</i></span> ᠬᠡᠮ ᠦᠨ ᠡᠷᠭᠢᠯᠲᠡ ᠪᠣᠯᠤᠨ ᠠ᠋᠃ <br />
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<span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠲᠣᠭ ᠠ᠋ ᠪᠡᠷ ᠦᠷᠡᠵᠢᠯᠭᠦᠯᠬᠦ ᠢᠨᠦ ᠨᠠᠷᠠ ᠪᠤᠷᠤᠭᠤ “ᠡᠶᠡᠷᠭᠦ” ᠴᠢᠭ᠌ ᠲᠦ <span style="writing-mode: horizontal-tb;"><i>90</i></span> ᠬᠡᠮ ᠦᠨ ᠡᠷᠭᠢᠯᠲᠡ ᠪᠣᠯᠬᠤ ᠪᠠ ᠬᠡᠷᠪᠡ <span style="writing-mode: horizontal-tb;"><i>i2 = -1</i></span> ᠭᠡᠰᠡᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢᠨᠦ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ᠢ ᠲᠣᠭᠤᠷᠢᠭᠰᠠᠨ <span style="writing-mode: horizontal-tb;"><i>90</i></span> ᠬᠡᠮ ᠦᠨ ᠬᠣᠶᠠᠷ ᠡᠷᠭᠢᠯᠲᠡ ᠭᠡᠵᠦ ᠦᠵᠡᠪᠡᠯ ᠦᠷ ᠡ᠋ ᠳ᠋ᠦ᠋ᠩ ᠢᠨᠦ ᠭᠠᠭᠴᠠ <span style="writing-mode: horizontal-tb;"><i>180</i></span> ᠬᠡᠮ ᠦᠨ ᠡᠷᠭᠢᠯᠲᠡ ᠪᠣᠯᠤᠨ ᠠ᠋᠃ <br />
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“ᠰᠥᠷᠭᠦ” (ᠨᠠᠷᠠ ᠵᠥᠪ) ᠴᠢᠭ᠌ ᠲᠦ <span style="writing-mode: horizontal-tb;"><i>90</i></span> ᠬᠡᠮ ᠡᠷᠭᠢᠬᠦ ᠢᠨᠦ ᠮᠥᠨ ᠡᠨᠡ ᠳᠦᠷᠰᠦᠯᠡᠯ ᠳ᠋ᠦ ᠨᠡᠶ᠋ᠢᠴᠡᠬᠦ ᠪᠠ <span style="writing-mode: horizontal-tb;"><i>x2 = -1</i></span> ᠭᠡᠰᠡᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠶ᠋ᠢᠳ <span style="writing-mode: horizontal-tb;"><i>−i</i></span> ᠪᠣᠯᠬᠤ ᠢ ᠬᠠᠷᠠᠭᠤᠯᠵᠤ ᠪᠠᠢᠢᠨ ᠠ᠋᠃ <br />
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ᠶᠡᠷᠦᠩᠬᠡᠢᠢ ᠳ᠋ᠦ ᠪᠡᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ ᠦᠷᠡᠵᠢᠯᠭᠦᠯᠬᠦ ᠢᠨᠦ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ᠢ ᠲᠣᠭᠤᠷᠢᠭᠤᠯᠤᠨ ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭᠠᠨ ᠤ ᠠᠷᠭᠤᠮᠧᠨ᠍ᠲ<sup>argument</sup> ᠢᠢᠠᠷ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ᠂ ᠴᠠᠷ ᠬᠡᠮᠵᠢᠶ ᠡ᠋ ᠪᠡᠷ ᠢᠨᠦ ᠤᠷᠲᠤ ᠢᠢ ᠴᠢᠴᠠᠤᠯᠠᠬᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠪᠣᠯᠤᠨᠠ᠃ <br />
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ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭᠠᠨ ᠤ ᠠᠷᠭᠤᠮᠧᠨ᠍ᠲ ᠭᠡᠳᠡᠭ ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠡᠬᠢᠯᠠᠯ ᠴᠡᠭ ᠢ ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭᠠ z ᠲᠡᠢ ᠬᠣᠯᠪᠤᠵᠤ ᠪᠠᠢᠢᠭᠠ ᠱᠤᠭᠤᠮ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠢᠢᠨ ᠥᠨᠴᠦᠭ <span style="writing-mode: horizontal-tb;"><i>φ</i></span> ᠶᠤᠮ᠃<br />
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{| class="wikitable"<br />
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|[[File:Complex_number_illustration_modarg.svg|thumb]]<br />
|style = "height: 150px;"|ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭᠠ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ<sup>complex</sup> ᠲᠣᠭᠠ ᠢᠢᠨ ᠠᠷᠭᠤᠮᠧᠨ᠍ᠲ<sup>argument</sup> ᠨᠢ ᠥᠨᠴᠦᠭ <span style="writing-mode: horizontal-tb;"><i>φ</i></span>᠃<br />
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==ᠬᠡᠷᠡᠭᠯᠡᠭᠡ==<br />
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[[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠬᠦ ᠠᠭᠠᠳ ᠧᠯᠧᠺᠲ᠋ᠷᠣᠨᠢᠭ᠌ ᠲᠦ<sup>electronic</sup> <span style="writing-mode: horizontal-tb;"><i>j</i></span> ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠨ ᠡ᠋᠃ <br />
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ᠡᠨᠡ ᠢᠨᠦ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠳᠡᠭ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠭᠦᠶᠦᠳᠡᠯ ᠡᠴᠡ ᠢᠯᠭᠠᠬᠤ ᠱᠠᠭᠠᠷᠳᠠᠯᠭ ᠠ᠋ ᠡᠴᠡ ᠤᠷᠭᠤᠨ ᠭᠠᠷᠤᠭᠰᠠᠨ ᠤᠯᠠᠮᠵᠢᠯᠠᠯ ᠶᠤᠮ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠧᠯᠧᠺᠲ᠋ᠷᠣᠨᠢᠭ᠌ ᠲᠦ᠂ ᠢᠯᠠᠩᠭᠤᠶ ᠠ᠋ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠡᠯ ᠳ᠋ᠦ ᠣᠨᠴᠤᠭᠤᠢ ᠠᠴᠢ ᠬᠣᠯᠪᠤᠭᠳᠠᠯ ᠲᠠᠢ᠃ <br />
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ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠡᠯ ᠰᠢᠨᠦ᠋ᠰ<sup>sinus</sup> ᠳᠣᠯᠭᠢᠶᠠᠨ ᠳᠤ ᠨᠡᠮᠡᠬᠦ ᠪᠠ ᠬᠠᠰᠤᠬᠤ ᠢᠢᠨ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠰᠣᠯᠢᠭᠳᠠᠳᠠᠭ᠃<ref>https://www.electricaltechnology.org/2013/07/active-reactive-apparent-and-complex.html Electrical Technology. Active, Reactive, Apparent and Complex Power. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/07/31</ref><br />
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{| class="wikitable"<br />
|+<br />
|[[File:Cmplxpower.svg|thumb]]<br />
|style = "height: 150px;"|ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠡᠯ ᠪᠦᠬᠦᠢ ᠡᠷᠴᠢᠮ ᠬᠦᠴᠦ᠃ <span style="writing-mode: horizontal-tb;"><i>P</i></span> ᠢᠨᠦ ᠢᠳᠡᠪᠬᠢᠲᠦ ᠴᠢᠳᠠᠯ᠂ <span style="writing-mode: horizontal-tb;"><i>Q</i></span> ᠢᠨᠦ ᠬᠠᠷᠢᠭᠤ ᠢᠳᠡᠪᠬᠢᠲᠦ ᠴᠢᠳᠠᠯ᠂ <span style="writing-mode: horizontal-tb;"><i>S</i></span> ᠨᠡᠶ᠋ᠢᠯᠡᠪᠦᠷᠢ ᠴᠢᠳᠠᠯ᠃<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠢ ᠶᠠᠪ ᠴᠠᠪ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠬᠤ ᠬᠡᠴᠡᠭᠦᠦ ᠲᠤᠯᠠ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠡᠯ ᠦᠳ ᠢ ᠳᠠᠪᠬᠤᠷᠯᠠᠬᠤ ᠨᠡᠨ ᠲᠦᠪᠡᠭ ᠲᠡᠢ᠃<br />
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ᠬᠠᠷᠢᠨ ᠲᠥᠰᠦᠭᠡᠯᠡᠯ ᠭᠦᠶᠦᠳᠡᠯ ᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠨᠤᠭᠤᠳ ᠲᠠᠶ ᠬᠣᠣᠰᠮᠠᠭᠯᠠᠭᠤᠯᠤᠭᠰᠠᠨ ᠢᠢᠠᠷ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠡᠯ ᠦᠨ ᠲᠣᠭᠠᠴᠠᠭ ᠠ᠋ ᠬᠢᠵᠦ᠂ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠤ ᠣᠰᠤᠯ ᠡᠴᠡ ᠤᠷᠢᠳᠤᠴᠢᠯᠠᠨ ᠰᠡᠷᠭᠡᠢᠢᠯᠡᠬᠦ ᠲᠤᠰᠠ ᠲᠠᠢ ᠪᠠᠢᠢᠳᠠᠭ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠮᠥᠨ ᠰᠢᠭ᠌ᠨᠠᠯ<sup>signal</sup> ᠪᠣᠯᠪᠠᠰᠤᠷᠠᠭᠤᠯᠬᠤ ᠲᠧᠺᠨᠣᠯᠣᠭᠢ ᠳ᠋ᠤ<sup>technology</sup> ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ <br />
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ᠡᠨᠡ ᠨᠢ ᠡᠭᠦᠷ ᠦᠨ ᠬᠣᠯᠪᠤᠭ ᠠ᠋᠂ ᠤᠲᠠᠰᠤ ᠦᠭᠡᠢ ᠳᠠᠮᠵᠢᠭᠤᠯᠤᠯᠲᠠ᠂ ᠷᠠᠳᠠᠷ<sup>radar</sup>᠂ ᠪᠢᠣᠯᠣᠭᠢ<sup>biology</sup> (ᠲᠠᠷᠢᠬᠢᠨ ᠤ ᠳᠣᠯᠭᠢᠶᠠᠨ) ᠰᠤᠳᠤᠯᠬᠤ ᠳ᠋ᠤ ᠬᠠᠮᠢᠶᠠᠷᠤᠨ ᠠ᠋᠃<br />
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ᠶᠡᠷᠦ᠂ ᠰᠢᠨᠦ᠋ᠰ ᠪᠠ ᠺᠣᠰᠢᠨᠦ᠋ᠰ<sup>cosinus</sup> ᠳᠣᠯᠭᠢᠶᠠᠨ ᠪᠦᠬᠦᠢ ᠵᠦᠢᠯ ᠢ ᠬᠡᠮᠵᠢᠬᠦ᠂ ᠲᠣᠭᠠᠴᠠᠭᠠᠯᠬᠤ ᠳ᠋ᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠴᠢᠬᠤᠯᠠ ᠬᠡᠷᠡᠭ ᠲᠡᠢ᠃<ref name=":0" /><br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠ - real number - действительное число<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ - imaginary unit - мнимая единица <br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭᠠ - imaginary number - чисто мнимое число<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ - complex number - комплексное число<br />
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ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ - real part - действительная часть<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ - imaginaty part - мнимая часть<br />
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ᠲᠡᠩᠬᠡᠯᠢᠭ - axis - ось<br />
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ᠣᠯᠠᠨᠯᠢᠭ - set - множество<br />
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ᠡᠶᠡᠷᠭᠦ - positive - положительный<br />
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ᠰᠥᠷᠭᠦ - negative - отрицательный<br />
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ᠲᠡᠭᠰᠢᠳᠬᠡᠯ - equation - уравнение<br />
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ᠡᠷᠭᠢᠯᠲᠡ - rotation - поворот<br />
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ᠬᠡᠮ - degree - градус<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==<br />
<div style="writing-mode: horizontal-tb; "><References/></div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%AC%E1%A0%A0%E1%A0%AD%E1%A0%A4%E1%A0%B7%E1%A0%AE%E1%A0%A0%E1%A0%AD_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%8A%E1%A0%A0%E1%A0%8B&diff=2720
ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠊ᠠ᠋
2021-08-03T11:09:52Z
<p>Angka: Edit</p>
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ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠢᠨᠦ ᠲᠡᠭ <span style="writing-mode: horizontal-tb;"><i>(0)</i></span> ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ [[ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ]] ᠢᠢ '''ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋''' ᠬᠠᠮᠠᠮᠦᠢ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠺᠸᠠᠲᠷᠠᠲ<sup>quadrate</sup> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠬᠠᠰᠤᠬᠤ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠬᠠᠷᠢᠭᠤ ᠭᠠᠷᠤᠨ ᠠ᠋᠃<ref name=":0">https://www.livescience.com/42748-imaginary-numbers.html Live Science. Elaine J. Hom. What Are Imaginary Numbers? 2014/01/22. </ref> <br />
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ᠲᠡᠭᠡᠬᠦ ᠪᠡᠷ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠠᠨᠤ ᠬᠠᠰᠤᠬᠤ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠶ ᠲᠣᠭᠠᠨ ᠤ ᠬᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠮᠥᠨ᠃ <br />
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ᠲᠡᠭ <span style="writing-mode: horizontal-tb;"><i>(0)</i></span> ᠲᠣᠭ ᠠ᠋ ᠪᠣᠳᠠᠲᠤ ᠪᠤᠶᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠠᠯᠢ ᠨᠢ ᠴᠤ ᠪᠠᠢᠢᠵᠤ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
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{| class="wikitable"<br />
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|[[File:Rotations_on_the_complex_plane.svg|thumb]]<br />
|style = "height: 150px;"|ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭᠠ ᠪᠦᠬᠦᠢ ᠲᠣᠭᠠᠴᠠᠭᠠᠯᠠᠯ ᠤᠨ ᠳᠦᠷᠰᠦᠯᠡᠯ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃<br />
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ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠳᠦᠷᠰᠦᠯᠡᠯ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;"><i>–1</i></span> ᠢᠢᠡᠷ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ᠢ ᠲᠣᠭᠤᠷᠢᠭᠰᠠᠨ <span style="writing-mode: horizontal-tb;"><i>180</i></span> ᠬᠡᠮ ᠦᠨ ᠡᠷᠭᠢᠯᠲᠡ ᠪᠣᠯᠤᠨ ᠠ᠋᠃ <br />
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<span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠲᠣᠭ ᠠ᠋ ᠪᠡᠷ ᠦᠷᠡᠵᠢᠯᠭᠦᠯᠬᠦ ᠢᠨᠦ ᠨᠠᠷᠠ ᠪᠤᠷᠤᠭᠤ “ᠡᠶᠡᠷᠭᠦ” ᠴᠢᠭ᠌ ᠲᠦ <span style="writing-mode: horizontal-tb;"><i>90</i></span> ᠬᠡᠮ ᠦᠨ ᠡᠷᠭᠢᠯᠲᠡ ᠪᠣᠯᠬᠤ ᠪᠠ ᠬᠡᠷᠪᠡ <span style="writing-mode: horizontal-tb;"><i>i2 = -1</i></span> ᠭᠡᠰᠡᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢᠨᠦ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ᠢ ᠲᠣᠭᠤᠷᠢᠭᠰᠠᠨ <span style="writing-mode: horizontal-tb;"><i>90</i></span> ᠬᠡᠮ ᠦᠨ ᠬᠣᠶᠠᠷ ᠡᠷᠭᠢᠯᠲᠡ ᠭᠡᠵᠦ ᠦᠵᠡᠪᠡᠯ ᠦᠷ ᠡ᠋ ᠳ᠋ᠦ᠋ᠩ ᠢᠨᠦ ᠭᠠᠭᠴᠠ <span style="writing-mode: horizontal-tb;"><i>180</i></span> ᠬᠡᠮ ᠦᠨ ᠡᠷᠭᠢᠯᠲᠡ ᠪᠣᠯᠤᠨ ᠠ᠋᠃ <br />
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“ᠰᠥᠷᠭᠦ” (ᠨᠠᠷᠠ ᠵᠥᠪ) ᠴᠢᠭ᠌ ᠲᠦ <span style="writing-mode: horizontal-tb;"><i>90</i></span> ᠬᠡᠮ ᠡᠷᠭᠢᠬᠦ ᠢᠨᠦ ᠮᠥᠨ ᠡᠨᠡ ᠳᠦᠷᠰᠦᠯᠡᠯ ᠳ᠋ᠦ ᠨᠡᠶ᠋ᠢᠴᠡᠬᠦ ᠪᠠ <span style="writing-mode: horizontal-tb;"><i>x2 = -1</i></span> ᠭᠡᠰᠡᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠶ᠋ᠢᠳ <span style="writing-mode: horizontal-tb;"><i>−i</i></span> ᠪᠣᠯᠬᠤ ᠢ ᠬᠠᠷᠠᠭᠤᠯᠵᠤ ᠪᠠᠢᠢᠨ ᠠ᠋᠃ <br />
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ᠶᠡᠷᠦᠩᠬᠡᠢᠢ ᠳ᠋ᠦ ᠪᠡᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ ᠦᠷᠡᠵᠢᠯᠭᠦᠯᠬᠦ ᠢᠨᠦ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ᠢ ᠲᠣᠭᠤᠷᠢᠭᠤᠯᠤᠨ ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭᠠᠨ ᠤ ᠠᠷᠭᠤᠮᠧᠨ᠍ᠲ<sup>argument</sup> ᠢᠢᠠᠷ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ᠂ ᠴᠠᠷ ᠬᠡᠮᠵᠢᠶ ᠡ᠋ ᠪᠡᠷ ᠢᠨᠦ ᠤᠷᠲᠤ ᠢᠢ ᠴᠢᠴᠠᠤᠯᠠᠬᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠪᠣᠯᠤᠨᠠ᠃ <br />
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ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭᠠᠨ ᠤ ᠠᠷᠭᠤᠮᠧᠨ᠍ᠲ ᠭᠡᠳᠡᠭ ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠡᠬᠢᠯᠠᠯ ᠴᠡᠭ ᠢ ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭᠠ z ᠲᠡᠢ ᠬᠣᠯᠪᠤᠵᠤ ᠪᠠᠢᠢᠭᠠ ᠱᠤᠭᠤᠮ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠢᠢᠨ ᠥᠨᠴᠦᠭ <span style="writing-mode: horizontal-tb;"><i>φ</i></span> ᠶᠤᠮ᠃<br />
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{| class="wikitable"<br />
|+<br />
|[[File:Complex_number_illustration_modarg.svg|thumb]]<br />
|style = "height: 150px;"|ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭᠠ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ<sup>complex</sup> ᠲᠣᠭᠠ ᠢᠢᠨ ᠠᠷᠭᠤᠮᠧᠨ᠍ᠲ<sup>argument</sup> ᠨᠢ ᠥᠨᠴᠦᠭ <span style="writing-mode: horizontal-tb;"><i>φ</i></span>᠃<br />
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==ᠬᠡᠷᠡᠭᠯᠡᠭᠡ==<br />
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[[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠬᠦ ᠠᠭᠠᠳ ᠧᠯᠧᠺᠲ᠋ᠷᠣᠨᠢᠭ᠌ ᠲᠦ<sup>electronic</sup> <span style="writing-mode: horizontal-tb;"><i>j</i></span> ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠨ ᠡ᠋᠃ <br />
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ᠡᠨᠡ ᠢᠨᠦ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠳᠡᠭ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠭᠦᠶᠦᠳᠡᠯ ᠡᠴᠡ ᠢᠯᠭᠠᠬᠤ ᠱᠠᠭᠠᠷᠳᠠᠯᠭ ᠠ᠋ ᠡᠴᠡ ᠤᠷᠭᠤᠨ ᠭᠠᠷᠤᠭᠰᠠᠨ ᠤᠯᠠᠮᠵᠢᠯᠠᠯ ᠶᠤᠮ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠧᠯᠧᠺᠲ᠋ᠷᠣᠨᠢᠭ᠌ ᠲᠦ᠂ ᠢᠯᠠᠩᠭᠤᠶ ᠠ᠋ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠡᠯ ᠳ᠋ᠦ ᠣᠨᠴᠤᠭᠤᠢ ᠠᠴᠢ ᠬᠣᠯᠪᠤᠭᠳᠠᠯ ᠲᠠᠢ᠃ <br />
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ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠡᠯ ᠰᠢᠨᠦ᠋ᠰ<sup>sinus</sup> ᠳᠣᠯᠭᠢᠶᠠᠨ ᠳᠤ ᠨᠡᠮᠡᠬᠦ ᠪᠠ ᠬᠠᠰᠤᠬᠤ ᠢᠢᠨ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠰᠣᠯᠢᠭᠳᠠᠳᠠᠭ᠃<ref>https://www.electricaltechnology.org/2013/07/active-reactive-apparent-and-complex.html Electrical Technology. Active, Reactive, Apparent and Complex Power. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/07/31</ref><br />
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{| class="wikitable"<br />
|+<br />
|[[File:Cmplxpower.svg|thumb]]<br />
