ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ: Засвар хоорондын ялгаа
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<div style="max-height: 80vh; writing-mode: vertical-lr; font-family: 'Noto Sans Mongolian'"> | <div style="max-height: 80vh; writing-mode: vertical-lr; font-family: 'Noto Sans Mongolian'"> | ||
'''ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ<sup>complex</sup> ᠲᠣᠭ ᠠ᠋''' ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ | '''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᠃ | ||
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]] | 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᠃ | ||
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|[[File:Komplexe_zahlenebene.svg|thumb]] | |[[File:Komplexe_zahlenebene.svg|thumb]] | ||
|style = "height: 225px;"| | |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᠃ | ||
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᠃ | 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᠃ | ||
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= ᠨᠡᠮᠡᠬᠦ = | = ᠨᠡᠮᠡᠬᠦ = | ||
ᠬᠣᠶᠠᠷ | 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> | ||
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= ᠬᠠᠰᠤᠬᠤ = | = ᠬᠠᠰᠤᠬᠤ = | ||
ᠨᠢᠭᠡ | 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" /> | ||
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= ᠦᠷᠡᠵᠢᠬᠦ = | = ᠦᠷᠡᠵᠢᠬᠦ = | ||
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> | |||
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|[[File:Complex_multi.svg|thumb]] | |[[File:Complex_multi.svg|thumb]] | ||
|style = "height: 225px;"|<span style="writing-mode: horizontal-tb;"><i>2 + i</i></span> ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭ ᠠ᠋ ᠢᠢ (ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) | |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᠃ | ||
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᠃ | 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᠃ | ||
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= ᠬᠤᠪᠢᠶᠠᠬᠤ = | = ᠬᠤᠪᠢᠶᠠᠬᠤ = | ||