|style = "height: 150px;"|ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠡᠯ ᠪᠦᠬᠦᠢ ᠡᠷᠴᠢᠮ ᠬᠦᠴᠦ᠃ <span style="writing-mode: horizontal-tb;"><i>P</i></span> ᠢᠨᠦ ᠢᠳᠡᠪᠬᠢᠲᠦ ᠴᠢᠳᠠᠯ᠂ <span style="writing-mode: horizontal-tb;"><i>Q</i></span> ᠢᠨᠦ ᠬᠠᠷᠢᠭᠤ ᠢᠳᠡᠪᠬᠢᠲᠦ ᠴᠢᠳᠠᠯ᠂ <span style="writing-mode: horizontal-tb;"><i>S</i></span> ᠨᠡᠶ᠋ᠢᠯᠡᠪᠦᠷᠢ ᠴᠢᠳᠠᠯ᠃<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠢ ᠶᠠᠪ ᠴᠠᠪ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠬᠤ ᠬᠡᠴᠡᠭᠦᠦ ᠲᠤᠯᠠ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠡᠯ ᠦᠳ ᠢ ᠳᠠᠪᠬᠤᠷᠯᠠᠬᠤ ᠨᠡᠨ ᠲᠦᠪᠡᠭ ᠲᠡᠢ᠃<br />
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ᠬᠠᠷᠢᠨ ᠲᠥᠰᠦᠭᠡᠯᠡᠯ ᠭᠦᠶᠦᠳᠡᠯ ᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠨᠤᠭᠤᠳ ᠲᠠᠶ ᠬᠣᠣᠰᠮᠠᠭᠯᠠᠭᠤᠯᠤᠭᠰᠠᠨ ᠢᠢᠠᠷ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠡᠯ ᠦᠨ ᠲᠣᠭᠠᠴᠠᠭ ᠠ᠋ ᠬᠢᠵᠦ᠂ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠤ ᠣᠰᠤᠯ ᠡᠴᠡ ᠤᠷᠢᠳᠤᠴᠢᠯᠠᠨ ᠰᠡᠷᠭᠡᠢᠢᠯᠡᠬᠦ ᠲᠤᠰᠠ ᠲᠠᠢ ᠪᠠᠢᠢᠳᠠᠭ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠮᠥᠨ ᠰᠢᠭ᠌ᠨᠠᠯ<sup>signal</sup> ᠪᠣᠯᠪᠠᠰᠤᠷᠠᠭᠤᠯᠬᠤ ᠲᠧᠺᠨᠣᠯᠣᠭᠢ ᠳ᠋ᠤ<sup>technology</sup> ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ <br />
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ᠡᠨᠡ ᠨᠢ ᠡᠭᠦᠷ ᠦᠨ ᠬᠣᠯᠪᠤᠭ ᠠ᠋᠂ ᠤᠲᠠᠰᠤ ᠦᠭᠡᠢ ᠳᠠᠮᠵᠢᠭᠤᠯᠤᠯᠲᠠ᠂ ᠷᠠᠳᠠᠷ<sup>radar</sup>᠂ ᠪᠢᠣᠯᠣᠭᠢ<sup>biology</sup> (ᠲᠠᠷᠢᠬᠢᠨ ᠤ ᠳᠣᠯᠭᠢᠶᠠᠨ) ᠰᠤᠳᠤᠯᠬᠤ ᠳ᠋ᠤ ᠬᠠᠮᠢᠶᠠᠷᠤᠨ ᠠ᠋᠃<br />
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ᠶᠡᠷᠦ᠂ ᠰᠢᠨᠦ᠋ᠰ ᠪᠠ ᠺᠣᠰᠢᠨᠦ᠋ᠰ<sup>cosinus</sup> ᠳᠣᠯᠭᠢᠶᠠᠨ ᠪᠦᠬᠦᠢ ᠵᠦᠢᠯ ᠢ ᠬᠡᠮᠵᠢᠬᠦ᠂ ᠲᠣᠭᠠᠴᠠᠭᠠᠯᠬᠤ ᠳ᠋ᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠴᠢᠬᠤᠯᠠ ᠬᠡᠷᠡᠭ ᠲᠡᠢ᠃<ref name=":0" /><br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠ - real number - действительное число<br />
ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ - imaginary unit - мнимая единица <br />
ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭᠠ - imaginary number - чисто мнимое число<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ - complex number - комплексное число<br />
ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ - real part - действительная часть<br />
ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ - imaginaty part - мнимая часть<br />
ᠲᠡᠩᠬᠡᠯᠢᠭ - axis - ось<br />
ᠣᠯᠠᠨᠯᠢᠭ - set - множество<br />
ᠡᠶᠡᠷᠭᠦ - positive - положительный<br />
ᠰᠥᠷᠭᠦ - negative - отрицательный<br />
ᠲᠡᠭᠰᠢᠳᠬᠡᠯ - equation - уравнение<br />
ᠡᠷᠭᠢᠯᠲᠡ - rotation - поворот<br />
ᠬᠡᠮ - degree - градус<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==<br />
<div style="writing-mode: horizontal-tb; "><References/></div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%AC%E1%A0%A0%E1%A0%AD%E1%A0%A4%E1%A0%B7%E1%A0%AE%E1%A0%A0%E1%A0%AD_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%8A%E1%A0%A0%E1%A0%8B&diff=2719
ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠊ᠠ᠋
2021-08-03T10:49:01Z
<p>Angka: Layout</p>
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<div><div style="max-height: 80vh; writing-mode: vertical-lr; font-family: 'Noto Sans Mongolian'"><br />
ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠢᠨᠦ ᠲᠡᠭ <span style="writing-mode: horizontal-tb;"><i>(0)</i></span> ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ [[ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ]] ᠢᠢ '''ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋''' ᠬᠠᠮᠠᠮᠦᠢ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠺᠸᠠᠲᠷᠠᠲ<sup>quadrate</sup> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠬᠠᠰᠤᠬᠤ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠬᠠᠷᠢᠭᠤ ᠭᠠᠷᠤᠨ ᠠ᠋᠃<ref name=":0">https://www.livescience.com/42748-imaginary-numbers.html Live Science. Elaine J. Hom. What Are Imaginary Numbers? 2014/01/22. </ref> <br />
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ᠲᠡᠭᠡᠬᠦ ᠪᠡᠷ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠠᠨᠤ ᠬᠠᠰᠤᠬᠤ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠶ ᠲᠣᠭᠠᠨ ᠤ ᠬᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠮᠥᠨ᠃ <br />
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ᠲᠡᠭ <span style="writing-mode: horizontal-tb;"><i>(0)</i></span> ᠲᠣᠭ ᠠ᠋ ᠪᠣᠳᠠᠲᠤ ᠪᠤᠶᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠠᠯᠢ ᠨᠢ ᠴᠤ ᠪᠠᠢᠢᠵᠤ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
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{| class="wikitable"<br />
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|[[File:Rotations_on_the_complex_plane.svg|thumb]]<br />
|style = "height: 150px;"|ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭᠠ ᠪᠦᠬᠦᠢ ᠲᠣᠭᠠᠴᠠᠭᠠᠯᠠᠯ ᠤᠨ ᠳᠦᠷᠰᠦᠯᠡᠯ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃<br />
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ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠳᠦᠷᠰᠦᠯᠡᠯ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;"><i>–1</i></span> ᠢᠢᠡᠷ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ᠢ ᠲᠣᠭᠤᠷᠢᠭᠰᠠᠨ <span style="writing-mode: horizontal-tb;"><i>180</i></span> ᠬᠡᠮ ᠦᠨ ᠡᠷᠭᠢᠯᠲᠡ ᠪᠣᠯᠤᠨ ᠠ᠋᠃ <br />
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<span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠲᠣᠭ ᠠ᠋ ᠪᠡᠷ ᠦᠷᠡᠵᠢᠯᠭᠦᠯᠬᠦ ᠢᠨᠦ ᠨᠠᠷᠠ ᠪᠤᠷᠤᠭᠤ “ᠡᠶᠡᠷᠭᠦ” ᠴᠢᠭ᠌ ᠲᠦ <span style="writing-mode: horizontal-tb;"><i>90</i></span> ᠬᠡᠮ ᠦᠨ ᠡᠷᠭᠢᠯᠲᠡ ᠪᠣᠯᠬᠤ ᠪᠠ ᠬᠡᠷᠪᠡ <span style="writing-mode: horizontal-tb;"><i>i2 = -1</i></span> ᠭᠡᠰᠡᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢᠨᠦ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ᠢ ᠲᠣᠭᠤᠷᠢᠭᠰᠠᠨ <span style="writing-mode: horizontal-tb;"><i>90</i></span> ᠬᠡᠮ ᠦᠨ ᠬᠣᠶᠠᠷ ᠡᠷᠭᠢᠯᠲᠡ ᠭᠡᠵᠦ ᠦᠵᠡᠪᠡᠯ ᠦᠷ ᠡ᠋ ᠳ᠋ᠦ᠋ᠩ ᠢᠨᠦ ᠭᠠᠭᠴᠠ <span style="writing-mode: horizontal-tb;"><i>180</i></span> ᠬᠡᠮ ᠦᠨ ᠡᠷᠭᠢᠯᠲᠡ ᠪᠣᠯᠤᠨ ᠠ᠋᠃ <br />
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“ᠰᠥᠷᠭᠦ” (ᠨᠠᠷᠠ ᠵᠥᠪ) ᠴᠢᠭ᠌ ᠲᠦ <span style="writing-mode: horizontal-tb;"><i>90</i></span> ᠬᠡᠮ ᠡᠷᠭᠢᠬᠦ ᠢᠨᠦ ᠮᠥᠨ ᠡᠨᠡ ᠳᠦᠷᠰᠦᠯᠡᠯ ᠳ᠋ᠦ ᠨᠡᠶ᠋ᠢᠴᠡᠬᠦ ᠪᠠ <span style="writing-mode: horizontal-tb;"><i>x2 = -1</i></span> ᠭᠡᠰᠡᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠶ᠋ᠢᠳ <span style="writing-mode: horizontal-tb;"><i>−i</i></span> ᠪᠣᠯᠬᠤ ᠢ ᠬᠠᠷᠠᠭᠤᠯᠵᠤ ᠪᠠᠢᠢᠨ ᠠ᠋᠃ <br />
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ᠶᠡᠷᠦᠩᠬᠡᠢᠢ ᠳ᠋ᠦ ᠪᠡᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ ᠦᠷᠡᠵᠢᠯᠭᠦᠯᠬᠦ ᠢᠨᠦ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ᠢ ᠲᠣᠭᠤᠷᠢᠭᠤᠯᠤᠨ ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭᠠᠨ ᠤ ᠠᠷᠭᠤᠮᠧᠨ᠍ᠲ<sup>argument</sup> ᠢᠢᠠᠷ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ᠂ ᠴᠠᠷ ᠬᠡᠮᠵᠢᠶ ᠡ᠋ ᠪᠡᠷ ᠢᠨᠦ ᠤᠷᠲᠤ ᠢᠢ ᠴᠢᠴᠠᠤᠯᠠᠬᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠪᠣᠯᠤᠨᠠ᠃ <br />
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ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭᠠᠨ ᠤ ᠠᠷᠭᠤᠮᠧᠨ᠍ᠲ ᠭᠡᠳᠡᠭ ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠡᠬᠢᠯᠠᠯ ᠴᠡᠭ ᠢ ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭᠠ z ᠲᠡᠢ ᠬᠣᠯᠪᠤᠵᠤ ᠪᠠᠢᠢᠭᠠ ᠱᠤᠭᠤᠮ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠢᠢᠨ ᠥᠨᠴᠦᠭ <span style="writing-mode: horizontal-tb;"><i>φ</i></span> ᠶᠤᠮ᠃<br />
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{| class="wikitable"<br />
|+<br />
|[[File:Complex_number_illustration_modarg.svg|thumb]]<br />
|style = "height: 150px;"|ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭᠠ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ<sup>complex</sup> ᠲᠣᠭᠠ ᠢᠢᠨ ᠠᠷᠭᠤᠮᠧᠨ᠍ᠲ<sup>argument</sup> ᠨᠢ ᠥᠨᠴᠦᠭ <span style="writing-mode: horizontal-tb;"><i>φ</i></span>᠃<br />
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==ᠬᠡᠷᠡᠭᠯᠡᠭᠡ==<br />
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[[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠬᠦ ᠠᠭᠠᠳ ᠧᠯᠧᠺᠲ᠋ᠷᠣᠨᠢᠭ᠌ ᠲᠦ<sup>electronic</sup> <span style="writing-mode: horizontal-tb;"><i>j</i></span> ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠨ ᠡ᠋᠃ <br />
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ᠡᠨᠡ ᠢᠨᠦ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠳᠡᠭ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠭᠦᠶᠦᠳᠡᠯ ᠡᠴᠡ ᠢᠯᠭᠠᠬᠤ ᠱᠠᠭᠠᠷᠳᠠᠯᠭ ᠠ᠋ ᠡᠴᠡ ᠤᠷᠭᠤᠨ ᠭᠠᠷᠤᠭᠰᠠᠨ ᠤᠯᠠᠮᠵᠢᠯᠠᠯ ᠶᠤᠮ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠧᠯᠧᠺᠲ᠋ᠷᠣᠨᠢᠭ᠌ ᠲᠦ᠂ ᠢᠯᠠᠩᠭᠤᠶ ᠠ᠋ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠡᠯ ᠳ᠋ᠦ ᠣᠨᠴᠤᠭᠤᠢ ᠠᠴᠢ ᠬᠣᠯᠪᠤᠭᠳᠠᠯ ᠲᠠᠢ᠃ <br />
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ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠡᠯ ᠰᠢᠨᠦ᠋ᠰ<sup>sinus</sup> ᠳᠣᠯᠭᠢᠶᠠᠨ ᠳᠤ ᠨᠡᠮᠡᠬᠦ ᠪᠠ ᠬᠠᠰᠤᠬᠤ ᠢᠢᠨ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠰᠣᠯᠢᠭᠳᠠᠳᠠᠭ᠃<ref>https://www.electricaltechnology.org/2013/07/active-reactive-apparent-and-complex.html Electrical Technology. Active, Reactive, Apparent and Complex Power. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/07/31</ref><br />
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{| class="wikitable"<br />
|+<br />
|[[File:Cmplxpower.svg|thumb]]<br />
|style = "height: 150px;"|ᠬᠦᠪᠰᠬᠤ ᠭᠦᠶᠦᠳᠡᠯ ᠪᠦᠬᠦᠢ ᠡᠷᠴᠢᠮ ᠬᠦᠴᠦ᠃ <span style="writing-mode: horizontal-tb;"><i>P</i></span> ᠢᠨᠦ ᠢᠳᠡᠪᠬᠢᠲᠦ ᠴᠢᠳᠠᠯ᠂ <span style="writing-mode: horizontal-tb;"><i>Q</i></span> ᠢᠨᠦ ᠬᠠᠷᠢᠭᠤ ᠢᠳᠡᠪᠬᠢᠲᠦ ᠴᠢᠳᠠᠯ᠂ <span style="writing-mode: horizontal-tb;"><i>S</i></span> ᠨᠡᠶ᠋ᠢᠯᠡᠪᠦᠷᠢ ᠴᠢᠳᠠᠯ᠃<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠢ ᠶᠠᠪ ᠴᠠᠪ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠬᠤ ᠬᠡᠴᠡᠭᠦᠦ ᠲᠤᠯᠠ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠡᠯ ᠦᠳ ᠢ ᠳᠠᠪᠬᠤᠷᠯᠠᠬᠤ ᠨᠡᠨ ᠲᠦᠪᠡᠭ ᠲᠡᠢ᠃<br />
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ᠬᠠᠷᠢᠨ ᠲᠥᠰᠦᠭᠡᠯᠡᠯ ᠭᠦᠶᠦᠳᠡᠯ ᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠨᠤᠭᠤᠳ ᠲᠠᠶ ᠬᠣᠣᠰᠮᠠᠭᠯᠠᠭᠤᠯᠤᠭᠰᠠᠨ ᠢᠢᠠᠷ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠡᠯ ᠦᠨ ᠲᠣᠭᠠᠴᠠᠭ ᠠ᠋ ᠬᠢᠵᠦ᠂ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠤ ᠣᠰᠤᠯ ᠡᠴᠡ ᠤᠷᠢᠳᠤᠴᠢᠯᠠᠨ ᠰᠡᠷᠭᠡᠢᠢᠯᠡᠬᠦ ᠲᠤᠰᠠ ᠲᠠᠢ ᠪᠠᠢᠢᠳᠠᠭ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠮᠥᠨ ᠰᠢᠭ᠌ᠨᠠᠯ<sup>signal</sup> ᠪᠣᠯᠪᠠᠰᠤᠷᠠᠭᠤᠯᠬᠤ ᠲᠧᠺᠨᠣᠯᠣᠭᠢ ᠳ᠋ᠤ<sup>technology</sup> ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ <br />
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ᠡᠨᠡ ᠨᠢ ᠡᠭᠦᠷ ᠦᠨ ᠬᠣᠯᠪᠤᠭ ᠠ᠋᠂ ᠤᠲᠠᠰᠤ ᠦᠭᠡᠢ ᠳᠠᠮᠵᠢᠭᠤᠯᠤᠯᠲᠠ᠂ ᠷᠠᠳᠠᠷ<sup>radar</sup>᠂ ᠪᠢᠣᠯᠣᠭᠢ<sup>biology</sup> (ᠲᠠᠷᠢᠬᠢᠨ ᠤ ᠳᠣᠯᠭᠢᠶᠠᠨ) ᠰᠤᠳᠤᠯᠬᠤ ᠳ᠋ᠤ ᠬᠠᠮᠢᠶᠠᠷᠤᠨ ᠠ᠋᠃<br />
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ᠶᠡᠷᠦ᠂ ᠰᠢᠨᠦ᠋ᠰ ᠪᠠ ᠺᠣᠰᠢᠨᠦ᠋ᠰ<sup>cosinus</sup> ᠳᠣᠯᠭᠢᠶᠠᠨ ᠪᠦᠬᠦᠢ ᠵᠦᠢᠯ ᠢ ᠬᠡᠮᠵᠢᠬᠦ᠂ ᠲᠣᠭᠠᠴᠠᠭᠠᠯᠬᠤ ᠳ᠋ᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠴᠢᠬᠤᠯᠠ ᠬᠡᠷᠡᠭ ᠲᠡᠢ᠃<ref name=":0" /><br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==<br />
<div style="writing-mode: horizontal-tb; "><References/></div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%AC%E1%A0%A0%E1%A0%AD%E1%A0%A4%E1%A0%B7%E1%A0%AE%E1%A0%A0%E1%A0%AD_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%8A%E1%A0%A0%E1%A0%8B&diff=2718
ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠊ᠠ᠋
2021-08-03T10:45:26Z
<p>Angka: Spelling</p>
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ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠢᠨᠦ ᠲᠡᠭ <span style="writing-mode: horizontal-tb;"><i>(0)</i></span> ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ [[ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ]] ᠢᠢ '''ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋''' ᠬᠠᠮᠠᠮᠦᠢ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠺᠸᠠᠲᠷᠠᠲ<sup>quadrate</sup> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠬᠠᠰᠤᠬᠤ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠬᠠᠷᠢᠭᠤ ᠭᠠᠷᠤᠨ ᠠ᠋᠃<ref name=":0">https://www.livescience.com/42748-imaginary-numbers.html Live Science. Elaine J. Hom. What Are Imaginary Numbers? 2014/01/22. </ref> <br />
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ᠲᠡᠭᠡᠬᠦ ᠪᠡᠷ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠠᠨᠤ ᠬᠠᠰᠤᠬᠤ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠶ ᠲᠣᠭᠠᠨ ᠤ ᠬᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠮᠥᠨ᠃ <br />
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ᠲᠡᠭ <span style="writing-mode: horizontal-tb;"><i>(0)</i></span> ᠲᠣᠭ ᠠ᠋ ᠪᠣᠳᠠᠲᠤ ᠪᠤᠶᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠠᠯᠢ ᠨᠢ ᠴᠤ ᠪᠠᠢᠢᠵᠤ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
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{| class="wikitable"<br />
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|[[File:Rotations_on_the_complex_plane.svg|thumb]]<br />
|style = "height: 150px;"|ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭᠠ ᠪᠦᠬᠦᠢ ᠲᠣᠭᠠᠴᠠᠭᠠᠯᠠᠯ ᠤᠨ ᠳᠦᠷᠰᠦᠯᠡᠯ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃<br />
<br />
ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠳᠦᠷᠰᠦᠯᠡᠯ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;"><i>–1</i></span> ᠢᠢᠡᠷ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ᠢ ᠲᠣᠭᠤᠷᠢᠭᠰᠠᠨ <span style="writing-mode: horizontal-tb;"><i>180</i></span> ᠬᠡᠮ ᠦᠨ ᠡᠷᠭᠢᠯᠲᠡ ᠪᠣᠯᠤᠨ ᠠ᠋᠃ <br />
<br />
i ᠲᠣᠭ ᠠ᠋ ᠪᠡᠷ ᠦᠷᠡᠵᠢᠯᠭᠦᠯᠬᠦ ᠢᠨᠦ ᠨᠠᠷᠠ ᠪᠤᠷᠤᠭᠤ “ᠡᠶᠡᠷᠭᠦ” ᠴᠢᠭ᠌ ᠲᠦ <span style="writing-mode: horizontal-tb;"><i>90</i></span> ᠬᠡᠮ ᠦᠨ ᠡᠷᠭᠢᠯᠲᠡ ᠪᠣᠯᠬᠤ ᠪᠠ ᠬᠡᠷᠪᠡ <span style="writing-mode: horizontal-tb;"><i>i2 = -1</i></span> ᠭᠡᠰᠡᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢᠨᠦ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ᠢ ᠲᠣᠭᠤᠷᠢᠭᠰᠠᠨ <span style="writing-mode: horizontal-tb;"><i>90</i></span> ᠬᠡᠮ ᠦᠨ ᠬᠣᠶᠠᠷ ᠡᠷᠭᠢᠯᠲᠡ ᠭᠡᠵᠦ ᠦᠵᠡᠪᠡᠯ ᠦᠷ ᠡ᠋ ᠳ᠋ᠦ᠋ᠩ ᠢᠨᠦ ᠭᠠᠭᠴᠠ <span style="writing-mode: horizontal-tb;"><i>180</i></span> ᠬᠡᠮ ᠦᠨ ᠡᠷᠭᠢᠯᠲᠡ ᠪᠣᠯᠤᠨ ᠠ᠋᠃ <br />
<br />
“ᠰᠥᠷᠭᠦ” (ᠨᠠᠷᠠ ᠵᠥᠪ) ᠴᠢᠭ᠌ ᠲᠦ <span style="writing-mode: horizontal-tb;"><i>90</i></span> ᠬᠡᠮ ᠡᠷᠭᠢᠬᠦ ᠢᠨᠦ ᠮᠥᠨ ᠡᠨᠡ ᠳᠦᠷᠰᠦᠯᠡᠯ ᠳ᠋ᠦ ᠨᠡᠶ᠋ᠢᠴᠡᠬᠦ ᠪᠠ <span style="writing-mode: horizontal-tb;"><i>x2 = -1</i></span> ᠭᠡᠰᠡᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠶ᠋ᠢᠳ <span style="writing-mode: horizontal-tb;"><i>−i</i></span> ᠪᠣᠯᠬᠤ ᠢ ᠬᠠᠷᠠᠭᠤᠯᠵᠤ ᠪᠠᠢᠢᠨ ᠠ᠋᠃ <br />
<br />
ᠶᠡᠷᠦᠩᠬᠡᠢᠢ ᠳ᠋ᠦ ᠪᠡᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ ᠦᠷᠡᠵᠢᠯᠭᠦᠯᠬᠦ ᠢᠨᠦ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ᠢ ᠲᠣᠭᠤᠷᠢᠭᠤᠯᠤᠨ ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭᠠᠨ ᠤ ᠠᠷᠭᠤᠮᠧᠨ᠍ᠲ<sup>argument</sup> ᠢᠢᠠᠷ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ᠂ ᠴᠠᠷ ᠬᠡᠮᠵᠢᠶ ᠡ᠋ ᠪᠡᠷ ᠢᠨᠦ ᠤᠷᠲᠤ ᠢᠢ ᠴᠢᠴᠠᠤᠯᠠᠬᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠪᠣᠯᠤᠨᠠ᠃ <br />
<br />
ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭᠠᠨ ᠤ ᠠᠷᠭᠤᠮᠧᠨ᠍ᠲ ᠭᠡᠳᠡᠭ ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠡᠬᠢᠯᠠᠯ ᠴᠡᠭ ᠢ ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭᠠ z ᠲᠡᠢ ᠬᠣᠯᠪᠤᠵᠤ ᠪᠠᠢᠢᠭᠠ ᠱᠤᠭᠤᠮ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠢᠢᠨ ᠥᠨᠴᠦᠭ <span style="writing-mode: horizontal-tb;"><i>φ</i></span> ᠶᠤᠮ᠃<br />
<br />
<br><br />
{| class="wikitable"<br />
|+<br />
|[[File:Complex_number_illustration_modarg.svg|thumb]]<br />
|style = "height: 150px;"|ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭᠠ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ<sup>complex</sup> ᠲᠣᠭᠠ ᠢᠢᠨ ᠠᠷᠭᠤᠮᠧᠨ᠍ᠲ<sup>argument</sup> ᠨᠢ ᠥᠨᠴᠦᠭ <span style="writing-mode: horizontal-tb;"><i>φ</i></span>᠃<br />
|}<br />
<br><br />
<br />
==ᠬᠡᠷᠡᠭᠯᠡᠭᠡ==<br />
<br />
[[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠢᠢ ''i'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠬᠦ ᠠᠭᠠᠳ ᠧᠯᠧᠺᠲ᠋ᠷᠣᠨᠢᠭ᠌ ᠲᠦ<sup>electronic</sup> ''j'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠨ ᠡ᠋᠃ <br />
<br />
ᠡᠨᠡ ᠢᠨᠦ ''i'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠳᠡᠭ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠭᠦᠶᠦᠳᠡᠯ ᠡᠴᠡ ᠢᠯᠭᠠᠬᠤ ᠱᠠᠭᠠᠷᠳᠠᠯᠭ ᠠ᠋ ᠡᠴᠡ ᠤᠷᠭᠤᠨ ᠭᠠᠷᠤᠭᠰᠠᠨ ᠤᠯᠠᠮᠵᠢᠯᠠᠯ ᠶᠤᠮ᠃<br />
<br />
ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠧᠯᠧᠺᠲ᠋ᠷᠣᠨᠢᠭ᠌ ᠲᠦ᠂ ᠢᠯᠠᠩᠭᠤᠶ ᠠ᠋ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠡᠯ ᠳ᠋ᠦ ᠣᠨᠴᠤᠭᠤᠢ ᠠᠴᠢ ᠬᠣᠯᠪᠤᠭᠳᠠᠯ ᠲᠠᠢ᠃ <br />
<br />
ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠡᠯ ᠰᠢᠨᠦ᠋ᠰ<sup>sinus</sup> ᠳᠣᠯᠭᠢᠶᠠᠨ ᠳᠤ ᠨᠡᠮᠡᠬᠦ ᠪᠠ ᠬᠠᠰᠤᠬᠤ ᠢᠢᠨ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠰᠣᠯᠢᠭᠳᠠᠳᠠᠭ᠃<ref>https://www.electricaltechnology.org/2013/07/active-reactive-apparent-and-complex.html Electrical Technology. Active, Reactive, Apparent and Complex Power. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/07/31</ref><br />
<br />
<br><br />
{| class="wikitable"<br />
|+<br />
|[[File:Cmplxpower.svg|thumb]]<br />
|style = "height: 150px;"|ᠬᠦᠪᠰᠬᠤ ᠭᠦᠶᠦᠳᠡᠯ ᠪᠦᠬᠦᠢ ᠡᠷᠴᠢᠮ ᠬᠦᠴᠦ᠃ <span style="writing-mode: horizontal-tb;"><i>P</i></span> ᠢᠨᠦ ᠢᠳᠡᠪᠬᠢᠲᠦ ᠴᠢᠳᠠᠯ᠂ <span style="writing-mode: horizontal-tb;"><i>Q</i></span> ᠢᠨᠦ ᠬᠠᠷᠢᠭᠤ ᠢᠳᠡᠪᠬᠢᠲᠦ ᠴᠢᠳᠠᠯ᠂ <span style="writing-mode: horizontal-tb;"><i>S</i></span> ᠨᠡᠶ᠋ᠢᠯᠡᠪᠦᠷᠢ ᠴᠢᠳᠠᠯ᠃<br />
|}<br />
<br><br />
<br />
ᠳᠣᠯᠭᠢᠶᠠᠨ ᠢ ᠶᠠᠪ ᠴᠠᠪ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠬᠤ ᠬᠡᠴᠡᠭᠦᠦ ᠲᠤᠯᠠ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠡᠯ ᠦᠳ ᠢ ᠳᠠᠪᠬᠤᠷᠯᠠᠬᠤ ᠨᠡᠨ ᠲᠦᠪᠡᠭ ᠲᠡᠢ᠃<br />
<br />
ᠬᠠᠷᠢᠨ ᠲᠥᠰᠦᠭᠡᠯᠡᠯ ᠭᠦᠶᠦᠳᠡᠯ ᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠨᠤᠭᠤᠳ ᠲᠠᠶ ᠬᠣᠣᠰᠮᠠᠭᠯᠠᠭᠤᠯᠤᠭᠰᠠᠨ ᠢᠢᠠᠷ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠡᠯ ᠦᠨ ᠲᠣᠭᠠᠴᠠᠭ ᠠ᠋ ᠬᠢᠵᠦ᠂ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠤ ᠣᠰᠤᠯ ᠡᠴᠡ ᠤᠷᠢᠳᠤᠴᠢᠯᠠᠨ ᠰᠡᠷᠭᠡᠢᠢᠯᠡᠬᠦ ᠲᠤᠰᠠ ᠲᠠᠢ ᠪᠠᠢᠢᠳᠠᠭ᠃<br />
<br />
ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠮᠥᠨ ᠰᠢᠭ᠌ᠨᠠᠯ<sup>signal</sup> ᠪᠣᠯᠪᠠᠰᠤᠷᠠᠭᠤᠯᠬᠤ ᠲᠧᠺᠨᠣᠯᠣᠭᠢ ᠳ᠋ᠤ<sup>technology</sup> ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ <br />
<br />
ᠡᠨᠡ ᠨᠢ ᠡᠭᠦᠷ ᠦᠨ ᠬᠣᠯᠪᠤᠭ ᠠ᠋᠂ ᠤᠲᠠᠰᠤ ᠦᠭᠡᠢ ᠳᠠᠮᠵᠢᠭᠤᠯᠤᠯᠲᠠ᠂ ᠷᠠᠳᠠᠷ<sup>radar</sup>᠂ ᠪᠢᠣᠯᠣᠭᠢ<sup>biology</sup> (ᠲᠠᠷᠢᠬᠢᠨ ᠤ ᠳᠣᠯᠭᠢᠶᠠᠨ) ᠰᠤᠳᠤᠯᠬᠤ ᠳ᠋ᠤ ᠬᠠᠮᠢᠶᠠᠷᠤᠨ ᠠ᠋᠃<br />
<br />
ᠶᠡᠷᠦ᠂ ᠰᠢᠨᠦ᠋ᠰ ᠪᠠ ᠺᠣᠰᠢᠨᠦ᠋ᠰ<sup>cosinus</sup> ᠳᠣᠯᠭᠢᠶᠠᠨ ᠪᠦᠬᠦᠢ ᠵᠦᠢᠯ ᠢ ᠬᠡᠮᠵᠢᠬᠦ᠂ ᠲᠣᠭᠠᠴᠠᠭᠠᠯᠬᠤ ᠳ᠋ᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠴᠢᠬᠤᠯᠠ ᠬᠡᠷᠡᠭ ᠲᠡᠢ᠃<ref name=":0" /><br />
<br />
<br />
==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==<br />
<div style="writing-mode: horizontal-tb; "><References/></div>
Angka
https://wikibilig.mn/index.php?title=%D0%9D%D2%AF%D2%AF%D1%80_%D1%85%D1%83%D1%83%D0%B4%D0%B0%D1%81&diff=2715
Нүүр хуудас
2021-07-31T12:11:20Z
<p>Angka: Expanded the list</p>
<hr />
<div><span style="writing-mode: vertical-lr; ">᠀᠋</span><br />
[[File:HomePage.JPG|center]]<br />
<div style="text-align:center"><br />
Encyclopædia Mongolica Grandis : Большая монгольская энциклопедия : <br><br />
༄།།ཆེན་པོ་ཧོར་གྱི་ཤེས་བྱ་ཀུན་བཏུས་ཚིག་མཛོད་ཆེན་མོ་བཞུགས་སོ།། : Ұлы Моңғол үлкен энциклопедиясы.<br />
<inputbox><br />
type=create<br />
width=100<br />
break=no<br />
buttonlabel=Шинэ нийтлэл үүсгэх<br />
default=(ᠨᠡᠢᠲᠡᠯᠡᠯ ᠊ᠦᠨ ᠨᠡᠷ ᠊ᠡ᠋)<br />
</inputbox><br />
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<div style="text-align:center"><br />
[https://wikibilig.mn/index.php?title=%E1%A0%B0%E1%A0%A2%E1%A0%A8_%E2%80%8D%E1%A0%A1%E1%A0%8B_%E1%A0%A8%E1%A0%A1%E1%A0%A2%E1%A0%A2%E1%A0%B2%E1%A0%A1%E1%A0%AF%E1%A0%A1%E1%A0%AF&veaction=edit <span style="color:Blue">'''Шинэ нийтлэл бичих'''] [https://www.youtube.com/watch?v=naD5wFtDDEQ&t=1s <span style="color:Blue">'''Шинэ нийтлэл бичих заавар''']<br />
</div><br />
{{Mongolian}}<big>ᠨᠡᠢᠢᠲᠡᠯᠡᠯ ᠦᠳ</big><br><br><br />
{{Mongolian}}<br />
[[ᠠᠪᠠᠲᠠᠢ ᠬᠠᠨ]]<br><br />
[[Авшиг|ᠠᠪᠢᠱᠢᠭ]]<br><br />
[[ᠠᠪᠯᠢᠭ ᠊ᠠ᠋|ᠠᠪᠯᠢᠭ᠍ ᠊ᠠ᠋]]<br><br />
[[ᠠᠯᠻᠦ᠋ᠪᠢᠶᠧᠷ ᠊ᠦᠨ ᠠᠶᠠᠯᠠᠯ]]<br><br />
[[ᠠᠷᠬᠠᠭ ᠶᠠᠳᠠᠷᠠᠭᠠᠨ ᠊ᠤ ᠬᠠᠮ ᠰᠢᠨᠵᠢ]]<br><br />
[[ᠡᠨ᠍ᠽᠢᠮ]]<br><br />
[[Энгийн герпес|ᠡᠩ ᠊ᠦᠨ ᠾᠧᠷᠫᠧᠰ]]<br><br />
[[ᠡᠭ ᠊ᠦᠨ ᠭᠣᠣᠯ ᠊ᠤᠨ ᠤᠰᠤᠨ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠰᠲ᠋ᠠᠨᠼ ᠲᠥᠰᠦᠯ]]<br><br />
[[ᠡᠰ ( ᠪᠢᠣᠯᠣᠭᠢ)]]<br><br />
[[ᠡᠰᠡᠷᠬᠦ ᠪᠢᠢ ᠡ᠋]]<br><br />
[[ᠢᠣᠳ᠋]]<br><br />
[[ᠢᠨ᠍ᠲ᠋ᠧᠷᠻᠣᠰᠮᠣᠰ]]<br><br />
[[Идэж болдог ус|ᠢᠳᠡᠵᠥ ᠪᠣᠯᠳᠠᠭ ᠣᠰᠣ]]<br><br />
[[ᠣᠭᠲᠠᠷᠭᠤᠢ]]<br><br />
[[ᠤᠪᠰᠤ ᠨᠠᠭᠤᠷ]]<br><br />
[[ᠤᠭᠤᠷᠠᠭ]]<br><br />
[[ᠤᠮᠠᠷᠠᠳᠤ ᠠᠮᠧᠷᠢᠻ ᠊ᠥᠨ ᠴᠢᠯᠦᠭᠡᠲᠦ ᠬᠤᠳᠠᠯᠳᠤᠭᠠᠨ ᠊ᠣ ᠬᠡᠯᠡᠯᠴᠡᠭᠡᠷ]]<br><br />
[[Улаан аварга|ᠤᠯᠠᠭᠠᠨ ᠠᠪᠤᠷᠭᠤ]]<br><br />
[[ᠤᠯᠠᠭᠠᠨ ᠴᠢᠳᠬᠤᠯᠠᠩ]]<br><br />
[[ᠤᠯᠠᠭᠠᠨᠪᠠᠭᠠᠲᠤᠷ]]<br><br />
[[ᠤᠯᠠᠭᠠᠭᠴᠢ ᠊ᠢᠢᠨ ᠬᠠᠷ ᠊ᠠ᠋ ᠨᠠᠭᠤᠷ]]<br><br />
[[ᠤᠰᠤᠨ ᠊ᠤ ᠬᠠᠨ ᠭᠦᠷᠪᠡᠯ]]<br><br />
[[ᠮᠦᠯᠲ᠋ᠢᠯᠢᠩ ᠤᠲᠠᠰᠤᠨ ᠤ ᠭᠠᠷ|ᠤᠲᠠᠰᠤᠨ ᠤ ᠮᠦᠯᠲ᠋ᠢᠯᠢᠩ 0 ᠭᠠᠷ]]<br><br />
[[ᠤᠷᠠᠨ ᠪᠠᠷᠢᠯᠭ ᠊ᠠ᠋ ᠊ᠢᠢᠨ ᠲᠡᠦᠬᠡ]]<br><br />
[[ᠤᠷᠪᠠᠯ ᠤᠨ ᠬᠤᠷᠳᠤ]]<br><br />
[[ᠥᠭᠡᠯᠦᠨ ᠬᠠᠲᠤᠨ]]<br><br />
[[Өлөн гэдэс|ᠥᠯᠦᠨ ᠭᠡᠳᠡᠰᠦ]]<br><br />
[[ᠦᠢᠯᠡ ᠦᠭᠡ]]<br><br />
[[Үелэх систем|ᠦᠶᠡᠯᠡᠬᠦ ᠰᠢᠰᠲ᠋ᠧᠮ]]<br><br />
[[ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋ ᠊ᠦᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ|ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋ ᠦᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ]]<br><br />
[[ᠨᠡᠭᠦᠷᠡᠰᠦ ᠬᠡᠶᠢᠵᠢᠭᠦᠯᠬᠦ ᠲᠧᠭᠨᠣᠯᠣᠭᠢ]]<br><br />
[[ᠨᠢᠩᠵᠢᠪᠠᠳᠭᠠᠷ᠂ ᠰᠠᠩᠵᠢᠮᠢᠲᠤᠪ ᠊ᠤᠨ]]<br><br />
[[ᠨᠢᠮᠪᠡᠭᠡ ᠶ᠋ᠢᠨ ᠬᠦᠴᠢᠯ ᠦᠨ ᠮᠥᠴᠡᠯᠭᠡ]]<br><br />
[[ᠨᠤᠭᠤ]]<br><br />
[[ᠨᠥᠬᠦᠴᠡᠯᠳᠡᠭᠦᠯᠦᠨ ᠬᠣᠯᠪᠤᠬᠤ ᠨᠥᠬᠦᠴᠡᠯ]]<br><br />
[[Ньютоны хөдөлгөөний хуулиуд|ᠨᠧᠸᠲᠣᠨ ᠊ᠤ ᠬᠥᠳᠡᠯᠭᠡᠭᠡᠨ ᠊ᠦ ᠬᠠᠤᠯᠢ ᠨᠤᠭᠤᠳ]]<br><br />
[[Баартагчин|ᠪᠠᠭᠠᠷᠲᠠᠭᠴᠢᠨ]]<br><br />
[[Байгалийн Түүхийн Музей|ᠪᠠᠢᠢᠭᠠᠯᠢ ᠊ᠢᠢᠨ ᠲᠡᠦᠬᠡ ᠊ᠢᠢᠨ ᠮᠦᠽᠧᠢ]]<br><br />
[[ᠪᠠᠲᠤᠲᠤᠯᠭ ᠠ᠂ ᠪᠣᠷᠵᠢᠬᠢᠨ ᠬᠠᠯᠲᠠᠮ ᠠ|ᠪᠠᠲᠤᠲᠤᠯᠭ᠍ ᠊ᠠ᠋᠂ ᠪᠣᠷᠵᠢᠭᠢᠨ ᠬᠠᠯᠲᠠᠮ ᠊ᠠ᠋]]<br><br />
[[ᠪᠠᠳᠮ ᠊ᠠ᠋]]<br><br />
[[ᠪᠠᠷᠰᠪᠣᠯᠤᠳ᠂ ᠷᠢᠨᠴᠡᠨ ᠊ᠦ]]<br><br />
[[ᠪᠣᠭᠳᠠ ᠬᠠᠨ ᠠᠭᠤᠯᠠ]]<br><br />
[[ᠪᠣᠯᠣᠷ]]<br><br />
[[ᠪᠥᠬᠡᠩ]]<br><br />
[[ᠪᠤᠷᠠᠲᠠᠶᠧᠸᠠ᠂ ᠠᠯᠧᠻᠰᠠᠨ᠍ᠲᠷᠠ ᠮᠠᠨᠵᠢᠶᠧᠹᠨᠠ]]<br><br />
[[ᠬᠣᠶᠠᠷᠲᠤ ᠶᠢᠨ ᠠᠯᠭᠧᠪᠷᠠ|ᠪᠦᠦᠯᠢ ᠢᠢᠨ ᠠᠯᠭᠧᠪᠷᠠ]]<br><br />
[[ᠪᠯᠠᠻᠫᠢᠨ᠍ᠻ]]<br><br />
[[ᠪᠷᠣᠨ᠍ᠲ᠋ᠣᠽᠠᠸᠷ]]<br><br />
[[ᠬᠠᠨ ᠪᠤᠮᠪᠠᠳᠤ ᠨᠢᠰᠬᠦ ᠪᠠᠭᠤᠳᠠᠯ]]<br><br />
[[Khanhiin gazar hudlult|ᠬᠠᠩᠬ ᠠ᠋ ᠢᠢᠨ ᠭᠠᠵᠠᠷ ᠬᠥᠳᠡᠯᠦᠯᠲᠡ]]<br><br />
[[ᠬᠠᠪᠤᠩ]]<br><br />
[[ᠬᠠᠪᠲᠠᠩ ᠊ᠤᠨ ᠲᠧᠻᠲ᠋ᠣᠨᠢᠻ]]<br><br />
[[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]]<br><br />
[[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠊ᠠ᠋]]<br><br />
[[Халиун буга|ᠬᠠᠯᠢᠭᠤᠨ ᠪᠤᠭᠤ]]<br><br />
[[Хадны суусар|ᠬᠠᠳᠠᠨ ᠊ᠤ ᠰᠤᠤᠰᠠᠷ]]<br><br />
[[Хар нүх|ᠬᠠᠷ ᠊ᠠ᠋ ᠨᠦᠬᠡ]]<br><br />
[[Харьцангуйн онол|ᠬᠠᠷᠢᠴᠠᠩᠭᠤᠢ ᠶ᠋ᠢᠨ ᠣᠨᠤᠯ]]<br><br />
[[ᠬᠠᠷᠢᠴᠠᠩᠭᠤᠢ ᠊ᠢᠢᠨ ᠶᠡᠷᠦᠩᠬᠡᠢ ᠣᠨᠤᠯ]]<br><br />
[[Хархорины усан цахилгаан станц|ᠬᠠᠷᠠᠬᠣᠷᠢᠨ ᠣ ᠣᠰᠣᠨ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠰᠲ᠋ᠠᠨ᠍ᠴᠢ]]<br><br />
[[Хархорины усжуулах байгууламж|ᠬᠠᠷᠠᠬᠣᠷᠢᠨ ᠣ ᠣᠰᠣᠵᠢᠭᠣᠯᠬᠣ ᠪᠠᠢᠢᠭᠤᠯᠤᠮᠵᠢ]]<br><br />
[[Харанхуй матери|ᠬᠠᠷᠠᠩᠭᠤᠶ ᠮᠠᠲ᠌ᠧᠷᠢ]]<br><br />
[[ᠬᠡᠦᠬᠡᠳ ᠊ᠦᠨ ᠡᠷᠬᠡ ᠊ᠢᠢᠨ ᠲᠤᠬᠠᠢ ᠻᠣᠨᠸᠧᠨᠼ]]<br><br />
[[Хэлмэгдэгсдийн хөшөө|ᠬᠡᠯᠮᠡᠭᠳᠡᠭᠰᠡᠳ ᠊ᠦᠨ ᠬᠥᠰᠢᠶ ᠊ᠡ᠋]]<br><br />
[[ᠬᠢᠮᠢ ᠊ᠢᠢᠨ ᠬᠥᠳᠡᠯᠭᠡᠭᠦᠷᠲᠦ ᠫᠤᠤᠵᠢᠩ]]<br><br />
[[ᠬᠢᠶᠠᠭᠲᠤ ᠢᠢᠨ ᠭᠡᠷ ᠡ᠋ ᠋(1915)]]<br><br />
[[ᠬᠣᠭᠣᠰᠤᠨ ᠴᠢᠨᠠᠷ]]<br><br />
[[Хоёртын тооллын систем|ᠬᠣᠶᠠᠷᠲᠣ ᠊ᠢᠢᠨ ᠲᠣᠭᠠᠯᠠᠯ ᠊ᠤᠨ ᠰᠢᠰᠲ᠋ᠡᠮ]]<br><br />
[[ᠬᠤᠪᠢᠰᠤᠯ ᠊ᠦᠨ ᠣᠨᠣᠯ]]<br><br />
[[Хутагтын дөрвөн үнэн|ᠬᠣᠲᠣᠬᠲᠣ ᠊ᠢᠢᠨ ᠳᠥᠷᠪᠡᠨ ᠦᠨᠡᠨ]]<br><br />
[[Хутагтын найман мөр|ᠬᠤᠲᠦᠬᠲᠦ ᠊ᠢᠢᠨ ᠨᠠᠢᠮᠠᠨ ᠮᠥᠷ]]<br><br />
[[ᠬᠤᠷᠢᠴᠠᠯ]]<br><br />
[[ᠬᠦᠴᠢᠯ]]<br><br />
[[Гангар хун|ᠭᠠᠩᠭᠠᠷ ᠬᠤᠨ]]<br><br />
[[ᠭᠠᠲᠤᠯᠭ ᠊ᠠ᠋]]<br><br />
[[ᠭᠠᠳᠠᠭᠠᠳᠤ ᠳ᠋ᠠᠬᠢ ᠭᠠᠵᠠᠷ ᠵᠦᠢ ᠊ᠢᠢᠨ ᠤᠯᠨᠮᠵᠢᠯᠠᠯᠲᠤ ᠮᠣᠩᠭᠤᠯ ᠨᠡᠷᠡᠰ ᠊ᠦᠨ ᠵᠢᠭᠰᠠᠭᠠᠯᠲᠠ]]<br><br />
[[Галактик|ᠭᠠᠯᠠᠻᠲ᠋ᠢᠻ]]<br><br />
[[ᠭᠠᠵᠠᠷ ᠵᠦᠢ ᠊ᠢᠢᠨ ᠲᠥᠪᠡᠳ ᠰᠠᠮᠭᠠᠷᠳᠢ ᠨᠡᠷᠡᠰ ᠦᠨ ᠵᠢᠭᠰᠠᠭᠠᠯᠲᠠ]]<br><br />
[[ᠭᠡᠷᠡᠯ ᠊ᠢᠢᠡᠷ ᠨᠡᠶ᠋ᠢᠯᠡᠭᠵᠢᠬᠦᠢ (ᠹᠣᠲ᠋ᠣᠰᠢᠨ᠍ᠲ᠋ᠸᠽ)]]<br><br />
[[ᠲᠷᠢᠨᠠᠷᠢ ᠰᠢᠰᠲ᠋ᠧᠮ|ᠭᠤᠷᠪᠠᠲᠤ ᠶ᠋ᠢᠨ ᠲᠣᠭᠠᠯᠠᠯ ᠦ᠋ᠨ ᠰᠢᠰᠲ᠋ᠧᠮ]]<br><br />
[[Геологийн цаг тоолол (босоо хүснэгт)|ᠭᠧᠣᠯᠣᠭᠢ ᠊ᠢᠢᠨ ᠴᠠᠭ ᠲᠣᠭᠠᠯᠠᠯ (ᠪᠣᠰᠤᠭ ᠠ ᠬᠦᠰᠦᠨᠦᠭᠲᠦ)]]<br><br />
[[Геологийн цаг тоолол|ᠭᠧᠣᠯᠣᠭᠢ ᠊ᠢᠢᠨ ᠴᠠᠭ ᠲᠣᠭᠠᠯᠠᠯ]]<br><br />
[[Мануул|ᠮᠠᠨᠤᠤᠯ]]<br><br />
[[Мазаалай|ᠮᠠᠵᠠᠭᠠᠯᠠᠢ]]<br><br />
[[ᠮᠠᠵᠠᠭᠠᠯᠠᠶ ᠬᠢᠮᠡᠯ ᠳᠠᠭᠠᠭᠤᠯᠢ]]<br><br />
[[ᠮᠠᠷᠰ-V]]<br><br />
[[Мартан|ᠮᠠᠷᠲᠠᠨ]]<br><br />
[[Мэдэмсэр|ᠮᠡᠳᠡᠮᠰᠡᠷ]]<br><br />
[[ᠮᠢᠩᠭᠠᠲᠤ ᠪᠠᠭ ᠊ᠠ᠋ ᠭᠠᠷᠠᠭ]]<br><br />
[[ᠮᠣᠩᠭᠣᠯ ᠤᠯᠤᠰ]]<br><br />
[[ᠮᠣᠩᠭᠤᠯ ᠤᠯᠤᠰ ᠊ᠤᠨ ᠵᠡᠪᠰᠡᠭᠲᠦ ᠬᠦᠴᠦᠨ|ᠮᠣᠩᠭᠤᠯ ᠤᠯᠤᠰ ᠤᠨ ᠵᠡᠪᠰᠡᠭᠲᠦ ᠬᠦᠴᠦᠨ]]<br><br />
[[Монголын газрын гадаргын геологийн түүх|ᠮᠣᠩᠭᠣᠯ ᠊ᠤᠨ ᠭᠠᠵᠠᠷ ᠊ᠤᠨ ᠭᠠᠳᠠᠷᠭᠤ ᠊ᠢᠢᠨ ᠭᠧᠣᠯᠣᠭᠢ ᠊ᠢᠢᠨ ᠲᠡᠦᠬᠡ]]<br><br />
[[ᠮᠣᠩᠭᠣᠯ ᠬᠡᠯᠡᠨ ᠦ ᠵᠥᠪ ᠪᠢᠴᠢᠬᠦ ᠳᠦᠷᠢᠮ ᠦᠨ ᠵᠢᠷᠤᠮᠯᠠᠭᠰᠠᠨ ᠲᠣᠯᠢ]]<br><br />
[[Монголын түүхийн цаг тоолол|ᠮᠣᠩᠭᠣᠯ ᠊ᠤᠨ ᠲᠡᠦᠬᠡ ᠊ᠢᠢᠨ ᠴᠠᠭ ᠲᠣᠭᠠᠯᠠᠯ]]<br><br />
[[ᠮᠣᠩᠭᠤᠯ ᠊ᠤᠨ ᠴᠠᠭᠳᠠᠭ ᠊ᠠ᠋]]<br><br />
[[ᠮᠣᠩᠭᠣᠯᠣᠷ ᠊ᠤᠨ ᠪᠠᠢᠢᠰᠢᠩ]]<br><br />
[[ᠮᠣᠩᠭᠤᠯᠰᠠᠲ-1 ᠬᠢᠮᠡᠯ ᠳᠠᠭᠠᠭᠤᠯᠢ]]<br><br />
[[ᠮᠥᠩᠭᠥᠯᠢᠭ ᠡᠭᠦᠯᠡ]]<br><br />
[[ᠯᠣᠪᠠᠴᠧᠹᠰᠻᠢ ᠊ᠢᠢᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ]]<br><br />
[[ᠰᠠᠭᠤᠷᠢ]]<br><br />
[[Сэтгэл төдийтний тогтсон таалал|ᠰᠡᠳᠬᠢᠯ ᠲᠡᠳᠥᠢᠢᠲᠡᠨ ᠥ ᠲᠣᠭᠳᠠᠭᠰᠠᠨ ᠲᠠᠭᠠᠯᠠᠯ]]<br><br />
[[ᠰᠡᠯᠡᠩᠭᠡ ᠠᠢᠢᠮᠠᠭ]]<br><br />
[[Шилүүс|ᠰᠢᠯᠢᠭᠦᠰᠦ]]<br><br />
[[ᠰᠢᠯᠲᠦᠭᠡᠨ]]<br><br />
[[ᠰᠣᠶᠣᠩᠪᠣ ᠦᠰᠦᠭ]]<br><br />
[[ᠰᠣᠷᠢᠨᠵᠢᠨ ᠢᠨ᠍ᠳᠦ᠋ᠭ᠌ᠼ]]<br><br />
[[ᠱᠦᠯᠲᠦ]]<br><br />
[[ᠲᠠᠢᠢᠭ ᠊ᠠ᠋]]<br><br />
[[Тарваган|ᠲᠠᠷᠪᠠᠭ ᠊ᠠ᠋]]<br><br />
[[ᠲᠡᠮᠡᠦᠯᠦᠯ ᠬᠢᠮᠡᠯ ᠳᠠᠭᠠᠭᠤᠯᠢ]]<br><br />
[[ᠲᠣᠰᠤᠲᠤ ᠶ᠋ᠢᠨ ᠨᠢᠷᠤᠭᠤ]]<br><br />
[[ᠲᠣᠳᠤ ᠪᠢᠴᠢᠭ]]<br><br />
[[Тусгай хамгаалалттай газар|ᠲᠤᠰᠬᠠᠢ ᠬᠠᠮᠠᠭᠠᠯᠠᠯᠲᠠ ᠲᠠᠢ ᠭᠠᠵᠠᠷ]]<br><br />
[[ᠲᠦᠪᠳᠡᠨᠳᠣᠷᠵᠢ᠂ ᠳ᠂]]<br><br />
[[Термодинамикийн хуулиуд|ᠲᠧᠷᠮᠣᠳᠢᠨᠠᠮᠢᠻ ᠊ᠤᠨ ᠬᠠᠦᠯᠢ ᠨᠤᠭᠤᠳ]]<br><br />
[[ᠳᠠᠪᠤᠰᠤ]]<br><br />
[[ᠳᠠᠭᠢᠨᠢ]]<br><br />
[[ᠳᠠᠮᠪᠠᠢᠢᠨᠣᠸᠠ᠂ ᠰᠸᠧᠲ᠋ᠯᠠᠨᠠ ᠠᠯᠧᠻᠰᠠᠨ᠍ᠲᠷᠣᠸᠨᠠ]]<br><br />
[[ᠳᠡᠮᠴᠣᠭᠳᠣᠨᠷᠤᠪ᠂ ᠪᠣᠷᠵᠢᠭᠢᠨ ᠨᠠᠮᠵᠠᠯᠸᠠᠩᠴᠣᠭ]]<br><br />
[[Дэлхийн Ван|ᠳᠡᠯᠡᠬᠡᠢ ᠊ᠢᠢᠨ ᠸᠠᠩ]]<br><br />
[[Дэг|ᠳᠢᠭ᠌]]<br><br />
[[ᠳᠣᠯᠭᠠᠷ᠂ ᠴᠣᠢᠢᠮᠪᠢᠯ ᠊ᠤᠨ]]<br><br />
[[ᠳᠥᠷᠪᠡᠯᠵᠢᠨ ᠪᠢᠴᠢᠭ]]<br><br />
[[ᠳᠤᠷ ᠊ᠠ᠋ ᠲᠠᠯᠪᠢᠯᠲᠠ]]<br><br />
[[Дурдал|ᠳᠤᠷᠠᠳᠤᠯ]]<br><br />
[[ᠴᠠᠬᠢᠭᠤᠷ ᠰᠠᠭᠤᠷᠢᠲᠤ ᠠᠮᠢᠳᠤᠷᠠᠯ]]<br><br />
[[ᠴᠠᠭᠢᠯᠭᠠᠨ ᠢᠨ᠍ᠳᠦ᠋ᠭ᠌ᠼ]]<br><br />
[[ᠴᠠᠬᠢᠯᠭᠠᠨ ᠰᠣᠷᠢᠨᠵᠢᠨ ᠢᠨ᠍ᠳᠦ᠋ᠭ᠌ᠼ]]<br><br />
[[Цахилгаан соронзон долгионы хуваарь|ᠴᠠᠬᠢᠯᠭᠠᠨ ᠰᠣᠷᠢᠨᠵᠢᠨ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠊ᠤ ᠬᠤᠪᠢᠶᠠᠷᠢ]]<br><br />
[[ᠴᠠᠬᠢᠯᠭᠠᠨ ᠰᠲ᠋ᠠᠲ᠋ᠢᠭ ᠢᠨ᠍ᠳᠦ᠋ᠭ᠌ᠼ]]<br><br />
[[ᠴᠠᠬᠢᠯᠭᠠᠨ ᠲᠡᠢᠢᠷᠦᠯᠲᠡᠲᠦ ᠫᠤᠤᠵᠢᠩ]]<br><br />
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[[ᠴᠠᠭᠠᠨ ᠵᠢᠮᠪᠠ]]<br><br />
[[Цагаан нүх|ᠴᠠᠭᠠᠨ ᠨᠦᠬᠡ]]<br><br />
[[Цагаан кола|ᠴᠠᠭᠠᠨ ᠻᠣᠯᠠ]]<br><br />
[[Цоохондой|ᠴᠣᠬᠣᠨᠳᠠᠢ]]<br><br />
[[Цоохор ирвэс|ᠴᠣᠬᠣᠷ ᠢᠷᠪᠢᠰ]]<br><br />
[[ᠴᠥᠮ ᠊ᠡ᠋ ᠊ᠢᠢᠨ ᠬᠥᠳᠡᠯᠭᠡᠭᠦᠷᠲᠦ ᠫᠤᠤᠵᠢᠩ]]<br><br />
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[[ᠴᠦᠻᠴᠠ ᠨᠠᠷ]]<br><br />
[[Заг хүйтэн|ᠵᠠᠭ ᠬᠦᠢᠲᠡᠨ]]<br><br />
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[[ᠵᠠᠶ ᠊ᠠ᠋ ᠊ᠢᠢᠨ ᠬᠦᠷᠢᠶ ᠊ᠡ᠋]]<br><br />
[[ᠵᠢᠵᠢᠭ ᠳᠤᠮᠳᠠ ᠦᠢᠯᠡᠳᠪᠦᠷᠢ]]<br><br />
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[[ᠶᠡᠭᠦᠯᠡᠭᠦᠷ]]<br><br />
[[ᠶᠢᠷᠲᠢᠨᠴᠦ ᠊ᠢ ᠦᠵᠡᠬᠦ ᠦᠵᠡᠯ]]<br><br />
[[Ёл|ᠶᠣᠯᠤ ᠰᠢᠪᠠᠭᠤ]]<br><br />
[[ᠶᠤᠸᠠᠮᠪᠤᠤ]]<br><br />
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[[ᠫᠾᠣᠲ᠋ᠣᠨ ᠊ᠤ ᠲᠡᠢᠢᠷᠥᠯᠲᠡᠲᠥ ᠬᠥᠳᠡᠯᠭᠡᠭᠦᠷ]]<br><br />
[[ᠷᠢᠮᠠᠨ ᠊ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ]]<br><br />
[[Робот|ᠷᠣᠪᠣᠲ]]<br><br />
[[ᠷᠣᠻ]]<br><br />
[[ᠹᠢᠽᠢᠣᠯᠣᠭᠢ]]<br><br />
[[Калашниковын автомат|ᠻᠠᠯᠠᠱᠨᠢᠻᠣᠹ ᠊ᠤᠨ ᠠᠦ᠋ᠲ᠋ᠣᠮᠠᠲ]]<br><br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠊ᠠ᠋
2021-07-31T12:07:14Z
<p>Angka: Link</p>