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠬᠤᠪᠢᠶᠠᠬᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃ | |||
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==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ== | ==ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ== | ||
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> | |||
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==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ== | ==ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ== | ||
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 | |||
<math> \pm (\gamma + \delta i)</math> ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄ | <math> \pm (\gamma + \delta i)</math> ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄ | ||
<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᠃ | <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᠃ | ||
<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᠃ | <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᠃ | ||
ᠡᠭᠦᠨ ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ | 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᠃ | ||
| 209-р мөр: | 209-р мөр: | ||
ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ - real part - действительная часть | ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ - real part - действительная часть | ||
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ - complex plane - комлексная плоскость | |||
ᠲᠡᠩᠬᠡᠯᠢᠭ - axis - ось | ᠲᠡᠩᠬᠡᠯᠢᠭ - axis - ось | ||
23:20, 10 Аравдугаар сар 2021-ий байдлаарх засвар
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰcomplex ᠲᠣᠭ ᠠ᠋ ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠋ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠥᠷᠭᠡᠵᠢᠭᠦᠯᠵᠦ᠂ $ x^{2}+1=0 $ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠢ ᠰᠢᠢᠳᠦᠯ ᠲᠡᠢ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠶᠤᠮ᠃
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ a + bi ᠬᠡᠯᠪᠡᠷᠢ ᠪᠡᠷ ᠢᠯᠡᠷᠡᠬᠡᠢᠢᠯᠡᠵᠦ ᠪᠣᠯᠬᠤ ᠪᠡ ᠡᠭᠦᠨ ᠳᠦ ᠨᠢ a ᠪᠣᠯᠤᠨ b ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋᠂ i ᠨᠢ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠪᠥᠭᠡᠳ i2 = −1 ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠰᠢᠢᠳᠦᠯ ᠢ ᠬᠠᠩᠭᠤᠭᠰᠠᠨ ᠤᠳᠬ ᠠ᠋ ᠲᠠᠢ ᠪᠠᠢᠢᠨ ᠠ᠋᠃