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ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠢᠨᠦ ᠲᠡᠭ <span style="writing-mode: horizontal-tb;"><i>(0)</i></span> ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ [[ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ]] ᠢᠢ '''ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋''' ᠬᠠᠮᠠᠮᠦᠢ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠺᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠬᠠᠰᠤᠬᠤ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠬᠠᠷᠢᠭᠤ ᠭᠠᠷᠤᠨ ᠠ᠋᠃<ref name=":0">https://www.livescience.com/42748-imaginary-numbers.html Live Science. Elaine J. Hom. What Are Imaginary Numbers? 2014/01/22. </ref> <br />
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ᠲᠡᠭᠡᠬᠦ ᠪᠡᠷ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠠᠭ ᠠ᠋ ᠠᠨᠤ ᠬᠠᠰᠤᠬᠤ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠶ ᠲᠣᠭᠠᠨ ᠤ ᠬᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠮᠥᠨ᠃ <br />
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ᠲᠡᠭ <span style="writing-mode: horizontal-tb;"><i>(0)</i></span> ᠲᠣᠭ ᠠ᠋ ᠪᠣᠳᠠᠲᠤ ᠪᠤᠶᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠠᠯᠢ ᠨᠢ ᠴᠤ ᠪᠠᠢᠢᠵᠤ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
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__TOC__<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
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{| class="wikitable"<br />
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|[[File:Rotations_on_the_complex_plane.svg|thumb]]<br />
|style = "height: 150px;"|ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭᠠ ᠪᠦᠬᠦᠢ ᠲᠣᠭᠠᠴᠠᠭᠠᠯᠠᠯ ᠤᠨ ᠳᠦᠷᠰᠦᠯᠡᠯ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠳ᠋ᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠳ᠋ᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃<br />
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ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠳᠦᠷᠰᠦᠯᠡᠯ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;"><i>–1</i></span> ᠢᠢᠡᠷ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ᠢ ᠲᠣᠭᠤᠷᠢᠭᠰᠠᠨ <span style="writing-mode: horizontal-tb;"><i>180</i></span> ᠬᠡᠮ ᠦᠨ ᠡᠷᠭᠢᠯᠲᠡ ᠪᠣᠯᠤᠨ ᠠ᠋᠃ <br />
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i ᠲᠣᠭ ᠠ᠋ ᠪᠡᠷ ᠦᠷᠡᠵᠢᠯᠭᠦᠯᠬᠦ ᠢᠨᠦ ᠨᠠᠷᠠ ᠪᠤᠷᠤᠭᠤ “ᠡᠷᠡᠭᠦ” ᠴᠢᠭ᠌ ᠲᠦ <span style="writing-mode: horizontal-tb;"><i>90</i></span> ᠬᠡᠮ ᠦᠨ ᠡᠷᠭᠢᠯᠲᠡ ᠪᠣᠯᠬᠤ ᠪᠠ ᠬᠡᠷᠪᠡ <span style="writing-mode: horizontal-tb;"><i>i2 = -1</i></span> ᠭᠡᠰᠡᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢᠨᠦ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ᠢ ᠲᠣᠭᠤᠷᠢᠭᠰᠠᠨ <span style="writing-mode: horizontal-tb;"><i>90</i></span> ᠬᠡᠮ ᠦᠨ ᠬᠣᠶᠠᠷ ᠡᠷᠭᠢᠯᠲᠡ ᠭᠡᠵᠦ ᠦᠵᠡᠪᠡᠯ ᠦᠷ ᠡ᠋ ᠳ᠋ᠦ᠋ᠩ ᠢᠨᠦ ᠭᠠᠭᠴᠠ <span style="writing-mode: horizontal-tb;"><i>180</i></span> ᠬᠡᠮ ᠦᠨ ᠡᠷᠭᠢᠯᠲᠡ ᠪᠣᠯᠤᠨ ᠠ᠋᠃ <br />
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“ᠰᠥᠷᠭᠦ” (ᠨᠠᠷᠠ ᠵᠥᠪ) ᠴᠢᠭ᠌ ᠲᠦ <span style="writing-mode: horizontal-tb;"><i>90</i></span> ᠬᠡᠮ ᠡᠷᠭᠢᠬᠦ ᠢᠨᠦ ᠮᠥᠨ ᠡᠨᠡ ᠳᠦᠷᠰᠦᠯᠡᠯ ᠳ᠋ᠦ ᠨᠡᠶᠢᠴᠡᠬᠦ ᠪᠠ <span style="writing-mode: horizontal-tb;"><i>x2 = -1</i></span> ᠭᠡᠰᠡᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠶ᠋ᠢᠳ <span style="writing-mode: horizontal-tb;"><i>−i</i></span> ᠪᠣᠯᠬᠤ ᠢ ᠬᠠᠷᠠᠭᠤᠯᠵᠤ ᠪᠠᠢᠢᠨ ᠠ᠋᠃ <br />
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ᠶᠡᠷᠦᠩᠬᠡᠢᠢ ᠳ᠋ᠦ ᠪᠡᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ ᠦᠷᠡᠵᠢᠯᠭᠦᠯᠬᠦ ᠢᠨᠦ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ᠢ ᠲᠣᠭᠤᠷᠢᠭᠤᠯᠤᠨ ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭᠠᠨ ᠤ ᠠᠷᠭᠤᠮᠧᠨ᠍ᠲ ᠢᠢᠠᠷ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ᠂ ᠴᠠᠷ ᠬᠡᠮᠵᠢᠶ ᠡ᠋ ᠪᠡᠷ ᠢᠨᠦ ᠤᠷᠲᠤ ᠢᠢ ᠴᠢᠴᠠᠤᠯᠠᠬᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠪᠣᠯᠤᠨᠠ᠃ <br />
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ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭᠠᠨ ᠤ ᠠᠷᠭᠤᠮᠧᠨ᠍ᠲ ᠭᠡᠳᠡᠭ ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠡᠬᠢᠯᠠᠯ ᠴᠡᠭ ᠢ ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭᠠᠨ z ᠲᠡᠢ ᠬᠣᠯᠪᠤᠵᠤ ᠪᠠᠢᠢᠭᠠ ᠱᠤᠭᠤᠮ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠢᠢᠨ ᠥᠨᠴᠦᠭ <span style="writing-mode: horizontal-tb;"><i>φ</i></span> ᠶᠤᠮ᠃<br />
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{| class="wikitable"<br />
|+<br />
|[[File:Complex_number_illustration_modarg.svg|thumb]]<br />
|style = "height: 150px;"|ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭᠠ ᠢᠢᠨ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ<sup>complex</sup> ᠲᠣᠭᠠ ᠢᠢᠨ ᠠᠷᠭᠤᠮᠧᠨ᠍ᠲ<sup>argument</sup> ᠨᠢ ᠥᠨᠴᠦᠭ <span style="writing-mode: horizontal-tb;"><i>φ</i></span>᠃<br />
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==ᠬᠡᠷᠡᠭᠯᠡᠭᠡ==<br />
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[[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠢᠢ ''i'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠬᠦ ᠠᠭᠠᠳ ᠧᠯᠧᠺᠲ᠋ᠷᠣᠨᠢᠭ᠌ ᠲᠦ<sup>electronic</sup> ''j'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠨ ᠡ᠋᠃ <br />
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ᠡᠨᠡ ᠢᠨᠦ ''i'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠳᠡᠭ ᠴᠠᠬᠢᠯᠭᠠᠨᠭᠦᠶᠦᠳᠦᠯ ᠡᠴᠡ ᠢᠯᠭᠠᠬᠤ ᠱᠠᠭᠠᠷᠳᠠᠯᠭ ᠠ᠋ ᠡᠴᠡ ᠤᠷᠭᠠᠨ ᠭᠠᠷᠤᠭᠰᠠᠨ ᠤᠯᠠᠮᠵᠢᠯᠠᠯ ᠶᠤᠮ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠧᠯᠧᠺᠲ᠋ᠷᠣᠨᠢᠭ᠌ ᠲᠦ᠂ ᠢᠯᠠᠩᠭᠤᠶ ᠠ᠋ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠳ᠋ᠦ ᠣᠩᠴᠤᠭᠤᠢ ᠠᠴᠢ ᠬᠣᠯᠪᠤᠭᠳᠠᠯ ᠲᠠᠢ᠃ <br />
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ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠰᠢᠨᠦ᠋ᠰ<sup>sinus</sup> ᠳᠣᠯᠭᠢᠶᠠᠨ ᠳᠤ ᠨᠡᠮᠡᠬᠦ ᠪᠠ ᠬᠠᠰᠤᠬᠤ ᠢᠢᠨ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠰᠣᠯᠢᠭᠳᠠᠳᠠᠭ᠃<ref>https://www.electricaltechnology.org/2013/07/active-reactive-apparent-and-complex.html Electrical Technology. Active, Reactive, Apparent and Complex Power. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/07/31</ref><br />
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{| class="wikitable"<br />
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|style = "height: 150px;"|ᠬᠦᠪᠰᠬᠤ ᠭᠦᠶᠦᠳᠡᠯ ᠪᠦᠬᠦᠢ ᠡᠷᠴᠢᠮ ᠬᠦᠴᠦ᠃ <span style="writing-mode: horizontal-tb;"><i>P</i></span> ᠢᠨᠦ ᠢᠳᠡᠪᠬᠢᠲᠦ ᠴᠢᠳᠠᠯ᠂ <span style="writing-mode: horizontal-tb;"><i>Q</i></span> ᠢᠨᠦ ᠬᠠᠷᠢᠭᠤ ᠢᠳᠡᠪᠬᠢᠲᠦ ᠴᠢᠳᠠᠯ᠂ <span style="writing-mode: horizontal-tb;"><i>S</i></span> ᠨᠡᠶᠢᠯᠡᠪᠦᠷᠢ ᠴᠢᠳᠠᠯ᠃<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠢ ᠶᠠᠪ ᠴᠠᠪ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠬᠤ ᠬᠡᠴᠡᠭᠦ ᠲᠤᠯᠠ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠦᠳ ᠢ ᠳᠠᠪᠬᠤᠷᠯᠠᠬᠤ ᠨᠡᠨ ᠲᠦᠪᠡᠭ ᠲᠡᠢ᠃<br />
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ᠬᠠᠷᠢᠨ ᠲᠥᠰᠦᠭᠡᠯᠡᠯ ᠭᠦᠶᠦᠳᠦᠯ ᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠨᠤᠭᠤᠳ ᠲᠠᠶ ᠬᠣᠣᠰᠮᠠᠭᠯᠠᠭᠤᠯᠤᠭᠰᠠᠨ ᠢᠢᠠᠷ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠦᠨ ᠲᠣᠭᠠᠴᠠᠭ ᠠ᠋ ᠬᠢᠵᠦ᠂ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠤ ᠣᠰᠤᠯ ᠡᠴᠡ ᠤᠷᠢᠳᠤᠴᠢᠯᠠᠨᠰᠡᠷᠭᠡᠢᠢᠯᠬᠦ ᠲᠤᠰᠠ ᠲᠠᠢ ᠪᠠᠢᠢᠳᠠᠭ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠠᠭ ᠠ᠋ ᠢᠢ ᠮᠥᠨ ᠰᠢᠭ᠌ᠨᠠᠯ<sup>signal</sup> ᠪᠣᠯᠪᠠᠰᠤᠷᠠᠭᠤᠯᠬᠤ ᠲᠧᠺᠨᠣᠯᠣᠭᠢ ᠳ᠋ᠤ<sup>technology</sup> ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ <br />
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ᠡᠨᠡ ᠨᠢ ᠡᠭᠦᠷᠦᠨ ᠬᠣᠯᠪᠤᠭ ᠠ᠋᠂ ᠤᠲᠠᠰᠤ ᠦᠭᠡᠢ ᠳᠠᠮᠵᠢᠭᠤᠯᠤᠯᠲᠠ᠂ ᠷᠠᠳᠠᠷ᠂ ᠪᠢᠣᠯᠣᠭᠢ (ᠲᠠᠷᠢᠬᠢᠨ ᠤ ᠳᠣᠯᠭᠢᠶᠠᠨ) ᠰᠤᠳᠤᠯᠬᠤ ᠳ᠋ᠤ ᠬᠠᠮᠢᠶᠠᠷᠠᠨ ᠠ᠋᠃<br />
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ᠶᠡᠷᠦ᠂ ᠰᠢᠨᠦ᠋ᠰ ᠪᠠ ᠺᠣᠰᠢᠨᠦ᠋ᠰ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠪᠦᠬᠦᠢ ᠵᠦᠢᠯ ᠢ ᠬᠡᠮᠵᠢᠬᠦ᠂ ᠲᠣᠭᠠᠴᠠᠭᠠᠯᠬᠤ ᠳ᠋ᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠴᠢᠬᠤᠯᠠ ᠬᠡᠷᠡᠭ ᠲᠡᠢ᠃<ref name=":0" /><br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==<br />
<div style="writing-mode: horizontal-tb; "><References/></div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%AC%E1%A0%A0%E1%A0%AD%E1%A0%A4%E1%A0%B7%E1%A0%AE%E1%A0%A0%E1%A0%AD_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%8A%E1%A0%A0%E1%A0%8B&diff=2713
ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠊ᠠ᠋
2021-07-31T11:47:29Z
<p>Angka: Images</p>
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ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠢᠨᠦ ᠲᠡᠭ <span style="writing-mode: horizontal-tb;"><i>(0)</i></span> ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ [[ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ]] ᠢᠢ '''ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋''' ᠬᠠᠮᠠᠮᠦᠢ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠺᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠬᠠᠰᠤᠬᠤ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠬᠠᠷᠢᠭᠤ ᠭᠠᠷᠤᠨ ᠠ᠋᠃<ref name=":0">https://www.livescience.com/42748-imaginary-numbers.html Live Science. Elaine J. Hom. What Are Imaginary Numbers? 2014/01/22. </ref> <br />
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ᠲᠡᠭᠡᠬᠦ ᠪᠡᠷ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠠᠭ ᠠ᠋ ᠠᠨᠤ ᠬᠠᠰᠤᠬᠤ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠶ ᠲᠣᠭᠠᠨ ᠤ ᠬᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠮᠥᠨ᠃ <br />
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ᠲᠡᠭ <span style="writing-mode: horizontal-tb;"><i>(0)</i></span> ᠲᠣᠭ ᠠ᠋ ᠪᠣᠳᠠᠲᠤ ᠪᠤᠶᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠠᠯᠢ ᠨᠢ ᠴᠤ ᠪᠠᠢᠢᠵᠤ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
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|style = "height: 150px;"|ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭᠠ ᠪᠦᠬᠦᠢ ᠲᠣᠭᠠᠴᠠᠭᠠᠯᠠᠯ ᠤᠨ ᠳᠦᠷᠰᠦᠯᠡᠯ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠳ᠋ᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠳ᠋ᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃<br />
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ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠳᠦᠷᠰᠦᠯᠡᠯ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;"><i>–1</i></span> ᠢᠢᠡᠷ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ᠢ ᠲᠣᠭᠤᠷᠢᠭᠰᠠᠨ <span style="writing-mode: horizontal-tb;"><i>180</i></span> ᠬᠡᠮ ᠦᠨ ᠡᠷᠭᠢᠯᠲᠡ ᠪᠣᠯᠤᠨ ᠠ᠋᠃ <br />
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i ᠲᠣᠭ ᠠ᠋ ᠪᠡᠷ ᠦᠷᠡᠵᠢᠯᠭᠦᠯᠬᠦ ᠢᠨᠦ ᠨᠠᠷᠠ ᠪᠤᠷᠤᠭᠤ “ᠡᠷᠡᠭᠦ” ᠴᠢᠭ᠌ ᠲᠦ <span style="writing-mode: horizontal-tb;"><i>90</i></span> ᠬᠡᠮ ᠦᠨ ᠡᠷᠭᠢᠯᠲᠡ ᠪᠣᠯᠬᠤ ᠪᠠ ᠬᠡᠷᠪᠡ <span style="writing-mode: horizontal-tb;"><i>i2 = -1</i></span> ᠭᠡᠰᠡᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢᠨᠦ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ᠢ ᠲᠣᠭᠤᠷᠢᠭᠰᠠᠨ <span style="writing-mode: horizontal-tb;"><i>90</i></span> ᠬᠡᠮ ᠦᠨ ᠬᠣᠶᠠᠷ ᠡᠷᠭᠢᠯᠲᠡ ᠭᠡᠵᠦ ᠦᠵᠡᠪᠡᠯ ᠦᠷ ᠡ᠋ ᠳ᠋ᠦ᠋ᠩ ᠢᠨᠦ ᠭᠠᠭᠴᠠ <span style="writing-mode: horizontal-tb;"><i>180</i></span> ᠬᠡᠮ ᠦᠨ ᠡᠷᠭᠢᠯᠲᠡ ᠪᠣᠯᠤᠨ ᠠ᠋᠃ <br />
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“ᠰᠥᠷᠭᠦ” (ᠨᠠᠷᠠ ᠵᠥᠪ) ᠴᠢᠭ᠌ ᠲᠦ <span style="writing-mode: horizontal-tb;"><i>90</i></span> ᠬᠡᠮ ᠡᠷᠭᠢᠬᠦ ᠢᠨᠦ ᠮᠥᠨ ᠡᠨᠡ ᠳᠦᠷᠰᠦᠯᠡᠯ ᠳ᠋ᠦ ᠨᠡᠶᠢᠴᠡᠬᠦ ᠪᠠ <span style="writing-mode: horizontal-tb;"><i>x2 = -1</i></span> ᠭᠡᠰᠡᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠶ᠋ᠢᠳ <span style="writing-mode: horizontal-tb;"><i>−i</i></span> ᠪᠣᠯᠬᠤ ᠢ ᠬᠠᠷᠠᠭᠤᠯᠵᠤ ᠪᠠᠢᠢᠨ ᠠ᠋᠃ <br />
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ᠶᠡᠷᠦᠩᠬᠡᠢᠢ ᠳ᠋ᠦ ᠪᠡᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ ᠦᠷᠡᠵᠢᠯᠭᠦᠯᠬᠦ ᠢᠨᠦ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ᠢ ᠲᠣᠭᠤᠷᠢᠭᠤᠯᠤᠨ ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭᠠᠨ ᠤ ᠠᠷᠭᠤᠮᠧᠨ᠍ᠲ ᠢᠢᠠᠷ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ᠂ ᠴᠠᠷ ᠬᠡᠮᠵᠢᠶ ᠡ᠋ ᠪᠡᠷ ᠢᠨᠦ ᠤᠷᠲᠤ ᠢᠢ ᠴᠢᠴᠠᠤᠯᠠᠬᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠪᠣᠯᠤᠨᠠ᠃ <br />
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ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭᠠᠨ ᠤ ᠠᠷᠭᠤᠮᠧᠨ᠍ᠲ ᠭᠡᠳᠡᠭ ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠡᠬᠢᠯᠠᠯ ᠴᠡᠭ ᠢ ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭᠠᠨ z ᠲᠡᠢ ᠬᠣᠯᠪᠤᠵᠤ ᠪᠠᠢᠢᠭᠠ ᠱᠤᠭᠤᠮ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠢᠢᠨ ᠥᠨᠴᠦᠭ <span style="writing-mode: horizontal-tb;"><i>φ</i></span> ᠶᠤᠮ᠃<br />
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==ᠬᠡᠷᠡᠭᠯᠡᠭᠡ==<br />
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[[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠢᠢ ''i'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠬᠦ ᠠᠭᠠᠳ ᠧᠯᠧᠺᠲ᠋ᠷᠣᠨᠢᠭ᠌ ᠲᠦ<sup>electronic</sup> ''j'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠨ ᠡ᠋᠃ <br />
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ᠡᠨᠡ ᠢᠨᠦ ''i'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠳᠡᠭ ᠴᠠᠬᠢᠯᠭᠠᠨᠭᠦᠶᠦᠳᠦᠯ ᠡᠴᠡ ᠢᠯᠭᠠᠬᠤ ᠱᠠᠭᠠᠷᠳᠠᠯᠭ ᠠ᠋ ᠡᠴᠡ ᠤᠷᠭᠠᠨ ᠭᠠᠷᠤᠭᠰᠠᠨ ᠤᠯᠠᠮᠵᠢᠯᠠᠯ ᠶᠤᠮ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠧᠯᠧᠺᠲ᠋ᠷᠣᠨᠢᠭ᠌ ᠲᠦ᠂ ᠢᠯᠠᠩᠭᠤᠶ ᠠ᠋ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠳ᠋ᠦ ᠣᠩᠴᠤᠭᠤᠢ ᠠᠴᠢ ᠬᠣᠯᠪᠤᠭᠳᠠᠯ ᠲᠠᠢ᠃ <br />
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ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠰᠢᠨᠦ᠋ᠰ<sup>sinus</sup> ᠳᠣᠯᠭᠢᠶᠠᠨ ᠳᠤ ᠨᠡᠮᠡᠬᠦ ᠪᠠ ᠬᠠᠰᠤᠬᠤ ᠢᠢᠨ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠰᠣᠯᠢᠭᠳᠠᠳᠠᠭ᠃<br />
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{| class="wikitable"<br />
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|style = "height: 150px;"|ᠬᠦᠪᠰᠬᠤ ᠭᠦᠶᠦᠳᠡᠯ ᠪᠦᠬᠦᠢ ᠡᠷᠴᠢᠮ ᠬᠦᠴᠦ᠃ <span style="writing-mode: horizontal-tb;"><i>P</i></span> ᠢᠨᠦ ᠢᠳᠡᠪᠬᠢᠲᠦ ᠴᠢᠳᠠᠯ᠂ <span style="writing-mode: horizontal-tb;"><i>Q</i></span> ᠢᠨᠦ ᠬᠠᠷᠢᠭᠤ ᠢᠳᠡᠪᠬᠢᠲᠦ ᠴᠢᠳᠠᠯ᠂ <span style="writing-mode: horizontal-tb;"><i>S</i></span> ᠨᠡᠶᠢᠯᠡᠪᠦᠷᠢ ᠴᠢᠳᠠᠯ᠃<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠢ ᠶᠠᠪ ᠴᠠᠪ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠬᠤ ᠬᠡᠴᠡᠭᠦ ᠲᠤᠯᠠ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠦᠳ ᠢ ᠳᠠᠪᠬᠤᠷᠯᠠᠬᠤ ᠨᠡᠨ ᠲᠦᠪᠡᠭ ᠲᠡᠢ᠃<br />
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ᠬᠠᠷᠢᠨ ᠲᠥᠰᠦᠭᠡᠯᠡᠯ ᠭᠦᠶᠦᠳᠦᠯ ᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠨᠤᠭᠤᠳ ᠲᠠᠶ ᠬᠣᠣᠰᠮᠠᠭᠯᠠᠭᠤᠯᠤᠭᠰᠠᠨ ᠢᠢᠠᠷ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠦᠨ ᠲᠣᠭᠠᠴᠠᠭ ᠠ᠋ ᠬᠢᠵᠦ᠂ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠤ ᠣᠰᠤᠯ ᠡᠴᠡ ᠤᠷᠢᠳᠤᠴᠢᠯᠠᠨᠰᠡᠷᠭᠡᠢᠢᠯᠬᠦ ᠲᠤᠰᠠ ᠲᠠᠢ ᠪᠠᠢᠢᠳᠠᠭ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠠᠭ ᠠ᠋ ᠢᠢ ᠮᠥᠨ ᠰᠢᠭ᠌ᠨᠠᠯ<sup>signal</sup> ᠪᠣᠯᠪᠠᠰᠤᠷᠠᠭᠤᠯᠬᠤ ᠲᠧᠺᠨᠣᠯᠣᠭᠢ ᠳ᠋ᠤ<sup>technology</sup> ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ <br />
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ᠡᠨᠡ ᠨᠢ ᠡᠭᠦᠷᠦᠨ ᠬᠣᠯᠪᠤᠭ ᠠ᠋᠂ ᠤᠲᠠᠰᠤ ᠦᠭᠡᠢ ᠳᠠᠮᠵᠢᠭᠤᠯᠤᠯᠲᠠ᠂ ᠷᠠᠳᠠᠷ᠂ ᠪᠢᠣᠯᠣᠭᠢ (ᠲᠠᠷᠢᠬᠢᠨ ᠤ ᠳᠣᠯᠭᠢᠶᠠᠨ) ᠰᠤᠳᠤᠯᠬᠤ ᠳ᠋ᠤ ᠬᠠᠮᠢᠶᠠᠷᠠᠨ ᠠ᠋᠃<br />
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ᠶᠡᠷᠦ᠂ ᠰᠢᠨᠦ᠋ᠰ ᠪᠠ ᠺᠣᠰᠢᠨᠦ᠋ᠰ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠪᠦᠬᠦᠢ ᠵᠦᠢᠯ ᠢ ᠬᠡᠮᠵᠢᠬᠦ᠂ ᠲᠣᠭᠠᠴᠠᠭᠠᠯᠬᠤ ᠳ᠋ᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠴᠢᠬᠤᠯᠠ ᠬᠡᠷᠡᠭ ᠲᠡᠢ᠃<ref name=":0" /><br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==<br />
<div style="writing-mode: horizontal-tb; "><References/></div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%AC%E1%A0%A0%E1%A0%AD%E1%A0%A4%E1%A0%B7%E1%A0%AE%E1%A0%A0%E1%A0%AD_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%8A%E1%A0%A0%E1%A0%8B&diff=2712
ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠊ᠠ᠋
2021-07-31T10:35:45Z
<p>Angka: Edit</p>
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ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠢᠨᠦ ᠲᠡᠭ <span style="writing-mode: horizontal-tb;"><i>(0)</i></span> ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ [[ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ]] ᠢᠢ '''ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋''' ᠬᠠᠮᠠᠮᠦᠢ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠺᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠬᠠᠰᠤᠬᠤ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠬᠠᠷᠢᠭᠤ ᠭᠠᠷᠤᠨ ᠠ᠋᠃<ref name=":0">https://www.livescience.com/42748-imaginary-numbers.html Live Science. Elaine J. Hom. What Are Imaginary Numbers? 2014/01/22. </ref> <br />
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ᠲᠡᠭᠡᠬᠦ ᠪᠡᠷ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠠᠭ ᠠ᠋ ᠠᠨᠤ ᠬᠠᠰᠤᠬᠤ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠶ ᠲᠣᠭᠠᠨ ᠤ ᠬᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠮᠥᠨ᠃ <br />
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ᠲᠡᠭ <span style="writing-mode: horizontal-tb;"><i>(0)</i></span> ᠲᠣᠭ ᠠ᠋ ᠪᠣᠳᠠᠲᠤ ᠪᠤᠶᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠠᠯᠢ ᠨᠢ ᠴᠤ ᠪᠠᠢᠢᠵᠤ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠳ᠋ᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠳ᠋ᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃<br />
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ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠳᠦᠷᠰᠦᠯᠡᠯ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;"><i>–1</i></span> ᠢᠢᠡᠷ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ᠢ ᠲᠣᠭᠤᠷᠢᠭᠰᠠᠨ <span style="writing-mode: horizontal-tb;"><i>180</i></span> ᠬᠡᠮ ᠦᠨ ᠡᠷᠭᠢᠯᠲᠡ ᠪᠣᠯᠤᠨ ᠠ᠋᠃ <br />
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i ᠲᠣᠭ ᠠ᠋ ᠪᠡᠷ ᠦᠷᠡᠵᠢᠯᠭᠦᠯᠬᠦ ᠢᠨᠦ ᠨᠠᠷᠠ ᠪᠤᠷᠤᠭᠤ “ᠡᠷᠡᠭᠦ” ᠴᠢᠭ᠌ ᠲᠦ <span style="writing-mode: horizontal-tb;"><i>90</i></span> ᠬᠡᠮ ᠦᠨ ᠡᠷᠭᠢᠯᠲᠡ ᠪᠣᠯᠬᠤ ᠪᠠ ᠬᠡᠷᠪᠡ <span style="writing-mode: horizontal-tb;"><i>i2 = -1</i></span> ᠭᠡᠰᠡᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢᠨᠦ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ᠢ ᠲᠣᠭᠤᠷᠢᠭᠰᠠᠨ <span style="writing-mode: horizontal-tb;"><i>90</i></span> ᠬᠡᠮ ᠦᠨ ᠬᠣᠶᠠᠷ ᠡᠷᠭᠢᠯᠲᠡ ᠭᠡᠵᠦ ᠦᠵᠡᠪᠡᠯ ᠦᠷ ᠡ᠋ ᠳ᠋ᠦ᠋ᠩ ᠢᠨᠦ ᠭᠠᠭᠴᠠ <span style="writing-mode: horizontal-tb;"><i>180</i></span> ᠬᠡᠮ ᠦᠨ ᠡᠷᠭᠢᠯᠲᠡ ᠪᠣᠯᠤᠨ ᠠ᠋᠃ <br />
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“ᠰᠥᠷᠭᠦ” (ᠨᠠᠷᠠ ᠵᠥᠪ) ᠴᠢᠭ᠌ ᠲᠦ <span style="writing-mode: horizontal-tb;"><i>90</i></span> ᠬᠡᠮ ᠡᠷᠭᠢᠬᠦ ᠢᠨᠦ ᠮᠥᠨ ᠡᠨᠡ ᠳᠦᠷᠰᠦᠯᠡᠯ ᠳ᠋ᠦ ᠨᠡᠶᠢᠴᠡᠬᠦ ᠪᠠ <span style="writing-mode: horizontal-tb;"><i>x2 = -1</i></span> ᠭᠡᠰᠡᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠶ᠋ᠢᠳ <span style="writing-mode: horizontal-tb;"><i>−i</i></span> ᠪᠣᠯᠬᠤ ᠢ ᠬᠠᠷᠠᠭᠤᠯᠵᠤ ᠪᠠᠢᠢᠨ ᠠ᠋᠃ <br />
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ᠶᠡᠷᠦᠩᠬᠡᠢᠢ ᠳ᠋ᠦ ᠪᠡᠨ ᠺᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ ᠦᠷᠡᠵᠢᠯᠭᠦᠯᠬᠦ ᠢᠨᠦ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ᠢ ᠲᠣᠭᠤᠷᠢᠭᠤᠯᠤᠨ ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭᠠᠨ ᠤ ᠠᠷᠭᠤᠮᠧᠨ᠍ᠲ ᠢᠢᠠᠷ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ᠂ ᠴᠠᠷ ᠬᠡᠮᠵᠢᠶ ᠡ᠋ ᠪᠡᠷ ᠢᠨᠦ ᠤᠷᠲᠤ ᠢᠢ ᠴᠢᠴᠠᠤᠯᠠᠬᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠪᠣᠯᠤᠨᠠ᠃ <br />
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ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭᠠᠨ ᠤ ᠠᠷᠭᠤᠮᠧᠨ᠍ᠲ ᠭᠡᠳᠡᠭ ᠢᠨᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠡᠬᠢᠯᠠᠯ ᠴᠡᠭ ᠢ ᠻᠣᠮᠫ᠊ᠯᠧᠻᠰ ᠲᠣᠭᠠᠨ z ᠲᠡᠢ ᠬᠣᠯᠪᠤᠵᠤ ᠪᠠᠢᠢᠭᠠ ᠱᠤᠭᠤᠮ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠢᠢᠨ ᠥᠨᠴᠦᠭ <span style="writing-mode: horizontal-tb;"><i>φ</i></span> ᠶᠤᠮ᠃<br />
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==ᠬᠡᠷᠡᠭᠯᠡᠭᠡ==<br />
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[[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠢᠢ ''i'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠬᠦ ᠠᠭᠠᠳ ᠧᠯᠧᠺᠲ᠋ᠷᠣᠨᠢᠭ᠌ ᠲᠦ<sup>electronic</sup> ''j'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠨ ᠡ᠋᠃ <br />
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ᠡᠨᠡ ᠢᠨᠦ ''i'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠳᠡᠭ ᠴᠠᠬᠢᠯᠭᠠᠨᠭᠦᠶᠦᠳᠦᠯ ᠡᠴᠡ ᠢᠯᠭᠠᠬᠤ ᠱᠠᠭᠠᠷᠳᠠᠯᠭ ᠠ᠋ ᠡᠴᠡ ᠤᠷᠭᠠᠨ ᠭᠠᠷᠤᠭᠰᠠᠨ ᠤᠯᠠᠮᠵᠢᠯᠠᠯ ᠶᠤᠮ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠧᠯᠧᠺᠲ᠋ᠷᠣᠨᠢᠭ᠌ ᠲᠦ᠂ ᠢᠯᠠᠩᠭᠤᠶ ᠠ᠋ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠳ᠋ᠦ ᠣᠩᠴᠤᠭᠤᠢ ᠠᠴᠢ ᠬᠣᠯᠪᠤᠭᠳᠠᠯ ᠲᠠᠢ᠃ <br />
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ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠰᠢᠨᠦ᠋ᠰ<sup>sinus</sup> ᠳᠣᠯᠭᠢᠶᠠᠨ ᠳᠤ ᠨᠡᠮᠡᠬᠦ ᠪᠠ ᠬᠠᠰᠤᠬᠤ ᠢᠢᠨ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠰᠣᠯᠢᠭᠳᠠᠳᠠᠭ᠃<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠢ ᠶᠠᠪ ᠴᠠᠪ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠬᠤ ᠬᠡᠴᠡᠭᠦ ᠲᠤᠯᠠ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠦᠳ ᠢ ᠳᠠᠪᠬᠤᠷᠯᠠᠬᠤ ᠨᠡᠨ ᠲᠦᠪᠡᠭ ᠲᠡᠢ᠃<br />
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ᠬᠠᠷᠢᠨ ᠲᠥᠰᠦᠭᠡᠯᠡᠯ ᠭᠦᠶᠦᠳᠦᠯ ᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠨᠤᠭᠤᠳ ᠲᠠᠶ ᠬᠣᠣᠰᠮᠠᠭᠯᠠᠭᠤᠯᠤᠭᠰᠠᠨ ᠢᠢᠠᠷ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠦᠨ ᠲᠣᠭᠠᠴᠠᠭ ᠠ᠋ ᠬᠢᠵᠦ᠂ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠤ ᠣᠰᠤᠯ ᠡᠴᠡ ᠤᠷᠢᠳᠤᠴᠢᠯᠠᠨᠰᠡᠷᠭᠡᠢᠢᠯᠬᠦ ᠲᠤᠰᠠ ᠲᠠᠢ ᠪᠠᠢᠢᠳᠠᠭ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠠᠭ ᠠ᠋ ᠢᠢ ᠮᠥᠨ ᠰᠢᠭ᠌ᠨᠠᠯ<sup>signal</sup> ᠪᠣᠯᠪᠠᠰᠤᠷᠠᠭᠤᠯᠬᠤ ᠲᠧᠺᠨᠣᠯᠣᠭᠢ ᠳ᠋ᠤ<sup>technology</sup> ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ <br />
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ᠡᠨᠡ ᠨᠢ ᠡᠭᠦᠷᠦᠨ ᠬᠣᠯᠪᠤᠭ ᠠ᠋᠂ ᠤᠲᠠᠰᠤ ᠦᠭᠡᠢ ᠳᠠᠮᠵᠢᠭᠤᠯᠤᠯᠲᠠ᠂ ᠷᠠᠳᠠᠷ᠂ ᠪᠢᠣᠯᠣᠭᠢ (ᠲᠠᠷᠢᠬᠢᠨ ᠤ ᠳᠣᠯᠭᠢᠶᠠᠨ) ᠰᠤᠳᠤᠯᠬᠤ ᠳ᠋ᠤ ᠬᠠᠮᠢᠶᠠᠷᠠᠨ ᠠ᠋᠃<br />
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ᠶᠡᠷᠦ᠂ ᠰᠢᠨᠦ᠋ᠰ ᠪᠠ ᠺᠣᠰᠢᠨᠦ᠋ᠰ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠪᠦᠬᠦᠢ ᠵᠦᠢᠯ ᠢ ᠬᠡᠮᠵᠢᠬᠦ᠂ ᠲᠣᠭᠠᠴᠠᠭᠠᠯᠬᠤ ᠳ᠋ᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠴᠢᠬᠤᠯᠠ ᠬᠡᠷᠡᠭ ᠲᠡᠢ᠃<ref name=":0" /><br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==<br />
<div style="writing-mode: horizontal-tb; "><References/></div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%AD%E1%A0%A1%E1%A0%B7%E1%A0%A1%E1%A0%AF_%E1%A0%8A%E1%A0%A2%E1%A0%A2%E1%A0%A1%E1%A0%B7_%E1%A0%A8%E1%A0%A1%E1%A0%B6%E1%A0%8B%E1%A0%A2%E1%A0%AF%E1%A0%A1%E1%A0%AD%E1%A0%B5%E1%A0%A2%E1%A0%AC%E1%A0%A6%E1%A0%A2_(%E1%A0%B9%E1%A0%A3%E1%A0%B2%E1%A0%8B%E1%A0%A3%E1%A0%B0%E1%A0%A2%E1%A0%A8%E1%A0%8D%E1%A0%B2%E1%A0%8B%E1%A0%B8%E1%A0%BD)&diff=2709
ᠭᠡᠷᠡᠯ ᠊ᠢᠢᠡᠷ ᠨᠡᠶ᠋ᠢᠯᠡᠭᠵᠢᠬᠦᠢ (ᠹᠣᠲ᠋ᠣᠰᠢᠨ᠍ᠲ᠋ᠸᠽ)
2021-07-21T00:40:02Z
<p>Angka: Angka moved page ᠭᠡᠷᠡᠯ ᠊ᠢᠢᠡᠷ ᠨᠡᠶ᠋ᠢᠯᠡᠭᠵᠢᠬᠦᠢ (ᠹᠣᠲ᠋ᠣᠰᠢᠨ᠍ᠲ᠋ᠸᠽ) to ᠭᠡᠷᠡᠯ ᠊ᠢᠢᠡᠷ ᠨᠡᠶ᠋ᠢᠯᠡᠭᠵᠢᠬᠦᠢ: URL</p>
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<div>#ЧИГЛҮҮЛЭГ [[ᠭᠡᠷᠡᠯ ᠊ᠢᠢᠡᠷ ᠨᠡᠶ᠋ᠢᠯᠡᠭᠵᠢᠬᠦᠢ]]</div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%AD%E1%A0%A1%E1%A0%B7%E1%A0%A1%E1%A0%AF_%E1%A0%8A%E1%A0%A2%E1%A0%A2%E1%A0%A1%E1%A0%B7_%E1%A0%A8%E1%A0%A1%E1%A0%B6%E1%A0%8B%E1%A0%A2%E1%A0%AF%E1%A0%A1%E1%A0%AD%E1%A0%B5%E1%A0%A2%E1%A0%AC%E1%A0%A6%E1%A0%A2&diff=2708
ᠭᠡᠷᠡᠯ ᠊ᠢᠢᠡᠷ ᠨᠡᠶ᠋ᠢᠯᠡᠭᠵᠢᠬᠦᠢ
2021-07-21T00:39:59Z
<p>Angka: Angka moved page ᠭᠡᠷᠡᠯ ᠊ᠢᠢᠡᠷ ᠨᠡᠶ᠋ᠢᠯᠡᠭᠵᠢᠬᠦᠢ (ᠹᠣᠲ᠋ᠣᠰᠢᠨ᠍ᠲ᠋ᠸᠽ) to ᠭᠡᠷᠡᠯ ᠊ᠢᠢᠡᠷ ᠨᠡᠶ᠋ᠢᠯᠡᠭᠵᠢᠬᠦᠢ: URL</p>
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<div><div style = "writing-mode: vertical-lr"><br />
=ᠭᠡᠷᠡᠯ ᠢᠢᠡᠷ ᠨᠡᠶ᠋ᠢᠯᠡᠭᠵᠢᠬᠦᠢ<br>(ᠹᠣᠲ᠋ᠣᠰᠢᠨ᠍ᠲ᠋ᠸᠽ)=<br />
{{Horizontal}}[[File:Photosynthesis_en.svg|thumb|{{Mongolian}}ᠤᠷᠭᠤᠮᠠᠯ ᠤᠨ<br>ᠹᠣᠲ᠋ᠣᠰᠢᠨ᠍ᠲ᠋ᠸᠽ ᠦᠨ<br>ᠲᠣᠶ᠋ᠢᠮᠤ<br>ᠵᠢᠷᠤᠭ᠃<br>ᠡᠭᠦᠰᠦᠭᠰᠡᠨ<br>ᠨᠡᠭᠦᠷᠡᠰᠦ ᠤᠰᠤ<br>ᠠᠨᠤ ᠤᠷᠭᠤᠮᠠᠯ ᠳᠦ<br>ᠬᠠᠳᠭᠠᠯᠠᠭᠳᠠᠵᠤ᠂<br>ᠡᠰᠡᠬᠦᠯᠡ<br>ᠠᠰᠢᠭᠯᠠᠭᠳᠠᠨᠠ᠃]]{{Z}}<br><br><br />
'''ᠤᠷᠭᠤᠮᠤᠯ ᠤᠨ ᠹᠣᠲ᠋ᠣᠰᠢᠨ᠍ᠲᠸᠽ''' ᠪᠣᠯ ᠨᠠᠷᠠᠨ ᠤ ᠭᠡᠷᠡᠯ ᠦᠨ ᠴᠠᠬᠢᠯᠠᠭᠠᠨ ᠰᠣᠷᠢᠨᠵᠢᠨ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠢ ᠬᠢᠮᠢ ᠢᠢᠨ ᠡᠷᠴᠢᠮ ᠬᠦᠴᠦ ᠪᠣᠯᠭᠠᠨ ᠬᠤᠪᠢᠷᠠᠭᠠᠭᠠᠳ᠂ ᠠᠲ᠋ᠮᠣᠰᠹᠧᠷ ᠠᠴᠠ ᠰᠢᠩᠭᠡᠭᠡᠳᠡᠭ ᠨᠡᠭᠦᠷᠡᠰᠦᠬᠦᠴᠢᠯ ᠦᠨ ᠬᠡᠢ ᠢᠢ ᠵᠠᠳᠠᠯᠵᠤ᠂ ᠬᠦᠴᠢᠯᠲᠦ᠋ᠷᠦᠭᠴᠢ ᠢᠯᠭᠠᠷᠠᠭᠤᠯᠬᠤ ᠦᠢᠯᠡ ᠶᠠᠪᠤᠴᠠ ᠶᠤᠮ᠃ <br>ᠨᠠᠷᠠᠨ ᠤ ᠬᠤᠪᠢᠷᠠᠭᠰᠠᠨ ᠧᠨᠧᠷᠭᠢ ᠢᠨᠤ᠂ ᠨᠡᠭᠦᠷᠡᠰᠦᠬᠥᠴᠢᠯ ᠤᠰᠤ ᠬᠣᠶᠠᠷ ᠠᠴᠠ ᠡᠭᠦᠰᠳᠡᠭ ᠪᠣᠳᠠᠰ ᠪᠣᠯᠬᠤ ᠨᠡᠭᠦᠷᠡᠰᠦ-ᠤᠰᠤ ᠵᠡᠷᠭᠡ ᠳᠦ ᠬᠣᠪᠢᠯᠭᠠᠬᠣ ᠠᠮᠤᠢ᠃ <br>ᠨᠡᠷᠡ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠭᠠᠷᠪᠠᠯᠢ ᠶᠢ ᠶᠠᠷᠢᠪᠠᠰᠤ᠂ ᠭᠧᠷᠧᠭ ᠦᠨ phōs (φῶς)᠂ ᠭᠡᠷᠡᠯ᠂ sunthesis (σύνθεσις)᠂ ᠨᠡᠶ᠋ᠢᠯᠡᠭᠵᠢ᠂ ᠦᠭᠡᠰ ᠡᠴᠡ ᠡᠭᠦᠰᠴᠦᠬᠦᠢ᠃ <br>ᠬᠦᠴᠢᠯᠲᠥ᠋ᠷᠦᠭᠴᠢ ᠢᠨᠤ ᠶᠡᠬᠡ ᠲᠥᠯᠦᠪ ᠳᠠᠶᠢᠪᠤᠷᠢ ᠪᠦᠲᠦᠭᠡᠭᠳᠡᠬᠦᠨ ᠨᠢ ᠠᠵᠤᠭᠤ᠃ <br>'''ᠹᠣᠲ᠋ᠣ᠋-ᠠᠦ᠋ᠲ᠋ᠣᠲ᠋ᠷᠣᠹ''' ᠮᠠᠬᠠᠪᠤᠳ ᠪᠣᠯᠬᠤ᠂ ᠤᠷᠭᠤᠮᠠᠯ᠂ ᠶᠡᠬᠡᠩᠬᠢ ᠵᠠᠮᠠᠭ᠂ ᠬᠥᠬᠡ ᠨᠣᠭᠤᠭᠠᠨ ᠵᠠᠮᠠᠭ ᠲᠤ ᠭᠡᠷᠡᠯ ᠪᠡᠷ ᠨᠡᠶ᠋ᠢᠯᠡᠭᠵᠢᠬᠦᠢ ᠤᠷᠪᠠᠯ ᠶᠠᠪᠤᠭᠳᠠᠮᠤᠢ᠃ <br>ᠡᠨᠡ ᠬᠦ ᠮᠠᠬᠠᠪᠤᠳᠠᠰ ᠤᠨ ᠬᠣᠪᠢᠷᠠᠭᠠᠭᠰᠠᠨ ᠧᠨᠧᠷᠬᠢ ᠢᠨᠤ ᠳᠡᠯᠡᠬᠡᠶ ᠶᠢᠨ ᠠᠮᠢᠳᠤᠷᠠᠯ ᠳᠤ ᠮᠠᠰᠢ ᠴᠢᠬᠤᠯᠠ᠂ ᠳᠡᠭᠡᠷᠡ ᠡᠴᠡ ᠢᠨᠤ ᠳᠡᠯᠡᠬᠡᠶ ᠶᠢᠨ ᠬᠦᠴᠢᠯᠲᠥ᠋ᠷᠦᠭᠴᠢ ᠶᠢᠨ ᠳᠡᠶ᠋ᠢᠯᠦᠩᠬᠢ ᠶᠢ ᠢᠯᠭᠠᠷᠠᠭᠤᠯᠤᠮᠤᠢ᠃<br><br />
ᠬᠡᠳᠦᠢᠢᠪᠡᠷ ᠵᠦᠢᠯ ᠡᠴᠡ ᠬᠠᠮᠢᠶᠠᠷᠤᠨ᠂ ᠤᠷᠪᠠᠯ ᠤᠨ ᠶᠠᠪᠤᠬᠤ ᠵᠠᠮ ᠥᠭᠡᠷᠡ ᠪᠣᠯᠪᠠᠴᠤ᠂ <b>ᠻᠯᠣᠷᠣᠹᠢᠯᠯ</b> ᠨᠥᠰᠦᠭᠡ ᠶᠢ ᠠᠭᠤᠯᠤᠭᠰᠠᠨ᠂ ᠮᠠᠬᠠᠪᠤᠳ ᠲᠤ ᠣᠷᠤᠰᠢᠬᠤ [[ᠤᠭᠤᠷᠠᠭ]]᠂ <b>ᠤᠷᠪᠠᠯ ᠶᠢᠨ ᠲᠥᠪ</b> ᠬᠡᠮᠡᠬᠦ ᠬᠡᠰᠡᠭ ᠲᠤᠷ ᠨᠠᠷᠠᠨ ᠤ ᠭᠡᠷᠡᠯ ᠰᠢᠩᠭᠡᠭᠡᠭᠳᠡᠵᠦ᠂ ᠤᠷᠪᠠᠯ ᠡᠬᠢᠯᠡᠶᠦ᠃ <br>ᠤᠷᠭᠤᠮᠠᠯ ᠳᠦᠷ ᠡᠨᠡ ᠤᠭᠤᠷᠠᠭ ᠶᠡᠬᠡ ᠲᠥᠯᠦᠪ ᠨᠠᠪᠴᠢᠨ ᠤ ᠡᠰ ᠳᠣᠲᠤᠷᠠ ᠪᠠᠶ᠋ᠢᠬᠤ <b>ᠻᠯᠣᠷᠣᠫᠯᠠᠰᠲ</b> ᠡᠷᠬᠡᠲᠡᠨᠴᠡᠷ ᠲᠦ ᠪᠠᠶ᠋ᠢᠮᠤᠢ᠃ <br>ᠬᠠᠷᠢᠨ ᠪᠠᠻᠲᠧᠷᠢ ᠳᠤᠷ ᠡᠨᠡ ᠻᠯᠣᠷᠣᠫᠯᠠᠰᠲ ᠡᠷᠬᠡᠲᠡᠨᠴᠡᠷ ᠢᠨᠤ ᠫᠯᠠᠰᠮ ᠮᠧᠮᠪᠷᠠᠨ ᠳᠡᠭᠡᠷᠡ ᠬᠣᠯᠪᠤᠭᠳᠠᠭᠰᠠᠨ ᠪᠠᠶ᠋ᠢᠳᠠᠯᠲᠠᠢ᠃ <br><b>ᠭᠡᠷᠡᠯ ᠯᠦᠭᠡ ᠬᠠᠮᠢᠶᠠᠷᠠᠯᠲᠦ ᠤᠷᠪᠠᠯ</b> ᠶᠠᠪᠤᠬᠤ ᠭᠣᠣᠯ ᠵᠠᠮ ᠠᠨᠤ᠂ ᠨᠠᠷᠠᠨ ᠠᠴᠠ ᠰᠢᠩᠭᠡᠭᠡᠭᠳᠡᠭᠰᠡᠨ ᠧᠨᠧᠷᠬᠢ ᠢᠨᠤ ᠲᠣᠬᠢᠷᠠᠮᠵᠢᠲᠠᠢ ᠪᠣᠳᠠᠰ ᠠᠴᠠ ᠧᠯᠧᠻᠲ᠋ᠷᠣᠨ᠌ ᠢ ᠰᠠᠯᠭᠠᠵᠤ ᠠᠪᠤᠬᠤ ᠳᠤ ᠠᠰᠢᠭᠯᠠᠭᠳᠠᠮᠤᠢ᠂ ᠵᠢᠰᠢᠶᠡᠯᠡᠪᠡᠯ ᠤᠰᠤᠨ ᠤ ᠮᠣᠯᠧᠻᠤᠯ ᠠᠴᠠ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠣᠶᠠᠷ ᠬᠣᠶᠠᠷ ᠧᠯᠧᠻᠲ᠋ᠷᠣᠨ᠌ ᠰᠠᠯᠭᠠᠵᠤ ᠠᠪᠤᠨ᠌ ᠠ᠋᠃<br />
{{Horizontal}}[[File:Photosynthesis equation.svg|thumb|{{Mongolian}}ᠭᠡᠷᠡᠯ ᠢᠢᠡᠷ<br>ᠨᠡᠶ᠋ᠢᠯᠡᠭᠵᠢᠬᠦᠢ ᠶᠢᠨ᠂<br>ᠤᠷᠭᠤᠮᠠᠯᠠ<br>ᠶᠡᠬᠡ<br>ᠲᠥᠯᠦᠪ<br>ᠶᠠᠪᠤᠭᠳᠠᠳᠠᠭ<br>ᠤᠷᠪᠠᠯ ᠤᠨ<br>ᠲᠡᠭᠰᠢᠳᠬᠡᠯ᠃]]{{Z}}<br><br><br />
ᠤᠰᠤᠨ ᠤ ᠵᠠᠳᠠᠷᠠᠯ ᠢᠢᠠᠷ ᠡᠭᠦᠰᠦᠭᠰᠡᠨ ᠤᠰᠤᠲᠥ᠋ᠷᠦᠭᠴᠢ ᠶᠢᠨ ᠢᠣᠨ᠌᠂ ᠧᠯᠸᠻᠲ᠋ᠷᠣᠨ᠌ ᠢ ᠵᠥᠭᠡᠭᠡᠭᠴᠢ ᠪᠣᠳᠠᠰ ᠤᠳ ᠶᠢ ᠠᠨᠩᠭᠢᠵᠢᠷᠠᠭᠤᠯᠬᠤᠢ᠂ ᠡᠭᠦᠰᠬᠡᠬᠦᠢ ᠳᠤ ᠣᠷᠣᠯᠴᠠᠳᠠᠭ᠃ <br>ᠡᠳᠡᠭᠡᠷ ᠧᠯᠧᠻᠲ᠋ᠷᠣᠨ᠌ ᠢ ᠵᠥᠭᠡᠭᠡᠭᠴᠢ ᠪᠣᠳᠠᠰ ᠤᠨ ᠠᠩᠭᠢᠵᠢᠷᠠᠭᠰᠠᠨ ᠬᠡᠯᠪᠡᠷᠢ ᠢᠨᠤ᠂ ᠨᠢᠻᠣᠲ᠋ᠢᠨᠠᠮᠢᠳ᠋ ᠠᠳᠧᠨᠢᠨ᠌ ᠳ᠋ᠢᠨ᠍ᠦ᠋ᠻᠯᠧᠣᠲ᠋ᠢ᠌ᠳ᠋ (NADPH)᠂ ᠠᠳᠧᠨᠢᠨ᠌ ᠲᠷᠢᠹᠣᠰᠹᠠᠲ(ATP)᠃<br><br />
ᠤᠷᠭᠤᠮᠠᠯ᠂ ᠵᠠᠮᠠᠭ᠂ ᠬᠥᠬᠡ ᠨᠣᠭᠤᠭᠠᠨ ᠵᠠᠮᠠᠭ ᠲᠤᠷ '''ᠭᠡᠷᠡᠯ ᠡᠴᠡ ᠦᠯᠦ ᠬᠠᠮᠢᠶᠠᠷᠠᠬᠤ ᠤᠷᠪᠠᠯ''' ᠶᠠᠪᠤᠭᠳᠠᠨᠠ᠃ <br>ᠡᠨᠡ ᠢᠨᠤ '''[[ᠨᠢᠮᠪᠡᠭᠡ ᠶ᠋ᠢᠨ ᠬᠦᠴᠢᠯ ᠦᠨ ᠮᠥᠴᠡᠯᠭᠡ]]''' ᠪᠣᠯᠠᠢ᠃ <br>ᠡᠨᠡ ᠤᠷᠪᠠᠯ ᠠᠴᠠ ᠧᠨᠧᠷᠬᠢ ᠶᠢ ᠬᠠᠳᠠᠭᠠᠯᠠᠭᠰᠠᠨ ᠨᠡᠭᠦᠷᠡᠰᠦ ᠤᠰᠤ ᠨᠡᠶ᠋ᠢᠯᠡᠭᠵᠢᠳᠡᠭ᠃ <br>ᠵᠠᠷᠢᠮ ᠪᠠᠻᠲ᠋ᠧᠷᠢ ᠤᠳ ᠠᠷᠠᠢ ᠥᠭᠡᠷᠡ ᠤᠷᠪᠠᠯ ᠤᠨ ᠵᠠᠮ ᠢ᠂ '''ᠤᠷᠪᠠᠭᠤ ᠨᠢᠮᠪᠡᠭᠡ ᠶᠢᠨ ᠬᠦᠴᠢᠯ ᠦᠨ ᠮᠥᠴᠢᠯᠭᠡ''' ᠶᠢ ᠠᠰᠢᠭᠯᠠᠮᠤᠢ᠃ <br>ᠨᠡᠷᠡ ᠯᠡ ᠥᠭᠡᠷᠡ ᠪᠣᠯᠪᠠᠴᠤ᠂ ᠨᠡᠭᠦᠷᠡᠰᠦ ᠤᠰᠤ ᠡᠭᠦᠰᠳᠡᠭ ᠦᠷᠡ ᠳ᠋ᠦ᠋ᠩ ᠡᠰᠡ ᠥᠭᠡᠷᠡᠴᠢᠯᠡᠭᠳᠡᠮᠦᠢ᠃<br />
{{Horizontal}}[[File:Seawifs global biosphere.jpg|thumb|{{Mongolian}}ᠹᠣᠲ᠋ᠣᠰᠢᠨ᠍ᠲ᠋ᠸᠽ ᠤᠨ<br>ᠲᠠᠷᠬᠠᠴᠠ᠃<br>ᠬᠠᠷᠠ<br>ᠤᠯᠠᠭᠠᠨ ᠠᠨᠤ<br>ᠶᠡᠬᠡ<br>ᠳᠠᠯᠠᠢ ᠶᠢᠨ<br>ᠹᠣᠲ᠋ᠣᠰᠢᠨ᠍ᠲ᠋ᠧᠽ<br>ᠡᠷᠴᠢᠮ᠂<br>ᠬᠥᠬᠡ<br>ᠨᠣᠭᠤᠭᠠᠨ ᠠᠨᠤ<br>ᠡᠬᠡ<br>ᠭᠠᠵᠠᠷ ᠤᠨ<br>ᠹᠣᠲ᠋ᠣᠰᠢᠨ᠍ᠲᠧᠽ ᠦᠨ<br>ᠡᠷᠴᠢᠮ᠃]]{{Z}}<br><br><br />
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[[Category:ᠪᠢᠣᠯᠣᠭᠢ]]<br />
[[Category:ᠬᠢᠮᠢ]]</div>
Angka
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠊ᠠ᠋
2021-07-15T12:57:03Z
<p>Angka: /* ᠳᠦᠷᠰᠦᠯᠡᠯ */ Spelling</p>
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ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠢᠨᠦ ᠲᠡᠭ <span style="writing-mode: horizontal-tb;"><i>(0)</i></span> ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ [[ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ]] ᠢᠢ '''ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋''' ᠬᠠᠮᠠᠮᠦᠢ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠺᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠬᠠᠰᠤᠬᠤ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠬᠠᠷᠢᠭᠤ ᠭᠠᠷᠤᠨ ᠠ᠋᠃<ref name=":0">https://www.livescience.com/42748-imaginary-numbers.html Live Science. Elaine J. Hom. What Are Imaginary Numbers? 2014/01/22. </ref> <br />