ᠳᠦᠷᠰᠦᠯᠡᠯ
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠨᠢ ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠪᠠ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠳᠦᠳᠡᠭ᠃
ᠬᠠᠭᠤᠷᠮᠠᠬ ᠲᠣᠭ ᠠ᠋ ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠰᠤᠭ ᠠ᠋ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠠᠢᠢᠷᠢᠰᠢᠳᠠᠭ᠃
ᠬᠡᠪᠲᠡᠭᠡ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠪᠠᠢᠢᠷᠢᠰᠢᠭᠤᠯᠳᠠᠭ᠃
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ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠢᠢ ᠲᠥᠯᠦᠭᠡᠯᠡᠵᠦ ᠪᠤᠶ ᠠᠷᠭᠠᠨ᠋ᠳ᠋ ᠤᠨArgand ᠳᠢᠶᠠᠭ᠋ᠷᠠᠮ ᠳᠡᠭᠡᠷ ᠡ᠋ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠸᠧᠺᠲ᠋ᠣᠷvector ᠡᠭᠦᠰᠬᠡᠵᠦ ᠪᠤᠶ ᠬᠣᠣᠰ ᠲᠣᠭ ᠠ᠋ ᠪᠠᠷ (a, b) ᠳᠦᠷᠰᠦᠯᠡᠨ ᠦᠵᠡᠭᠦᠯᠵᠦ ᠪᠣᠯᠤᠨ ᠠ᠋᠃
ᠡᠭᠦᠨ ᠳ᠋ᠦ: |
ᠦᠢᠯᠡᠳᠦᠯ
ᠨᠡᠮᠡᠬᠦ
ᠬᠣᠶᠠᠷ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠵᠦ᠂ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠨᠢ ᠨᠡᠮᠡᠨ ᠡ᠋᠃[1]
$ (a+bi)+(c+di)= $ $ =(a+c)+(b+d)i $
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ᠬᠣᠶᠠᠷ ᠲᠣᠭ ᠠ᠋ ᠢᠢ ᠨᠡᠮᠡᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃ |
ᠬᠠᠰᠤᠬᠤ
ᠨᠢᠭᠡ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠡᠴᠡ ᠨᠥᠭᠦᠭᠡ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠢᠢ ᠬᠠᠰᠠᠬᠤ ᠳ᠋ᠦ ᠪᠡᠨ ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ ᠪᠠ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ ᠦᠳ ᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠠᠰᠤᠶᠤ᠃[1]
$ (a+bi)-(c+di)= $ $ =(a-c)+(b-d)i $
ᠦᠷᠡᠵᠢᠬᠦ
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠦᠨ ᠰᠡᠯᠭᠦᠬᠦ (commutative)᠂ ᠪᠦᠯᠦᠭᠯᠡᠬᠦ (associatove)᠂ ᠵᠠᠳᠠᠯᠬᠤ (distributive) ᠴᠢᠨᠠᠷ ᠤᠳ ᠦᠢᠯᠡᠴᠢᠯᠡᠳᠡᠭ ᠃[2]
$ (a+bi)(c+di)= $ $ =ac+bci-bd+adi= $ $ =(ac-bd)+(bc+ad)i $
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2 + i ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭ ᠠ᠋ ᠢᠢ (ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) 3 + i ᠬᠡᠮᠡᠬᠦ ᠲᠣᠭᠠᠨ ᠳᠤ (ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ) ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠦᠢᠯᠡᠳᠦᠯ ᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠢᠨᠦ᠃
ᠤᠯᠠᠭᠠᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠢ ᠡᠷᠭᠢᠭᠦᠯᠵᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠣᠷᠤᠶ ᠲᠠᠢ ᠲᠠᠭᠠᠷᠠᠭᠤᠯᠵᠤ᠂ √5 ᠢᠢᠠᠷ ᠰᠤᠩᠭᠠᠨ ᠠ᠋᠃ ᠡᠨᠡ ᠢᠨᠦ ᠴᠡᠩᠬᠡᠷ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠾᠢᠫᠣᠲ᠋ᠧᠨᠦᠽ ᠤᠨhypotenuse ᠤᠷᠲᠤ ᠪᠣᠯᠠᠢ᠃ |
ᠬᠤᠪᠢᠶᠠᠬᠤ
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳᠤ ᠬᠤᠪᠢᠶᠠᠬᠤ ᠦᠢᠯᠡᠳᠦᠯ ᠭᠦᠢᠴᠡᠳᠬᠡᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ᠃ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢᠨ ᠬᠣᠣᠰᠮᠠᠭ ᠢᠢᠠᠷ ᠬᠦᠷᠲᠡᠪᠦᠷᠢ ᠪᠠ ᠬᠤᠪᠢᠶᠠᠷᠢ ᠢᠢ ᠦᠷᠡᠵᠢᠭᠦᠯᠳᠡᠭ᠃
$ {\frac {a+bi}{c+di}}= $
$ ={\frac {\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}}= $
$ ={\frac {ac+bd}{c^{2}+d^{2}}}+\left({\frac {bc-ad}{c^{2}+d^{2}}}\right)i. $
ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ᠋ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲquadrate ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ ᠳ᠋ᠦ ᠪᠡᠨ ᠥᠪᠡᠷ ᠢ ᠢᠨᠦ ᠥᠪᠡᠷ ᠲᠦ ᠢᠨᠦ ᠦᠷᠡᠵᠢᠭᠦᠯᠦᠨᠡ᠋᠃[3]
$ (x+yi)^{2}=x^{2}-y^{2}+2xyi. $
ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ a + bi (b ≠ 0) ᠢᠢᠨ ᠻᠸᠠᠲᠷᠠᠲ ᠢᠵᠠᠭᠤᠷ ᠠᠨᠤ $ \pm (\gamma +\delta i) $ ᠪᠠᠢᠢᠬᠤ ᠠᠭᠠᠳ ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄ $ \gamma ={\sqrt {\frac {a+{\sqrt {a^{2}+b^{2}}}}{2}}} $ ᠪᠠ $ \delta =(\operatorname {sgn} b){\sqrt {\frac {-a+{\sqrt {a^{2}+b^{2}}}}{2}}} $ ᠪᠣᠯᠤᠮᠤᠢ᠃ sgn b ᠭᠡᠭᠴᠢ ᠢᠨᠦ $ \operatorname {sgn} b={\begin{cases}\ \ 1,&b>0\\\ \ 0,&b=0\\-1,&b<0\end{cases}} $ ᠭᠡᠵᠦ ᠲᠠᠢᠢᠯᠠᠭᠳᠠᠨᠠ᠃
ᠡᠭᠦᠨ ᠢ ᠪᠠᠲᠤᠯᠠᠬᠤ ᠢᠢᠨ ᠲᠤᠬᠠᠢᠢᠲᠠ ᠳ᠋ᠤ $ \pm (\gamma +\delta i) $ ᠲᠣᠭᠠ ᠢᠢ ᠻᠸᠠᠲᠷᠠᠲ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠦ a + bi ᠲᠣᠭᠠ ᠢᠢ ᠭᠠᠷᠭᠠᠵᠤ ᠢᠷᠡᠬᠦ ᠶᠣᠰᠤᠲᠠᠢ᠃
ᠬᠡᠷᠡᠭᠯᠡᠭᠡ
ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ
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ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠳᠠᠪᠲᠠᠯᠲᠠ ᠪᠠᠷ ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨMandelbrot ᠣᠯᠠᠨᠯᠢᠭ ᠪᠠᠢᠢᠭᠤᠯᠬᠤ᠃ |
ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨMandelbrot ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠯ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ ᠳ᠋ᠤ ᠡᠭᠦᠰᠬᠡᠳᠡᠭ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨfractal ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠵᠡᠰᠱᠢᠶ ᠡ᠋ ᠶᠤᠮ᠃
ᠡᠨᠡ ᠬᠦ ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ z = 0 ᠡᠴᠡ ᠡᠬᠢᠯᠡᠭᠡᠳ ᠢᠲ᠋ᠧᠷᠠᠼ ᠬᠢᠬᠦ ᠳ᠋ᠦ fc(z)=z2+c ᠹᠦᠨ᠍ᠻᠼ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠲᠠᠢ ᠪᠠᠢᠢᠬᠤ ᠨᠥᠬᠦᠴᠡᠯ ᠪᠦᠬᠦᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠤ ᠣᠯᠠᠨᠯᠢᠬ ᠪᠠᠢᠢᠳᠠᠭ᠃
ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨMandelbrot ᠣᠯᠠᠨᠯᠢᠭ ᠠᠨᠤ ᠪᠠᠢᠢᠭᠠᠯᠢ ᠢᠢᠨ ᠭᠠᠢᠢᠬᠠᠮᠰᠢᠭᠲᠤ ᠳᠦᠷᠰᠦ ᠪᠠ ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ᠌ ᠦᠨ ᠰᠠᠢᠢᠬᠠᠨ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠶᠡᠷᠦ ᠪᠤᠰᠤ ᠢᠢᠨ ᠭᠣᠶᠤᠮᠰᠠᠭ ᠬᠣᠣᠰᠯᠠᠯ ᠢ ᠢᠯᠡᠷᠬᠡᠶ᠋ᠢᠯᠡᠳᠡᠭ᠃[4]
ᠮᠠᠨ᠍ᠳᠧᠯᠪᠷᠣᠲ ᠤᠨ ᠹᠷᠠᠻᠲ᠋ᠠᠯ ᠤᠨ ᠵᠠᠭᠠᠭ ᠢ ᠳᠠᠪᠠᠬᠤ ᠳᠤ ᠪᠠᠨ ᠵᠦᠯᠢᠶ ᠠ᠋ ᠢᠢᠨJulia ᠹᠷᠠᠻᠲᠠᠯ ᠤᠳ ᠦᠷᠭᠡᠯᠵᠢ ᠬᠣᠯᠪᠤᠯᠲᠠ ᠪᠠᠨ ᠠᠯᠳᠠᠵᠤ᠂ ᠹᠠᠲᠣᠤ ᠢᠢᠨFatou ᠲᠣᠭᠤᠰᠤ ᠪᠣᠯᠤᠨ ᠬᠦᠪᠢᠷᠠᠳᠠᠭ᠃[5]
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ᠳᠦᠷᠰᠦ ᠠᠮᠢᠯᠠᠭᠤᠯᠤᠨ ᠵᠦᠯᠢᠶ ᠠ᠋ ᠢᠢᠨ Julia ᠣᠯᠠᠨᠯᠢᠭ ᠢ ᠦᠵᠡᠭᠦᠯᠦᠭᠰᠡᠨ ᠢᠨᠦ᠃ |
ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌
ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌quantum mechanics ᠠᠲ᠋ᠣᠮ ᠠᠴᠠ ᠵᠢᠵᠢᠭ ᠡᠭᠡᠯ ᠪᠥᠭᠡᠮᠰ᠂ ᠶᠠᠭᠤᠨ ᠤ ᠡᠮᠦᠨᠡ ᠪᠣᠽᠣᠨboson (ᠵᠢᠱᠢᠶᠡ ᠨᠢ ᠹᠣᠲ᠋ᠣᠨphotonᠪᠠ ᠹᠧᠷᠮᠢᠣᠨ ᠤfermion (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 Schrödinger ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠦᠨ ᠠᠴᠢ ᠬᠣᠯᠪᠣᠭᠳᠠᠯ ᠨᠢ ᠰᠣᠩᠭᠣᠳᠠᠭ ᠹᠢᠽᠢᡘ ᠳ᠋ᠡᠬᠢ ᠨᠧᠧᠲ᠋ᠣᠨ᠍ ᠤ ᠬᠣᠶᠠᠳᠤᠭᠠᠷ ᠬᠠᠤᠯᠢ ᠶᠢᠨ ᠠᠳᠠᠯᠢ ᠶᠤᠮ᠃
ᠠᠯᠢ ᠠᠯᠢ ᠨᠢ ᠪᠥᠭᠡᠮ ᠦᠨ ᠪᠠᠶ᠋ᠢᠷᠢᠯᠠᠯ᠂ ᠢᠮᠫᠦ᠋ᠯᠰ ᠢ impuls ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲ᠋ᠢᠭᠴᠢᠯᠠᠨ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠳᠠᠭ᠃
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠨᠢ ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠢfunction 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 ᠾᠠᠢᠢᠲ᠋ᠯᠧᠷWalter Heitler ᠂ ᠹᠷᠢᠼ ᠯᠣᠨ᠍ᠳᠣᠨFritz London 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᠄
$ {\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}} $
ᠻᠢᠨᠧᠲᠢᠭkinetic ᠧᠨᠧᠷᠭᠢenergy
ᠫᠣᠲ᠋ᠧᠨ᠍ᠼᠢᠶᠠᠯpotencial ᠧᠨᠧᠷᠭᠢ
ᠨᠡᠢᠢᠲᠡ ᠧᠨᠧᠷᠭᠢ