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ᠲᠡᠭᠡᠬᠦ ᠪᠡᠷ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠠᠭ ᠠ᠋ ᠠᠨᠤ ᠬᠠᠰᠤᠬᠤ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠶ ᠲᠣᠭᠠᠨ ᠤ ᠬᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠮᠥᠨ᠃ <br />
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ᠲᠡᠭ <span style="writing-mode: horizontal-tb;"><i>(0)</i></span> ᠲᠣᠭ ᠠ᠋ ᠪᠣᠳᠠᠲᠤ ᠪᠤᠶᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠠᠯᠢ ᠨᠢ ᠴᠤ ᠪᠠᠢᠢᠵᠤ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠳ᠋ᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠳ᠋ᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃<br />
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==ᠬᠡᠷᠡᠭᠯᠡᠭᠡ==<br />
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[[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠢᠢ ''i'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠬᠦ ᠠᠭᠠᠳ ᠧᠯᠧᠺᠲ᠋ᠷᠣᠨᠢᠭ᠌ ᠲᠦ<sup>electronic</sup> ''j'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠨ ᠡ᠋᠃ <br />
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ᠡᠨᠡ ᠢᠨᠦ ''i'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠳᠡᠭ ᠴᠠᠬᠢᠯᠭᠠᠨᠭᠦᠶᠦᠳᠦᠯ ᠡᠴᠡ ᠢᠯᠭᠠᠬᠤ ᠱᠠᠭᠠᠷᠳᠠᠯᠭ ᠠ᠋ ᠡᠴᠡ ᠤᠷᠭᠠᠨ ᠭᠠᠷᠤᠭᠰᠠᠨ ᠤᠯᠠᠮᠵᠢᠯᠠᠯ ᠶᠤᠮ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠧᠯᠧᠺᠲ᠋ᠷᠣᠨᠢᠭ᠌ ᠲᠦ᠂ ᠢᠯᠠᠩᠭᠤᠶ ᠠ᠋ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠳ᠋ᠦ ᠣᠩᠴᠤᠭᠤᠢ ᠠᠴᠢ ᠬᠣᠯᠪᠤᠭᠳᠠᠯ ᠲᠠᠢ᠃ <br />
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ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠰᠢᠨᠦ᠋ᠰ<sup>sinus</sup> ᠳᠣᠯᠭᠢᠶᠠᠨ ᠳᠤ ᠨᠡᠮᠡᠬᠦ ᠪᠠ ᠬᠠᠰᠤᠬᠤ ᠢᠢᠨ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠰᠣᠯᠢᠭᠳᠠᠳᠠᠭ᠃<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠢ ᠶᠠᠪ ᠴᠠᠪ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠬᠤ ᠬᠡᠴᠡᠭᠦ ᠲᠤᠯᠠ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠦᠳ ᠢ ᠳᠠᠪᠬᠤᠷᠯᠠᠬᠤ ᠨᠡᠨ ᠲᠦᠪᠡᠭ ᠲᠡᠢ᠃<br />
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ᠬᠠᠷᠢᠨ ᠲᠥᠰᠦᠭᠡᠯᠡᠯ ᠭᠦᠶᠦᠳᠦᠯ ᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠨᠤᠭᠤᠳ ᠲᠠᠶ ᠬᠣᠣᠰᠮᠠᠭᠯᠠᠭᠤᠯᠤᠭᠰᠠᠨ ᠢᠢᠠᠷ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠦᠨ ᠲᠣᠭᠠᠴᠠᠭ ᠠ᠋ ᠬᠢᠵᠦ᠂ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠤ ᠣᠰᠤᠯ ᠡᠴᠡ ᠤᠷᠢᠳᠤᠴᠢᠯᠠᠨᠰᠡᠷᠭᠡᠢᠢᠯᠬᠦ ᠲᠤᠰᠠ ᠲᠠᠢ ᠪᠠᠢᠢᠳᠠᠭ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠠᠭ ᠠ᠋ ᠢᠢ ᠮᠥᠨ ᠰᠢᠭ᠌ᠨᠠᠯ<sup>signal</sup> ᠪᠣᠯᠪᠠᠰᠤᠷᠠᠭᠤᠯᠬᠤ ᠲᠧᠺᠨᠣᠯᠣᠭᠢ ᠳ᠋ᠤ<sup>technology</sup> ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ <br />
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ᠡᠨᠡ ᠨᠢ ᠡᠭᠦᠷᠦᠨ ᠬᠣᠯᠪᠤᠭ ᠠ᠋᠂ ᠤᠲᠠᠰᠤ ᠦᠭᠡᠢ ᠳᠠᠮᠵᠢᠭᠤᠯᠤᠯᠲᠠ᠂ ᠷᠠᠳᠠᠷ᠂ ᠪᠢᠣᠯᠣᠭᠢ (ᠲᠠᠷᠢᠬᠢᠨ ᠤ ᠳᠣᠯᠭᠢᠶᠠᠨ) ᠰᠤᠳᠤᠯᠬᠤ ᠳ᠋ᠤ ᠬᠠᠮᠢᠶᠠᠷᠠᠨ ᠠ᠋᠃<br />
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ᠶᠡᠷᠦ᠂ ᠰᠢᠨᠦ᠋ᠰ ᠪᠠ ᠺᠣᠰᠢᠨᠦ᠋ᠰ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠪᠦᠬᠦᠢ ᠵᠦᠢᠯ ᠢ ᠬᠡᠮᠵᠢᠬᠦ᠂ ᠲᠣᠭᠠᠴᠠᠭᠠᠯᠬᠤ ᠳ᠋ᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠴᠢᠬᠤᠯᠠ ᠬᠡᠷᠡᠭ ᠲᠡᠢ᠃<ref name=":0" /><br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==<br />
<div style="writing-mode: horizontal-tb; "><References/></div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%AC%E1%A0%A0%E1%A0%AD%E1%A0%A4%E1%A0%B7%E1%A0%AE%E1%A0%A0%E1%A0%AD_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%8A%E1%A0%A0%E1%A0%8B&diff=2706
ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠊ᠠ᠋
2021-07-15T09:14:30Z
<p>Angka: Edit</p>
<hr />
<div><div style="max-height: 80vh; writing-mode: vertical-lr; font-family: 'Noto Sans Mongolian'"><br />
ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠢᠨᠦ ᠲᠡᠭ <span style="writing-mode: horizontal-tb;"><i>(0)</i></span> ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ [[ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ]] ᠢᠢ '''ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋''' ᠬᠠᠮᠠᠮᠦᠢ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠺᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠬᠠᠰᠤᠬᠤ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠬᠠᠷᠢᠭᠤ ᠭᠠᠷᠤᠨ ᠠ᠋᠃<ref name=":0">https://www.livescience.com/42748-imaginary-numbers.html Live Science. Elaine J. Hom. What Are Imaginary Numbers? 2014/01/22. </ref> <br />
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ᠲᠡᠭᠡᠬᠦ ᠪᠡᠷ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠠᠭ ᠠ᠋ ᠠᠨᠤ ᠬᠠᠰᠤᠬᠤ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠶ ᠲᠣᠭᠠᠨ ᠤ ᠬᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠮᠥᠨ᠃ <br />
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ᠲᠡᠭ <span style="writing-mode: horizontal-tb;"><i>(0)</i></span> ᠲᠣᠭ ᠠ᠋ ᠪᠣᠳᠠᠲᠤ ᠪᠤᠶᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠠᠯᠢ ᠨᠢ ᠴᠤ ᠪᠠᠢᠢᠵᠤ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
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__TOC__<br />
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== ᠳᠦᠷᠡᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠳ᠋ᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠳ᠋ᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃<br />
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==ᠬᠡᠷᠡᠭᠯᠡᠭᠡ==<br />
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[[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠢᠢ ''i'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠬᠦ ᠠᠭᠠᠳ ᠧᠯᠧᠺᠲ᠋ᠷᠣᠨᠢᠭ᠌ ᠲᠦ<sup>electronic</sup> ''j'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠨ ᠡ᠋᠃ <br />
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ᠡᠨᠡ ᠢᠨᠦ ''i'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠳᠡᠭ ᠴᠠᠬᠢᠯᠭᠠᠨᠭᠦᠶᠦᠳᠦᠯ ᠡᠴᠡ ᠢᠯᠭᠠᠬᠤ ᠱᠠᠭᠠᠷᠳᠠᠯᠭ ᠠ᠋ ᠡᠴᠡ ᠤᠷᠭᠠᠨ ᠭᠠᠷᠤᠭᠰᠠᠨ ᠤᠯᠠᠮᠵᠢᠯᠠᠯ ᠶᠤᠮ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠧᠯᠧᠺᠲ᠋ᠷᠣᠨᠢᠭ᠌ ᠲᠦ᠂ ᠢᠯᠠᠩᠭᠤᠶ ᠠ᠋ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠳ᠋ᠦ ᠣᠩᠴᠤᠭᠤᠢ ᠠᠴᠢ ᠬᠣᠯᠪᠤᠭᠳᠠᠯ ᠲᠠᠢ᠃ <br />
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ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠰᠢᠨᠦ᠋ᠰ<sup>sinus</sup> ᠳᠣᠯᠭᠢᠶᠠᠨ ᠳᠤ ᠨᠡᠮᠡᠬᠦ ᠪᠠ ᠬᠠᠰᠤᠬᠤ ᠢᠢᠨ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠰᠣᠯᠢᠭᠳᠠᠳᠠᠭ᠃<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠢ ᠶᠠᠪ ᠴᠠᠪ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠬᠤ ᠬᠡᠴᠡᠭᠦ ᠲᠤᠯᠠ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠦᠳ ᠢ ᠳᠠᠪᠬᠤᠷᠯᠠᠬᠤ ᠨᠡᠨ ᠲᠦᠪᠡᠭ ᠲᠡᠢ᠃<br />
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ᠬᠠᠷᠢᠨ ᠲᠥᠰᠦᠭᠡᠯᠡᠯ ᠭᠦᠶᠦᠳᠦᠯ ᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠨᠤᠭᠤᠳ ᠲᠠᠶ ᠬᠣᠣᠰᠮᠠᠭᠯᠠᠭᠤᠯᠤᠭᠰᠠᠨ ᠢᠢᠠᠷ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠦᠨ ᠲᠣᠭᠠᠴᠠᠭ ᠠ᠋ ᠬᠢᠵᠦ᠂ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠤ ᠣᠰᠤᠯ ᠡᠴᠡ ᠤᠷᠢᠳᠤᠴᠢᠯᠠᠨᠰᠡᠷᠭᠡᠢᠢᠯᠬᠦ ᠲᠤᠰᠠ ᠲᠠᠢ ᠪᠠᠢᠢᠳᠠᠭ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠠᠭ ᠠ᠋ ᠢᠢ ᠮᠥᠨ ᠰᠢᠭ᠌ᠨᠠᠯ<sup>signal</sup> ᠪᠣᠯᠪᠠᠰᠤᠷᠠᠭᠤᠯᠬᠤ ᠲᠧᠺᠨᠣᠯᠣᠭᠢ ᠳ᠋ᠤ<sup>technology</sup> ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ <br />
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ᠡᠨᠡ ᠨᠢ ᠡᠭᠦᠷᠦᠨ ᠬᠣᠯᠪᠤᠭ ᠠ᠋᠂ ᠤᠲᠠᠰᠤ ᠦᠭᠡᠢ ᠳᠠᠮᠵᠢᠭᠤᠯᠤᠯᠲᠠ᠂ ᠷᠠᠳᠠᠷ᠂ ᠪᠢᠣᠯᠣᠭᠢ (ᠲᠠᠷᠢᠬᠢᠨ ᠤ ᠳᠣᠯᠭᠢᠶᠠᠨ) ᠰᠤᠳᠤᠯᠬᠤ ᠳ᠋ᠤ ᠬᠠᠮᠢᠶᠠᠷᠠᠨ ᠠ᠋᠃<br />
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ᠶᠡᠷᠦ᠂ ᠰᠢᠨᠦ᠋ᠰ ᠪᠠ ᠺᠣᠰᠢᠨᠦ᠋ᠰ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠪᠦᠬᠦᠢ ᠵᠦᠢᠯ ᠢ ᠬᠡᠮᠵᠢᠬᠦ᠂ ᠲᠣᠭᠠᠴᠠᠭᠠᠯᠬᠤ ᠳ᠋ᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠴᠢᠬᠤᠯᠠ ᠬᠡᠷᠡᠭ ᠲᠡᠢ᠃<ref name=":0" /><br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==<br />
<div style="writing-mode: horizontal-tb; "><References/></div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%BA%E1%A0%A3%E1%A0%AE%E1%A0%AB%E1%A0%8A%E1%A0%AF%E1%A0%A7%E1%A0%BA%E1%A0%B0_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%A0&diff=2705
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ
2021-07-15T09:12:00Z
<p>Angka: Edit</p>
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<div><div style="max-height: 80vh; writing-mode: vertical-lr; font-family: 'Noto Sans Mongolian'"><br />
'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠯᠦᠯᠵᠦ᠂ <small><math>x ^ {2} + 1 = 0</math></small> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃<br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ <span style="writing-mode: horizontal-tb;"><i>a + bi</i></span> ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>a</i></span> ᠪᠣᠯᠤᠨ <span style="writing-mode: horizontal-tb;"><i>b</i></span> ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠨᠢ [[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠪᠢᠭᠡᠳ <span style="writing-mode: horizontal-tb;"><i>i<sup>2</sup> = −1</i></span> ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br />
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== ᠳᠦᠷᠡᠰᠦᠯᠡᠯ ==<br />
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠳ᠋ᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃ <br />
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ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠭᠡ ᠳ᠋ᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃</div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%AC%E1%A0%A0%E1%A0%AD%E1%A0%A4%E1%A0%B7%E1%A0%AE%E1%A0%A0%E1%A0%AD_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%8A%E1%A0%A0%E1%A0%8B&diff=2704
ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠊ᠠ᠋
2021-07-11T04:18:30Z
<p>Angka: Link</p>
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<div><div style="max-height: 80vh; writing-mode: vertical-lr; font-family: 'Noto Sans Mongolian'"><br />
ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠢᠨᠦ ᠲᠡᠭ <span style="writing-mode: horizontal-tb;"><i>(0)</i></span> ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ [[ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ]] ᠢᠢ '''ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋''' ᠬᠠᠮᠠᠮᠦᠢ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠺᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠬᠠᠰᠤᠬᠤ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠬᠠᠷᠢᠭᠤ ᠭᠠᠷᠤᠨ ᠠ᠋᠃<ref name=":0">https://www.livescience.com/42748-imaginary-numbers.html Live Science. Elaine J. Hom. What Are Imaginary Numbers? 2014/01/22. </ref> <br />
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ᠲᠡᠭᠡᠬᠦ ᠪᠡᠷ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠠᠭ ᠠ᠋ ᠠᠨᠤ ᠬᠠᠰᠤᠬᠤ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠶ ᠲᠣᠭᠠᠨ ᠤ ᠬᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠮᠥᠨ᠃ <br />
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ᠲᠡᠭ <span style="writing-mode: horizontal-tb;"><i>(0)</i></span> ᠲᠣᠭ ᠠ᠋ ᠪᠣᠳᠠᠲᠤ ᠪᠤᠶᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠠᠯᠢ ᠨᠢ ᠴᠤ ᠪᠠᠢᠢᠵᠤ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
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__TOC__<br />
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==ᠬᠡᠷᠡᠭᠯᠡᠭᠡ==<br />
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[[ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] ᠢᠢ ''i'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠬᠦ ᠠᠭᠠᠳ ᠧᠯᠧᠺᠲ᠋ᠷᠣᠨᠢᠭ᠌ ᠲᠦ<sup>electronic</sup> ''j'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠨ ᠡ᠋᠃ <br />
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ᠡᠨᠡ ᠢᠨᠦ ''i'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠳᠡᠭ ᠴᠠᠬᠢᠯᠭᠠᠨᠭᠦᠶᠦᠳᠦᠯ ᠡᠴᠡ ᠢᠯᠭᠠᠬᠤ ᠱᠠᠭᠠᠷᠳᠠᠯᠭ ᠠ᠋ ᠡᠴᠡ ᠤᠷᠭᠠᠨ ᠭᠠᠷᠤᠭᠰᠠᠨ ᠤᠯᠠᠮᠵᠢᠯᠠᠯ ᠶᠤᠮ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠧᠯᠧᠺᠲ᠋ᠷᠣᠨᠢᠭ᠌ ᠲᠦ᠂ ᠢᠯᠠᠩᠭᠤᠶ ᠠ᠋ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠳ᠋ᠦ ᠣᠩᠴᠤᠭᠤᠢ ᠠᠴᠢ ᠬᠣᠯᠪᠤᠭᠳᠠᠯ ᠲᠠᠢ᠃ <br />
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ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠰᠢᠨᠦ᠋ᠰ<sup>sinus</sup> ᠳᠣᠯᠭᠢᠶᠠᠨ ᠳᠤ ᠨᠡᠮᠡᠬᠦ ᠪᠠ ᠬᠠᠰᠤᠬᠤ ᠢᠢᠨ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠰᠣᠯᠢᠭᠳᠠᠳᠠᠭ᠃<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠢ ᠶᠠᠪ ᠴᠠᠪ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠬᠤ ᠬᠡᠴᠡᠭᠦ ᠲᠤᠯᠠ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠦᠳ ᠢ ᠳᠠᠪᠬᠤᠷᠯᠠᠬᠤ ᠨᠡᠨ ᠲᠦᠪᠡᠭ ᠲᠡᠢ᠃<br />
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ᠬᠠᠷᠢᠨ ᠲᠥᠰᠦᠭᠡᠯᠡᠯ ᠭᠦᠶᠦᠳᠦᠯ ᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠨᠤᠭᠤᠳ ᠲᠠᠶ ᠬᠣᠣᠰᠮᠠᠭᠯᠠᠭᠤᠯᠤᠭᠰᠠᠨ ᠢᠢᠠᠷ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠦᠨ ᠲᠣᠭᠠᠴᠠᠭ ᠠ᠋ ᠬᠢᠵᠦ᠂ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠤ ᠣᠰᠤᠯ ᠡᠴᠡ ᠤᠷᠢᠳᠤᠴᠢᠯᠠᠨᠰᠡᠷᠭᠡᠢᠢᠯᠬᠦ ᠲᠤᠰᠠ ᠲᠠᠢ ᠪᠠᠢᠢᠳᠠᠭ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠠᠭ ᠠ᠋ ᠢᠢ ᠮᠥᠨ ᠰᠢᠭ᠌ᠨᠠᠯ<sup>signal</sup> ᠪᠣᠯᠪᠠᠰᠤᠷᠠᠭᠤᠯᠬᠤ ᠲᠧᠺᠨᠣᠯᠣᠭᠢ ᠳ᠋ᠤ<sup>technology</sup> ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ <br />
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ᠡᠨᠡ ᠨᠢ ᠡᠭᠦᠷᠦᠨ ᠬᠣᠯᠪᠤᠭ ᠠ᠋᠂ ᠤᠲᠠᠰᠤ ᠦᠭᠡᠢ ᠳᠠᠮᠵᠢᠭᠤᠯᠤᠯᠲᠠ᠂ ᠷᠠᠳᠠᠷ᠂ ᠪᠢᠣᠯᠣᠭᠢ (ᠲᠠᠷᠢᠬᠢᠨ ᠤ ᠳᠣᠯᠭᠢᠶᠠᠨ) ᠰᠤᠳᠤᠯᠬᠤ ᠳ᠋ᠤ ᠬᠠᠮᠢᠶᠠᠷᠠᠨ ᠠ᠋᠃<br />
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ᠶᠡᠷᠦ᠂ ᠰᠢᠨᠦ᠋ᠰ ᠪᠠ ᠺᠣᠰᠢᠨᠦ᠋ᠰ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠪᠦᠬᠦᠢ ᠵᠦᠢᠯ ᠢ ᠬᠡᠮᠵᠢᠬᠦ᠂ ᠲᠣᠭᠠᠴᠠᠭᠠᠯᠬᠤ ᠳ᠋ᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠴᠢᠬᠤᠯᠠ ᠬᠡᠷᠡᠭ ᠲᠡᠢ᠃<ref name=":0" /><br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==<br />
<div style="writing-mode: horizontal-tb; "><References/></div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%AC%E1%A0%A0%E1%A0%AD%E1%A0%A4%E1%A0%B7%E1%A0%AE%E1%A0%A0%E1%A0%AD_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%8A%E1%A0%A0%E1%A0%8B&diff=2703
ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠊ᠠ᠋
2021-07-11T02:27:10Z
<p>Angka: Ed</p>
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<div><div style="max-height: 80vh; writing-mode: vertical-lr; font-family: 'Noto Sans Mongolian'"><br />
ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠢᠨᠦ ᠲᠡᠭ <span style="writing-mode: horizontal-tb;"><i>(0)</i></span> ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ [[ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ]] ᠢᠢ '''ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋''' ᠬᠠᠮᠠᠮᠦᠢ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠺᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠬᠠᠰᠤᠬᠤ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠬᠠᠷᠢᠭᠤ ᠭᠠᠷᠤᠨ ᠠ᠋᠃<ref name=":0">https://www.livescience.com/42748-imaginary-numbers.html Live Science. Elaine J. Hom. What Are Imaginary Numbers? 2014/01/22. </ref> <br />
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ᠲᠡᠭᠡᠬᠦ ᠪᠡᠷ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠠᠭ ᠠ᠋ ᠠᠨᠤ ᠬᠠᠰᠤᠬᠤ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠶ ᠲᠣᠭᠠᠨ ᠤ ᠬᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠮᠥᠨ᠃ <br />
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ᠲᠡᠭ <span style="writing-mode: horizontal-tb;"><i>(0)</i></span> ᠲᠣᠭ ᠠ᠋ ᠪᠣᠳᠠᠲᠤ ᠪᠤᠶᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠠᠯᠢ ᠨᠢ ᠴᠤ ᠪᠠᠢᠢᠵᠤ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
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__TOC__<br />
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<br />
==ᠬᠡᠷᠡᠭᠯᠡᠭᠡ==<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠡᠴᠡ ᠢᠢ ''i'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠬᠦ ᠠᠭᠠᠳ ᠧᠯᠧᠺᠲ᠋ᠷᠣᠨᠢᠭ᠌ ᠲᠦ<sup>electronic</sup> ''j'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠨ ᠡ᠋᠃ <br />
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ᠡᠨᠡ ᠢᠨᠦ ''i'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠳᠡᠭ ᠴᠠᠬᠢᠯᠭᠠᠨᠭᠦᠶᠦᠳᠦᠯ ᠡᠴᠡ ᠢᠯᠭᠠᠬᠤ ᠱᠠᠭᠠᠷᠳᠠᠯᠭ ᠠ᠋ ᠡᠴᠡ ᠤᠷᠭᠠᠨ ᠭᠠᠷᠤᠭᠰᠠᠨ ᠤᠯᠠᠮᠵᠢᠯᠠᠯ ᠶᠤᠮ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠧᠯᠧᠺᠲ᠋ᠷᠣᠨᠢᠭ᠌ ᠲᠦ᠂ ᠢᠯᠠᠩᠭᠤᠶ ᠠ᠋ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠳ᠋ᠦ ᠣᠩᠴᠤᠭᠤᠢ ᠠᠴᠢ ᠬᠣᠯᠪᠤᠭᠳᠠᠯ ᠲᠠᠢ᠃ <br />
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ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠰᠢᠨᠦ᠋ᠰ<sup>sinus</sup> ᠳᠣᠯᠭᠢᠶᠠᠨ ᠳᠤ ᠨᠡᠮᠡᠬᠦ ᠪᠠ ᠬᠠᠰᠤᠬᠤ ᠢᠢᠨ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠰᠣᠯᠢᠭᠳᠠᠳᠠᠭ᠃<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠢ ᠶᠠᠪ ᠴᠠᠪ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠬᠤ ᠬᠡᠴᠡᠭᠦ ᠲᠤᠯᠠ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠦᠳ ᠢ ᠳᠠᠪᠬᠤᠷᠯᠠᠬᠤ ᠨᠡᠨ ᠲᠦᠪᠡᠭ ᠲᠡᠢ᠃<br />
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ᠬᠠᠷᠢᠨ ᠲᠥᠰᠦᠭᠡᠯᠡᠯ ᠭᠦᠶᠦᠳᠦᠯ ᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠨᠤᠭᠤᠳ ᠲᠠᠶ ᠬᠣᠣᠰᠮᠠᠭᠯᠠᠭᠤᠯᠤᠭᠰᠠᠨ ᠢᠢᠠᠷ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠦᠨ ᠲᠣᠭᠠᠴᠠᠭ ᠠ᠋ ᠬᠢᠵᠦ᠂ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠤ ᠣᠰᠤᠯ ᠡᠴᠡ ᠤᠷᠢᠳᠤᠴᠢᠯᠠᠨᠰᠡᠷᠭᠡᠢᠢᠯᠬᠦ ᠲᠤᠰᠠ ᠲᠠᠢ ᠪᠠᠢᠢᠳᠠᠭ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠠᠭ ᠠ᠋ ᠢᠢ ᠮᠥᠨ ᠰᠢᠭ᠌ᠨᠠᠯ<sup>signal</sup> ᠪᠣᠯᠪᠠᠰᠤᠷᠠᠭᠤᠯᠬᠤ ᠲᠧᠺᠨᠣᠯᠣᠭᠢ ᠳ᠋ᠤ<sup>technology</sup> ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ <br />
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ᠡᠨᠡ ᠨᠢ ᠡᠭᠦᠷᠦᠨ ᠬᠣᠯᠪᠤᠭ ᠠ᠋᠂ ᠤᠲᠠᠰᠤ ᠦᠭᠡᠢ ᠳᠠᠮᠵᠢᠭᠤᠯᠤᠯᠲᠠ᠂ ᠷᠠᠳᠠᠷ᠂ ᠪᠢᠣᠯᠣᠭᠢ (ᠲᠠᠷᠢᠬᠢᠨ ᠤ ᠳᠣᠯᠭᠢᠶᠠᠨ) ᠰᠤᠳᠤᠯᠬᠤ ᠳ᠋ᠤ ᠬᠠᠮᠢᠶᠠᠷᠠᠨ ᠠ᠋᠃<br />
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ᠶᠡᠷᠦ᠂ ᠰᠢᠨᠦ᠋ᠰ ᠪᠠ ᠺᠣᠰᠢᠨᠦ᠋ᠰ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠪᠦᠬᠦᠢ ᠵᠦᠢᠯ ᠢ ᠬᠡᠮᠵᠢᠬᠦ᠂ ᠲᠣᠭᠠᠴᠠᠭᠠᠯᠬᠤ ᠳ᠋ᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠴᠢᠬᠤᠯᠠ ᠬᠡᠷᠡᠭ ᠲᠡᠢ᠃<ref name=":0" /><br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==<br />
<div style="writing-mode: horizontal-tb; "><References/></div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%AC%E1%A0%A0%E1%A0%AD%E1%A0%A4%E1%A0%B7%E1%A0%AE%E1%A0%A0%E1%A0%AD_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%8A%E1%A0%A0%E1%A0%8B&diff=2702
ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠊ᠠ᠋
2021-07-11T02:25:01Z
<p>Angka: Edit</p>
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<div><div style="max-height: 80vh; writing-mode: vertical-lr; font-family: 'Noto Sans Mongolian'"><br />
ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠢᠨᠦ ᠲᠡᠭ <span style="writing-mode: horizontal-tb;"><i>(0)</i></span> ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ [[ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ]] ᠢᠢ '''ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋''' ᠬᠠᠮᠠᠮᠦᠢ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠺᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠬᠠᠰᠤᠬᠤ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠬᠠᠷᠢᠭᠤ ᠭᠠᠷᠤᠨ ᠠ᠋᠃<ref name=":0">https://www.livescience.com/42748-imaginary-numbers.html Live Science. Elaine J. Hom. What Are Imaginary Numbers? 2014/01/22. </ref> <br />
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ᠲᠡᠭᠡᠬᠦ ᠪᠡᠷ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠠᠭ ᠠ᠋ ᠠᠨᠤ ᠬᠠᠰᠤᠬᠤ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠶ ᠲᠣᠭᠠᠨ ᠤ ᠬᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠮᠥᠨ᠃ <br />
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ᠲᠡᠭ ᠲᠣᠭ ᠠ᠋ ᠪᠣᠳᠠᠲᠤ ᠪᠤᠶᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠠᠯᠢ ᠨᠢ ᠴᠤ ᠪᠠᠢᠢᠵᠤ ᠪᠣᠯᠤᠨ ᠠ᠋᠃<br />
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__TOC__<br />
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==ᠬᠡᠷᠡᠭᠯᠡᠭᠡ==<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠡᠴᠡ ᠢᠢ ''i'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠬᠦ ᠠᠭᠠᠳ ᠧᠯᠧᠺᠲ᠋ᠷᠣᠨᠢᠭ᠌ ᠲᠦ<sup>electronic</sup> ''j'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠨ ᠡ᠋᠃ <br />
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ᠡᠨᠡ ᠢᠨᠦ ''i'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠳᠡᠭ ᠴᠠᠬᠢᠯᠭᠠᠨᠭᠦᠶᠦᠳᠦᠯ ᠡᠴᠡ ᠢᠯᠭᠠᠬᠤ ᠱᠠᠭᠠᠷᠳᠠᠯᠭ ᠠ᠋ ᠡᠴᠡ ᠤᠷᠭᠠᠨ ᠭᠠᠷᠤᠭᠰᠠᠨ ᠤᠯᠠᠮᠵᠢᠯᠠᠯ ᠶᠤᠮ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠧᠯᠧᠺᠲ᠋ᠷᠣᠨᠢᠭ᠌ ᠲᠦ᠂ ᠢᠯᠠᠩᠭᠤᠶ ᠠ᠋ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠳ᠋ᠦ ᠣᠩᠴᠤᠭᠤᠢ ᠠᠴᠢ ᠬᠣᠯᠪᠤᠭᠳᠠᠯ ᠲᠠᠢ᠃ <br />
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ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠰᠢᠨᠦ᠋ᠰ<sup>sinus</sup> ᠳᠣᠯᠭᠢᠶᠠᠨ ᠳᠤ ᠨᠡᠮᠡᠬᠦ ᠪᠠ ᠬᠠᠰᠤᠬᠤ ᠢᠢᠨ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠰᠣᠯᠢᠭᠳᠠᠳᠠᠭ᠃<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠢ ᠶᠠᠪ ᠴᠠᠪ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠬᠤ ᠬᠡᠴᠡᠭᠦ ᠲᠤᠯᠠ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠦᠳ ᠢ ᠳᠠᠪᠬᠤᠷᠯᠠᠬᠤ ᠨᠡᠨ ᠲᠦᠪᠡᠭ ᠲᠡᠢ᠃<br />
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ᠬᠠᠷᠢᠨ ᠲᠥᠰᠦᠭᠡᠯᠡᠯ ᠭᠦᠶᠦᠳᠦᠯ ᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠨᠤᠭᠤᠳ ᠲᠠᠶ ᠬᠣᠣᠰᠮᠠᠭᠯᠠᠭᠤᠯᠤᠭᠰᠠᠨ ᠢᠢᠠᠷ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠦᠨ ᠲᠣᠭᠠᠴᠠᠭ ᠠ᠋ ᠬᠢᠵᠦ᠂ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠤ ᠣᠰᠤᠯ ᠡᠴᠡ ᠤᠷᠢᠳᠤᠴᠢᠯᠠᠨᠰᠡᠷᠭᠡᠢᠢᠯᠬᠦ ᠲᠤᠰᠠ ᠲᠠᠢ ᠪᠠᠢᠢᠳᠠᠭ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠠᠭ ᠠ᠋ ᠢᠢ ᠮᠥᠨ ᠰᠢᠭ᠌ᠨᠠᠯ<sup>signal</sup> ᠪᠣᠯᠪᠠᠰᠤᠷᠠᠭᠤᠯᠬᠤ ᠲᠧᠺᠨᠣᠯᠣᠭᠢ ᠳ᠋ᠤ<sup>technology</sup> ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ <br />
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ᠡᠨᠡ ᠨᠢ ᠡᠭᠦᠷᠦᠨ ᠬᠣᠯᠪᠤᠭ ᠠ᠋᠂ ᠤᠲᠠᠰᠤ ᠦᠭᠡᠢ ᠳᠠᠮᠵᠢᠭᠤᠯᠤᠯᠲᠠ᠂ ᠷᠠᠳᠠᠷ᠂ ᠪᠢᠣᠯᠣᠭᠢ (ᠲᠠᠷᠢᠬᠢᠨ ᠤ ᠳᠣᠯᠭᠢᠶᠠᠨ) ᠰᠤᠳᠤᠯᠬᠤ ᠳ᠋ᠤ ᠬᠠᠮᠢᠶᠠᠷᠠᠨ ᠠ᠋᠃<br />
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ᠶᠡᠷᠦ᠂ ᠰᠢᠨᠦ᠋ᠰ ᠪᠠ ᠺᠣᠰᠢᠨᠦ᠋ᠰ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠪᠦᠬᠦᠢ ᠵᠦᠢᠯ ᠢ ᠬᠡᠮᠵᠢᠬᠦ᠂ ᠲᠣᠭᠠᠴᠠᠭᠠᠯᠬᠤ ᠳ᠋ᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠴᠢᠬᠤᠯᠠ ᠬᠡᠷᠡᠭ ᠲᠡᠢ᠃<ref name=":0" /><br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==<br />
<div style="writing-mode: horizontal-tb; "><References/></div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%AC%E1%A0%A0%E1%A0%AD%E1%A0%A4%E1%A0%B7%E1%A0%AE%E1%A0%A0%E1%A0%AD_%E1%A0%B2%E1%A0%A3%E1%A0%AD_%E1%A0%8A%E1%A0%A0%E1%A0%8B&diff=2701
ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠊ᠠ᠋
2021-07-10T15:59:26Z
<p>Angka: Edit</p>
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<div><div style="max-height: 80vh; writing-mode: vertical-lr; font-family: 'Noto Sans Mongolian'"><br />
ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠢᠨᠦ ᠲᠡᠭ <span style="writing-mode: horizontal-tb;"><i>(0)</i></span> ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ [[ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ]] ᠢᠢ '''ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋''' ᠬᠠᠮᠠᠮᠦᠢ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠺᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠬᠠᠰᠤᠬᠤ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠬᠠᠷᠢᠭᠤ ᠭᠠᠷᠤᠨ ᠠ᠋᠃<ref name=":0">https://www.livescience.com/42748-imaginary-numbers.html Live Science. Elaine J. Hom. What Are Imaginary Numbers? 2014/01/22. </ref> <br />
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ᠲᠡᠭᠡᠬᠦ ᠪᠡᠷ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠠᠭ ᠠ᠋ ᠬᠠᠰᠤᠬᠤ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠶ ᠲᠣᠭᠠᠨ ᠤ ᠬᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠮᠥᠨ᠃ <br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠡ ᠢᠢ ''i'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠬᠦ ᠠᠭᠠᠳ ᠧᠯᠧᠺᠲ᠋ᠷᠣᠨᠢᠭ᠌ ᠲᠦ ''j'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠨ ᠡ᠋᠃ <br />
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ᠡᠨᠡ ᠢᠨᠦ ''i'' ᠦᠰᠦᠭ ᠢᠢᠡᠷ ᠲᠡᠮᠳᠡᠭᠯᠡᠳᠡᠭ ᠴᠠᠬᠢᠯᠭᠠᠨᠭᠦᠶᠦᠳᠦᠯ ᠡᠴᠡ ᠢᠯᠭᠠᠬᠤ ᠱᠠᠭᠠᠷᠳᠠᠯᠭ ᠠ᠋ ᠡᠴᠡ ᠤᠷᠭᠠᠨ ᠭᠠᠷᠤᠭᠰᠠᠨ ᠤᠯᠠᠮᠵᠢᠯᠠᠯ ᠶᠤᠮ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠧᠯᠧᠺᠲ᠋ᠷᠣᠨᠢᠭ᠌ ᠲᠦ᠂ ᠢᠯᠠᠩᠭᠤᠶ ᠠ᠋ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠳ᠋ᠦ ᠣᠩᠴᠤᠭᠤᠢ ᠠᠴᠢ ᠬᠣᠯᠪᠤᠭᠳᠠᠯ ᠲᠠᠢ᠃ <br />
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ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠰᠢᠨᠦ᠋ᠰ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠳᠤ ᠨᠡᠮᠡᠬᠦ ᠪᠠ ᠬᠠᠰᠤᠬᠤ ᠢᠢᠨ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠰᠣᠯᠢᠭᠳᠠᠳᠠᠭ᠃<br />
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ᠳᠣᠯᠭᠢᠶᠠᠨ ᠢ ᠶᠠᠪ ᠴᠠᠪ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠬᠤ ᠬᠡᠴᠡᠭᠦ ᠲᠤᠯᠠ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠦᠳ ᠢ ᠳᠠᠪᠬᠤᠷᠯᠠᠬᠤ ᠨᠡᠨ ᠲᠦᠪᠡᠭ ᠲᠡᠢ᠃<br />
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ᠬᠠᠷᠢᠨ ᠲᠥᠰᠦᠭᠡᠯᠡᠯ ᠭᠦᠶᠦᠳᠦᠯ ᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠨᠤᠭᠤᠳ ᠲᠠᠶ ᠬᠣᠣᠰᠮᠠᠭᠯᠠᠭᠤᠯᠤᠭᠰᠠᠨ ᠢᠢᠠᠷ ᠬᠤᠪᠢᠰᠬᠤ ᠭᠦᠶᠦᠳᠦᠯ ᠦᠨ ᠲᠣᠭᠠᠴᠠᠭ ᠠ᠋ ᠬᠢᠵᠦ᠂ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠤ ᠣᠰᠤᠯ ᠡᠴᠡ ᠤᠷᠢᠳᠤᠴᠢᠯᠠᠨᠰᠡᠷᠭᠡᠢᠢᠯᠬᠦ ᠲᠤᠰᠠ ᠲᠠᠢ ᠪᠠᠢᠢᠳᠠᠭ᠃<br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠠᠭ ᠠ᠋ ᠢᠢ ᠮᠥᠨ ᠰᠢᠭᠨᠠᠯ ᠪᠣᠯᠪᠠᠰᠤᠷᠠᠭᠤᠯᠬᠤ ᠲᠧᠺᠨᠣᠯᠣᠭᠢ ᠳ᠋ᠤ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ <br />
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ᠡᠨᠡ ᠨᠢ ᠡᠭᠦᠷᠦᠨ ᠬᠣᠯᠪᠤᠭ ᠠ᠋᠂ ᠤᠲᠠᠰᠤ ᠦᠭᠡᠢ ᠳᠠᠮᠵᠢᠭᠤᠯᠤᠯᠲᠠ᠂ ᠷᠠᠳᠠᠷ᠂ ᠪᠢᠣᠯᠣᠭᠢ (ᠲᠠᠷᠢᠬᠢᠨ ᠤ ᠳᠣᠯᠭᠢᠶᠠᠨ) ᠰᠤᠳᠤᠯᠬᠤ ᠳ᠋ᠤ ᠬᠠᠮᠢᠶᠠᠷᠠᠨ ᠠ᠋᠃<br />
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ᠶᠡᠷᠦ᠂ ᠰᠢᠨᠦ᠋ᠰ ᠪᠠ ᠺᠣᠰᠢᠨᠦ᠋ᠰ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠪᠦᠬᠦᠢ ᠵᠦᠢᠯ ᠢ ᠬᠡᠮᠵᠢᠬᠦ᠂ ᠲᠣᠭᠠᠴᠠᠭᠠᠯᠬᠤ ᠳ᠋ᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠴᠢᠬᠦᠯᠠ ᠬᠡᠷᠡᠭ ᠲᠡᠢ᠃<ref name=":0" /></div>
Angka
https://wikibilig.mn/index.php?title=%E1%A0%AC%E1%A0%A0%E1%A0%AD%E1%A0%A4%E1%A0%B7%E1%A0%AE%E1%A0%A0%E1%A0%AD_%E1%A0%A8%E1%A0%A2%E1%A0%AD%E1%A0%A1%E1%A0%B4%E1%A0%A1&diff=2700
ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ
2021-07-07T05:55:26Z
<p>Angka: Layout</p>
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<div><div style="max-height: 80vh; writing-mode: vertical-lr; font-family: 'Noto Sans Mongolian'"><br />
<span style="color:Violet;">ᠦᠨᠳᠦᠰᠦᠨ ᠨᠡᠢᠢᠲᠡᠯᠡᠯ ᠢ ᠤᠩᠰᠢᠬᠤ ᠳ᠋ᠤ ᠰᠧᠷᠸᠧᠷ ᠦᠨ<sup>server</sup> ᠬᠦᠴᠦᠨ ᠴᠢᠳᠠᠯ ᠡᠴᠡ ᠰᠢᠯᠲᠠᠭᠠᠯᠠᠨ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠺ ᠲᠣᠮᠢᠶᠠᠨ ᠨᠤᠭᠤᠳ ᠡᠪᠳᠡᠷᠡᠵᠦ ᠬᠠᠷᠠᠭᠳᠠᠵᠤ ᠪᠠᠢᠢᠪᠠᠯ ᠳᠠᠷᠠᠭᠠᠬᠢ ᠬᠣᠯᠪᠤᠭᠠᠰᠤ ᠪᠠᠷ ᠣᠷᠤᠵᠤ ᠤᠩᠰᠢᠨᠠ ᠤᠤ᠄</span> [[Хэлэлцүүлэг:ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ]] <br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠪᠣᠯ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;"><i>-1</i></span> ᠭᠠᠷᠳᠠᠭ ᠲᠣᠭ ᠠ᠋ ᠶᠤᠮ᠃ <br />
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ᠥᠭᠡᠷ ᠡ᠋ ᠪᠡᠷ ᠬᠡᠯᠡᠪᠡᠯ ᠻᠸᠠᠲᠷᠠᠲ᠋<sup>quadratic</sup> ᠵᠡᠷᠭᠡ ᠢᠨᠦ <span style="writing-mode: horizontal-tb;" ><i>-1</i></span> ᠤᠳᠬ ᠠ᠋ ᠠᠪᠳᠠᠭ <span style="writing-mode: horizontal-tb;"><i>(i<sup>2</sup> = -1)</i></span> ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠭᠡᠨ ᠡ᠋᠃ <br />
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ᠪᠠᠰᠠ <span style="writing-mode: horizontal-tb;" ><i>-1</i></span> ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠭᠡᠵᠦ ᠬᠡᠯᠡᠵᠦ ᠪᠣᠯᠤᠨ ᠠ᠋: <br />
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:<math>i = \sqrt{-1}</math> <br />