ᠳᠣᠯᠭᠢᠶᠠᠨ ᠤ ᠹᠦᠨ᠍ᠻᠼ ᠲᠦ ᠰᠢᠲᠦᠭᠰᠡᠨ ᠻᠸᠠᠨ᠍ᠲ ᠮᠧᠻᠠᠨᠢᠭ᠌ ᠤᠨ ᠰᠠᠭᠤᠷᠢ ᠲᠣᠮᠢᠶᠠᠨ ᠤ ᠨᠢᠭᠡ ᠪᠣᠯ ᠳᠡᠭᠡᠷᠡ ᠳᠤᠷᠠᠳᠤᠭᠰᠠᠨ ᠱᠷᠥ᠋ᠲᠢᠩᠧᠷ ᠤᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠶᠤᠮ᠃
ᠡᠨᠡᠬᠦ ᠲᠣᠮᠢᠶᠠ ᠶᠢ ᠠᠰᠢᠭᠯᠠᠭᠰᠠᠨ ᠢᠶ᠋ᠠᠷ ᠤᠰᠤ ᠲᠥᠷᠦᠭᠴᠢ ᠶᠢᠨ ᠮᠣᠯᠧᠻᠤᠯ ᠳᠠᠬᠢ ᠬᠣᠶᠠᠷ ᠠᠲ᠋ᠣᠮ ᠨᠢ ᠻᠣᠸᠠᠯᠧᠨ᠍ᠲcovalent ᠬᠣᠯᠪᠣᠭᠠ ᠭᠡᠳᠡᠭ ᠢ ᠡᠭᠦᠰᠬᠡᠨ ᠧᠯᠧᠻᠲ᠋ᠷᠣᠨ ᠨᠤᠭᠤᠳ ᠢᠢᠠᠨ ᠬᠤᠪᠢᠶᠠᠯᠴᠠᠵᠤ ᠪᠠᠶ᠋ᠢᠳᠠᠭ ᠢ ᠪᠠᠲᠤᠯᠠᠭᠰᠠᠨ᠃[6]
ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠢᠢᠨ ᠣᠨᠤᠯ
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ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠢᠢᠨ ᠲᠣᠭᠲᠠᠯᠴᠠᠭᠠᠨ ᠤ ᠡᠩ ᠦᠨ ᠪᠦᠳᠦᠭᠦᠪᠴᠢ᠃ X(s) ᠣᠷᠤᠯᠲᠠ ᠢᠢᠨ ᠮᠡᠳᠡᠭᠡᠯᠡᠯ᠂ Y(s) ᠭᠠᠷᠤᠯᠲᠠ ᠢᠢᠨ ᠮᠡᠳᠡᠭᠡᠯᠡᠯ᠂ G(s) ᠤᠷᠤᠭᠰᠢ ᠳᠠᠮᠵᠢᠭᠤᠯᠬᠤ ᠹᠦᠨ᠍ᠻᠼfunction᠂ H(s) ᠡᠷᠭᠢᠬᠦ ᠬᠣᠯᠪᠤᠭᠠ |
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ ᠶᠢ ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ᠂ ᠢᠯᠠᠩᠭᠤᠶᠠ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠨ ᠲᠣᠭᠲᠠᠪᠤᠷᠢᠲᠠᠢ ᠪᠠᠶᠢᠳᠠᠯ ᠤᠨ ᠰᠢᠨᠵᠢᠯᠡᠭᠡᠨ ᠳᠤ ᠬᠡᠷᠡᠭᠯᠡᠳᠡᠭ᠃
ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠳᠤ 《ᠰᠢᠰᠲ᠋ᠧᠮ》 ᠭᠡᠳᠡᠭ ᠦᠭᠡ ᠶᠢ ᠲᠦᠭᠡᠭᠡᠮᠡᠯ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ ᠪᠥᠭᠡᠳ ᠡᠨᠡ ᠨᠢ ᠵᠠᠪᠠᠯ ᠴᠠᠬᠢᠯᠭᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠢ ᠬᠡᠯᠡᠳᠡᠭ ᠦᠭᠡᠢ᠃
ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠡᠭᠦᠨ ᠢ ᠬᠥᠷᠥᠩᠭᠡ ᠶᠢᠨ ᠵᠠᠬᠠ ᠵᠡᠭᠡᠯᠢ ᠳ᠋ᠡᠬᠢ ᠥᠭᠡᠷᠡᠴᠢᠯᠡᠯᠲᠡ᠂ ᠬᠢᠮᠢ ᠶᠢᠨ ᠤᠷᠪᠠᠯ᠂ ᠦᠢᠯᠡ ᠶᠠᠪᠤᠴᠠ ᠶᠢ ᠰᠤᠳᠤᠯᠬᠤ ᠳᠤ ᠬᠡᠷᠡᠭᠯᠡᠵᠦ ᠪᠣᠯᠤᠨᠠ᠃
ᠲᠡᠭᠦᠨᠴᠢᠯᠡᠨ ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠢ ᠷᠣᠪᠣᠲ ᠤᠨ ᠲᠧᠻᠨᠣᠯᠣᠭᠢ ᠳᠤ ᠥᠷᠭᠡᠨ ᠠᠰᠢᠭᠯᠠᠳᠠᠭ ᠲᠤᠯᠠ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠠᠨᠠᠯᠢᠽ ᠴᠤ ᠪᠠᠰᠠ ᠲᠡᠭᠦᠨ ᠳᠦ ᠬᠡᠷᠡᠭᠯᠡᠭᠳᠡᠨᠡ ᠭᠡᠰᠡᠨ ᠦᠭᠡ᠃[7]
ᠬᠢᠨᠠᠯᠲᠠ ᠶᠢᠨ ᠣᠨᠣᠯ ᠳᠤ ᠵᠠᠷᠢᠮ ᠳᠤ ᠪᠠᠨ ᠯᠠᠫ᠊ᠯᠠᠰ ᠤᠨLaplace ᠬᠤᠪᠢᠷᠠᠭᠠᠯᠲᠠ ᠶᠢ ᠠᠰᠢᠭᠯᠠᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠳ ᠢsystem ᠴᠠᠭ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ᠤ ᠮᠤᠵᠢ ᠠᠴᠠ ᠳᠠᠪᠲᠠᠮᠵᠢ ᠶᠢᠨ ᠮᠤᠵᠢ ᠳᠤ ᠰᠢᠯᠵᠢᠭᠦᠯᠳᠡᠭ᠃
ᠳᠠᠷᠠᠭᠠ ᠨᠢ ᠰᠢᠰᠲ᠋ᠧᠮ ᠦᠨ ᠲᠡᠭ ᠪᠠ ᠲᠤᠶ᠋ᠢᠯ ᠤᠳ ᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠰᠢᠨᠵᠢᠯᠡᠨᠡ᠃
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᠃
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᠄
- ᠪᠠᠷᠠᠭᠤᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠲᠤᠶ᠋ᠢᠯ ᠲᠠᠢ ᠪᠣᠯ ᠲᠣᠭᠲᠠᠪᠤᠷᠢ ᠦᠭᠡᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠂
- ᠵᠡᠭᠦᠨ ᠬᠠᠭᠠᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠤ ᠪᠦᠬᠦ ᠲᠤᠶ᠋ᠢᠯ ᠨᠢ ᠪᠠᠶ᠋ᠢᠪᠠᠯ ᠲᠣᠭᠲᠠᠪᠤᠷᠢᠲᠠᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠂
- ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠲᠤᠶ᠋ᠢᠯ ᠲᠠᠢ ᠪᠣᠯ ᠶᠠᠯᠢᠭᠦᠢ ᠲᠣᠭᠲᠠᠪᠤᠷᠢᠲᠠᠢ ᠪᠠᠶ᠋ᠢᠨᠠ᠃
ᠨᠡᠷ ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠ - complex number - комплексное число
ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ - imaginary unit - мнимая единица
ᠬᠠᠭᠤᠷᠮᠠᠭ ᠲᠣᠭᠠ - imaginary number - чисто мнимое число
ᠬᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰᠡᠭ - imaginaty part - мнимая часть
ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭᠠ - real number - действительное число
ᠪᠣᠳᠠᠲᠤ ᠬᠡᠰᠡᠭ - real part - действительная часть
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠶ - complex plane - комлексная плоскость
ᠲᠡᠩᠬᠡᠯᠢᠭ - axis - ось
ᠣᠯᠠᠨᠯᠢᠭ - set - множество
ᠡᠶᠡᠷᠭᠦ - positive - положительный
ᠰᠥᠷᠭᠦ - negative - отрицательный
ᠲᠡᠭᠰᠢᠳᠬᠡᠯ - equation - уравнение
ᠡᠷᠭᠢᠯᠲᠡ - rotation - поворот
ᠬᠡᠮ - degree - градус
ᠤᠳᠤᠷᠢᠳᠤᠯᠭᠠ ᠢᠢᠨ ᠣᠨᠤᠯ - control theory - теория управления
ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ
- ↑ Jump up to: 1.0 1.1 ᠰᠡᠷᠳᠠᠮᠪᠠ ᠪᠥᠬᠡᠪᠠᠲᠤ᠃ ᠺᠣᠮᠫᠯᠧᠺᠰ ᠲᠣᠭᠠ᠃ Geogebra. https://www.geogebra.org/m/CwryWtd9?fbclid=IwAR0i1VN-HH5hegFJoWMjgCM9si7zH7x9hHDuEbU6iROuW3gbqIAeMGngvAs#material/nLiRDrsU ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/08/17
- ↑ Виленкин Н. Я., Ивашов-Мусатов О. С., Шварцбурд С. И. Алгебра и математический анализ для 11 класса. Учебное пособие. — Изд. 6-е. — М.: Просвещение, 1998. — 288 с. — ISBN 5-09-008036-4.
- ↑ Math is Fun. https://www.mathsisfun.com/algebra/complex-number-multiply.html ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/09 01
- ↑ 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
- ↑ 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
- ↑ Josiah Wu. Real Life Applications of Complex Numbers. 2020 https://issuu.com/harrowhongkong/docs/final_scientific_harrovian_issue_vi-i/s/11488755
- ↑ 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