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ᠨᠥᠭᠦᠭᠡᠲᠡᠭᠦᠷ᠂ <small><small><math>\sqrt{-1}</math></small></small> ᠪᠤᠶᠤ ᠬᠠᠰᠠᠬᠤ ᠨᠢᠭᠡ ᠢᠢᠨ ᠢᠵᠠᠭᠤᠷ ᠤᠨ ᠰᠢᠢᠳᠦᠯ ᠢᠨᠦ <math>i</math> ᠲᠡᠳᠦᠢ ᠦᠭᠡᠢ ᠪᠠᠰᠠ <math>-i</math> ᠪᠠᠢᠢᠬᠤ ᠶᠤᠮ᠃<ref>James Tamton. Encyclopedia of Mathematics. Facts on File Inc. New York 2005. [[index.php?title=Special:BookSources/0816051240|ISBN 0-8160-5124-0]]</ref> <br />
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ᠠᠩᠭᠯᠢ ᠪᠠᠷ imaginary unit, ᠣᠷᠤᠰ ᠢᠢᠠᠷ мнимая единица ᠬᠡᠮᠡᠬᠦ ᠡᠨᠡ ᠤᠬᠠᠭᠳᠠᠬᠤᠨ ᠠᠨᠤ ᠲᠥᠰᠦᠭᠡᠯᠡᠯ ᠨᠢᠭᠡᠴᠡ᠂ ᠲᠥᠰᠦᠭᠡᠯᠡᠭᠰᠡᠨ ᠨᠢᠭᠡᠴᠡ᠂ ᠬᠣᠭᠤᠰᠤᠨ ᠨᠢᠭᠡᠴᠡ᠂ ᠬᠡᠢᠢᠰᠪᠦᠷᠢ ᠨᠢᠭᠡᠴᠡ ᠭᠡᠰᠡᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ᠃ <br />
<br />
ᠬᠠᠯᠢᠮᠠᠭ ᠲᠦᠮᠡᠨ imaginary number ᠬᠡᠮᠡᠬᠦᠢ ᠢᠢ ухалдаг тойг (ᠤᠬᠠᠭᠠᠯᠠᠳᠠᠭ ᠲᠣᠭᠠ) ᠬᠡᠮᠡᠨ ᠪᠠᠭᠤᠯᠭᠠᠭᠰᠠᠨ ᠪᠠᠢᠢᠬᠤ ᠲᠤᠯᠠ imaginary unit ᠭᠡᠳᠡᠭ ᠢ ᠪᠠᠰᠠ "ᠤᠬᠠᠭᠠᠯᠠᠳᠠᠭ ᠨᠢᠭᠡᠴᠡ" ᠭᠡᠵᠦ ᠪᠠᠭᠤᠯᠭᠠᠳᠠᠭ ᠪᠤᠢ ᠵᠠ᠃ <br />
<br />
ᠬᠠᠷᠢᠨ комплексное число ᠭᠡᠳᠡᠭ ᠢ ᠣᠷᠤᠰ-ᠮᠣᠩᠭᠤᠯ ᠨᠡᠷᠡ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠲᠣᠯᠢ ᠳ᠋ᠤ "ᠬᠠᠪᠰᠤᠷᠠᠭᠰᠠ ᠲᠣᠭᠠ" (ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ<sup>complex</sup> ᠲᠣᠭᠠ) ᠬᠡᠮᠡᠵᠦᠬᠦᠢ᠃<ref>ᠡ᠊᠂ ᠸᠠᠩᠳᠤᠢ᠃ ᠣᠷᠤᠰ-ᠮᠣᠩᠭᠤᠯ ᠨᠡᠷ ᠡ᠋ ᠮᠣᠮᠢᠶᠠᠨ ᠤ ᠲᠣᠯᠢ᠃ ᠪᠦ᠊᠂ ᠨᠠ᠊᠂ ᠮᠣ᠊᠂ ᠠ᠊᠂ ᠤ᠊᠂ ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠬᠡᠯᠡ ᠵᠣᠬᠢᠶᠠᠯ ᠤᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ ᠤᠯᠤᠰ ᠤᠨ ᠬᠡᠪᠯᠡᠯ ᠦᠨ ᠬᠡᠷᠡᠭ ᠡᠷᠬᠢᠯᠡᠬᠦ ᠬᠣᠷᠢᠶ ᠠ᠋᠃ ᠤᠯᠠᠭᠠᠨᠪᠠᠭᠠᠲᠤᠷ 1964᠃</ref><br />
<br />
<br><br />
{| class="wikitable"<br />
|+<br />
|[[File:Complex Plane number i and 1.svg|thumb]]<br />
|style = "height: 150px;"|ᠢᠵᠢ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠡᠭᠡᠷ ᠡ᠋ ᠦᠵᠡᠭᠦᠯᠦᠭᠰᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠬᠢᠭᠡᠳ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ <span style="writing-mode: horizontal-tb;"><i>1</i></span> ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ <span style="writing-mode: horizontal-tb;"><i>i</i></span>᠃<br />
|}<br />
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<br />
__TOC__<br />
<br><br />
<br />
== ᠵᠠᠷᠢᠮ ᠦᠢᠯᠡᠳᠦᠯ ==<br />
<br><br />
=== ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠬᠤᠪᠢᠶᠠᠬᠤ ===<br />
[[ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ]] ᠢᠢ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠪᠤᠶᠤ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠲᠣᠭᠠᠨ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠬᠦᠯᠦᠭᠰᠡᠨ ᠢᠢᠡᠷ᠄ <br />
<br><br><br />
<span style="writing-mode: horizontal-tb;"> <br />
<br><math>i\,(a + bi)</math><br />
<br><math> = ai + bi^2</math><br />
<br><math> = -b + ai</math><br />
</span><br />
<br />
<br />
ᠭᠠᠷᠤᠮᠤᠢ᠃<br />
<br />
ᠡᠨᠡ ᠨᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠡᠭᠡᠷᠡᠬᠢ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ᠢ ᠲᠣᠭᠤᠷᠢᠭᠤᠯᠤᠨ ᠸᠧᠺᠲ᠋ᠣᠷ ᠢ<sup>vector</sup> ᠨᠠᠷᠠ ᠪᠤᠷᠤᠭᠤ 90°ᠡᠷᠭᠢᠭᠦᠯᠦᠭᠰᠡᠨ ᠲᠡᠢ ᠠᠭᠠᠷ ᠨᠢᠭᠡᠨ ᠪᠤᠶ ᠠ᠋᠃ <br />
<br />
ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠳ᠋ᠦ ᠬᠤᠪᠢᠶᠠᠬᠤ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠲᠣᠭᠠᠨ ᠤ ᠤᠷᠪᠠᠭᠤ ᠲᠣᠭᠠᠨ ᠳ᠋ᠤ ᠦᠷᠵᠢᠭᠦᠯᠬᠦ ᠲᠡᠢ ᠠᠭᠠᠷ ᠨᠢᠭᠡᠨ:<br />
<br><br><br />
<span style="writing-mode: horizontal-tb;"><br />
<br><math>\frac{1}{i} = \frac{1}{i} \cdot \frac{i}{i}</math><br />
<br><math> = \frac{i}{i^2} = \frac{i}{-1}</math><br />
<br><math> = -i</math><br />
</span><br />
<br />
<br />
ᠡᠭᠦᠨ ᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳ᠋ᠤ ᠬᠡᠷᠡᠭᠯᠡᠪᠡᠰᠦ:<br />
<br><br><br />
<span style="writing-mode: horizontal-tb;"><br />
<br><math>\frac{a + bi}{i} = -i\,(a + bi)</math><br />
<br><math> = -a i - bi^2 = b - a i~.</math><br />
</span><br />
<br />
<br />
ᠡᠨᠡ ᠨᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠡᠭᠡᠷᠡᠬᠢ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ᠢ ᠲᠣᠭᠤᠷᠢᠭᠤᠯᠤᠨ ᠸᠧᠺᠲ᠋ᠣᠷ ᠢ<sup>vector</sup> ᠨᠠᠷᠠ ᠵᠥᠪ 90°ᠡᠷᠭᠢᠭᠦᠯᠦᠭᠰᠡᠨ ᠲᠡᠢ ᠠᠭᠠᠷ ᠨᠢᠭᠡᠨ ᠪᠤᠶ ᠠ᠋᠃ <br />
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<br />
===ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ===<br />
<math>i</math> ᠲᠣᠭᠠ ᠢᠢ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠨᠢ ᠮᠥᠴᠢᠯᠭᠡ ᠪᠡᠷ ᠳᠠᠪᠲᠠᠭᠳᠠᠬᠤ ᠰᠢᠨᠵᠢ ᠴᠢᠨᠠᠷ ᠲᠠᠢ:<ref>MathBits Notebook. Cyclic Nature of the Powers of "''i ".''https://mathbitsnotebook.com/Algebra2/ComplexNumbers/CPPowers.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/07/03</ref><br />
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<br />
<br />
{| class="wikitable"<br />
|+<br />
|<span style="writing-mode: horizontal-tb;"> i<sup>0</sup> = 1<br />
|<span style="writing-mode: horizontal-tb;"> i<sup>1</sup> = i<br />
|<span style="writing-mode: horizontal-tb;"> i<sup>2</sup> = -1<br />
|<span style="writing-mode: horizontal-tb;"> i<sup>3</sup> = i<sup>2</sup> • i = (-1) • i = -i<br />
|-<br />
|<span style="writing-mode: horizontal-tb;"> i<sup>4</sup> = i<sup>3</sup> • i = (-i) • i = -i<sup>2</sup> = 1<br />
|<span style="writing-mode: horizontal-tb;"> i<sup>5</sup> = i<sup>4</sup> • i = 1 • (i) = i<br />
|<span style="writing-mode: horizontal-tb;"> i<sup>6</sup> = i<sup>4</sup> • i<sup>2</sup> = 1 • (-1) =-1<br />
|<span style="writing-mode: horizontal-tb;"> i<sup>7</sup>= i<sup>4</sup> • i<sup>3</sup> = 1 • (-i) = -i<br />
|-<br />
|<span style="writing-mode: horizontal-tb;"> i<sup>8</sup> = i<sup>4</sup> • i<sup>4</sup> = 1 • 1 = 1<br />
|<span style="writing-mode: horizontal-tb;"> i<sup>9</sup>= i<sup>4</sup> • i<sup>4</sup> • i = 1 • 1• i = i<br />
|<span style="writing-mode: horizontal-tb;"> i<sup>10</sup> = (i<sup>4</sup>)<sup>2</sup> • i<sup>2</sup> = 1 • (-1) = -1<br />
|<span style="writing-mode: horizontal-tb;"> i<sup>11</sup> = (i<sup>4</sup>)<sup>2</sup> • i<sup>3</sup> = 1 • (-i) = -i<br />
|}<br />
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<br />
ᠡᠭᠦᠨ ᠢ ᠶᠡᠷᠦᠩᠬᠡᠢᠢᠯᠡᠭᠰᠡᠨ ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠪᠢᠴᠢᠪᠡᠯ᠄<br />
<br />
<br />
<span style="writing-mode: horizontal-tb;"><br />
<br><math>i^{4n} = 1</math><br />
<br><math>i^{4n+1} = i</math><br />
<br><math>i^{4n+2} = -1</math><br />
<br><math>i^{4n+3} = -i</math><br />
<br><br><span style="writing-mode: vertical-lr;"><br />
ᠡ᠊ᠨᠳᠡ ᠡᠴᠡ᠄ <br />
<math>i^n = i^{(n \bmod 4)}</math><br />
</span></span><br />
<br />
ᠡᠩ ᠦᠨ ᠢᠢᠡᠷ ᠠᠷᠭᠠᠴᠢᠯᠠᠪᠠᠰᠤ᠂ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠲᠣᠭᠠᠨ ᠤ ᠵᠡᠷᠭᠡ ᠢᠢ ᠲᠣᠳᠤᠷᠬᠠᠶ᠋ᠢᠯᠠᠬᠤ ᠢᠢᠨ ᠲᠤᠯᠠᠳᠠ ᠡᠬᠢᠯᠡᠭᠡᠳ ᠢᠯᠡᠳᠬᠡᠭᠴᠢ ᠢᠢ<ref>ᠡ᠊᠂ ᠸᠠᠩᠳᠤᠢ᠃ ᠣᠷᠤᠰ-ᠮᠣᠩᠭᠤᠯ ᠨᠡᠷ ᠡ᠋ ᠮᠣᠮᠢᠶᠠᠨ ᠤ ᠲᠣᠯᠢ᠃ ᠬᠣᠶᠠᠳᠤᠭᠠᠷ ᠪᠣᠲᠢ᠃ ᠪᠦ᠊᠂ ᠨᠠ᠊᠂ ᠮᠣ᠊᠂ ᠠ᠊᠂ ᠤ᠊᠂ ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠᠨ ᠤ ᠠᠻᠠᠳᠧᠮᠢ᠃ ᠬᠡᠯᠡ ᠵᠣᠬᠢᠶᠠᠯ ᠤᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠩ᠃ ᠤᠯᠤᠰ ᠤᠨ ᠬᠡᠪᠯᠡᠯ ᠦᠨ ᠭᠠᠵᠠᠷ᠃ ᠤᠯᠠᠭᠠᠨᠪᠠᠭᠠᠲᠤᠷ 1970᠃</ref> <span style="writing-mode: horizontal-tb;">4</span> ᠳ᠋ᠦ ᠬᠤᠪᠢᠶᠠᠮᠤᠢ᠃ <br />
<br />
<br />
ᠦᠯᠡᠳᠡᠭᠳᠡᠯ ᠢᠨᠦ <span style="writing-mode: horizontal-tb;">0</span> ᠲᠡᠢ ᠲᠡᠩᠴᠡᠬᠦ ᠠᠪᠠᠰᠤ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠦᠭᠰᠡᠨ ᠦ ᠬᠠᠷᠢᠭᠤ ᠠᠨᠤ <math>1</math>᠂<br />
<br />
ᠦᠯᠡᠳᠡᠭᠳᠡᠯ ᠢᠨᠦ <span style="writing-mode: horizontal-tb;">1</span> ᠲᠡᠢ ᠲᠡᠩᠴᠡᠬᠦ ᠠᠪᠠᠰᠤ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠦᠭᠰᠡᠨ ᠦ ᠬᠠᠷᠢᠭᠤ ᠠᠨᠤ <math>i</math>᠂<br />
<br />
ᠦᠯᠡᠳᠡᠭᠳᠡᠯ ᠢᠨᠦ <span style="writing-mode: horizontal-tb;">2</span> ᠲᠡᠢ ᠲᠡᠩᠴᠡᠬᠦ ᠠᠪᠠᠰᠤ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠦᠭᠰᠡᠨ ᠦ ᠬᠠᠷᠢᠭᠤ ᠠᠨᠤ <math>-1</math>᠂<br />
<br />
ᠦᠯᠡᠳᠡᠭᠳᠡᠯ ᠢᠨᠦ <span style="writing-mode: horizontal-tb;">3</span> ᠲᠡᠢ ᠲᠡᠩᠴᠡᠬᠦ ᠠᠪᠠᠰᠤ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠦᠭᠰᠡᠨ ᠦ ᠬᠠᠷᠢᠭᠤ ᠠᠨᠤ <math>-i</math><br />
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ᠪᠠᠢᠢᠨᠠ ᠬᠡᠮᠡᠨ ᠲᠣᠭᠲᠠᠭᠠᠵᠤ ᠪᠣᠯᠤᠨᠠ᠃<br />
<br />
ᠢᠢᠨ ᠬᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠢᠢ ᠠᠯᠢᠮᠠᠳ ᠪᠦᠬᠦᠯᠢ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠥᠭᠦᠯᠪᠡᠯ ᠬᠠᠷᠢᠭᠤ ᠠᠨᠤ <math>1</math>᠂ <math>i</math>᠂ <math>-1</math>᠂ <math>-i</math> - ᠡᠳᠡᠭᠡᠷ ᠦᠨ ᠨᠢᠭᠡ ᠢᠮᠠᠭᠲᠠ ᠪᠠᠢᠢᠬᠤ ᠠᠵᠤᠭᠤ᠃<br />
<br />
<br />
===<math>i</math> ᠲᠣᠭᠠ ᠢᠢ <math>i</math> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ===<br />
<br><br />
<span style="writing-mode: horizontal-tb;"><br />
<math>i^i = \left( e^{i (\pi/2 + 2k \pi)} \right)^i </math><br />
<br><br />
<math>= e^{i^2 (\pi/2 + 2k \pi)} </math><br />
<br><br />
<math>= e^{- (\pi/2 + 2k \pi)}</math><br />
</span><br />
<br />
ᠡᠭᠦᠨ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;">''k'' ∈ ℤ</span> ᠪᠤᠶᠤ ᠪᠦᠬᠦᠯᠢ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ᠃<br />
<br />
<span style="writing-mode: horizontal-tb;">''k'' = 0</span> ᠪᠠᠢᠢᠭ ᠠ᠋ ᠨᠥᠭᠦᠴᠡᠯ ᠳ᠋ᠦ ᠦᠨᠳᠦᠰᠦᠨ ᠤᠳᠬ ᠠ᠋ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>e</i><sup>−''π''/2</sup></span> ᠪᠤᠶᠤ ᠣᠢᠢᠷᠠᠯᠴᠠᠭ ᠠ᠋ ᠪᠠᠷ <span style="writing-mode: horizontal-tb;">0.207879576</span> ᠪᠣᠯᠤᠨ ᠠ᠋᠃<ref>David Wells. The Penguin Dictionary of Curious and Interesting Numbers. UK: Penguin Books 1997. ISBN=0-14-026149-4</ref><ref>Brilliant. What is i to the power of i. https://brilliant.org/discussions/thread/what-is-i-to-the-power-of-i-T. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/017/04</ref><br />
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===ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠡᠴᠡ ᠢᠵᠠᠭᠤᠷ ᠭᠠᠷᠭᠠᠬᠤ===<br />
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{| class="wikitable"<br />
|+<br />
|[[File:3-root of imaginary unit,.svg|thumb]]<br />
|style = "height: 150px;"|<span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠲᠣᠭᠠᠨ ᠤ ᠺᠦ᠋ᠪ<sup>cubic</sup> ᠢᠵᠠᠭᠤᠷ ᠨᠢ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠣᠷᠣᠢ ᠨᠤᠭᠤᠳ ᠋ᠲᠤ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠮᠤᠢ᠃<br />
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<math>i</math> ᠲᠣᠭᠠᠨ ᠤ <span style="writing-mode: horizontal-tb;"><i>n</i></span> ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠢᠵᠠᠭᠤᠷ ᠨᠢ <span style="writing-mode: horizontal-tb;"><i>n</i></span> ᠲᠣᠭᠠᠨ ᠤ ᠬᠠᠷᠢᠭᠤ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃ <br />
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ᠬᠡᠳᠦᠨ ᠬᠠᠷᠢᠭᠤ ᠲᠠᠢ ᠪᠠᠢᠢᠬᠤ ᠨᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳ᠋ᠤ ᠭᠠᠷᠬᠤ <span style="writing-mode: horizontal-tb;"><i>n</i></span>-ᠥᠨᠴᠦᠭᠲᠦ ᠢᠢᠨ ᠥᠨᠴᠦᠭ ᠦᠨ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ ᠪᠠᠢᠢᠨ ᠠ᠋᠃<br><br><br />
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<br><small><math>u_k=\cos {\frac{{\frac{\pi}{2}} + 2\pi k}{n}} +i\ \sin {\frac{{\frac{\pi}{2}} + 2\pi k}{n}},</math> </small> <br />
<br><br><small><math>\quad k=0,1,...,n-1</math> </small><br />
<br><br><span style="writing-mode: vertical-lr;"><math>i</math> ᠲᠣᠭᠠᠨ ᠤ <br>ᠺᠸᠠᠲᠷᠠᠲ <br>ᠢᠵᠠᠭᠤᠷ᠄</span><br />
<br><small><math>\{\sqrt i\} = \left\{\frac{1+i}\sqrt2; ~\frac{-1-i}\sqrt2 \right\}</math> </small><br />
<br><br><span style="writing-mode: vertical-lr;"><math>i</math> ᠲᠣᠭᠠᠨ ᠤ <br>ᠺᠦᠪ <br>ᠢᠵᠠᠭᠤᠷ᠄</span><br />
<br><small><math>\{\sqrt[3]i\} = \left\{-i;~\frac{i+\sqrt3}2;~ \frac{i-\sqrt3}2\right\}.</math></small><br />
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===<math>i</math> ᠲᠣᠭᠠᠨ ᠡᠴᠡ <math>i</math> ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠢᠵᠠᠭᠤᠷ ᠭᠠᠷᠭᠠᠬᠤ===<br />
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<br><math>\sqrt[i]{i}<br />
= i^{\frac{1}{i}\frac{i}{i}}<br />
= i^{-i}</math><br />
<br><math>= \biggl(e^{{i}\frac{\pi}{2}}\biggr)^{-i}</math><br />
<br><math>= e^{-i^2\frac{\pi}{2}}<br />
= e^{\frac{\pi}{2}}</math><br />
</span><br />
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==ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==<br />
ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ - imaginary unit - мнимая единица <br />
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ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭ ᠠ᠋ - imaginary number - чисто мнимое число <br />
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ (ᠬᠠᠪᠰᠤᠷᠠᠭᠰᠠᠨ ᠲᠣᠭ ᠠ᠋) - complex number - комплексное число <br />
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ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ - real number - вещественное число (действительное число) <br />
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ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ - multiplication - умножение <br />
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ᠬᠤᠪᠢᠶᠠᠬᠤ - division - деление <br />
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ᠵᠡᠷᠭᠡ - power - степень <br />
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ᠢᠵᠠᠭᠤᠷ - root - корень <br />
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ᠢᠯᠡᠳᠬᠡᠭᠴᠢ - exponent - экспонента <br />
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ᠦᠯᠡᠳᠡᠭᠳᠡᠯ - remainder - остаток<br />
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ᠣᠯᠠᠨᠯᠢᠭ - set - множество<br />
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ᠪᠦᠬᠦᠯᠢ ᠲᠣᠭ ᠠ᠋ - integer - целое число<br />
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==ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ==<br />
<references /><br />
[[Ангилал:ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ]]</div>
Angka