"ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ"-ны өөр хувилбарууд
ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ ᠠ (засварлах)
23:20, 10 Аравдугаар сар 2021-ий байдлаарх засвар
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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᠃ | ||
__TOC__ | __TOC__ | ||
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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> | ||
| 43-р мөр: | 43-р мөр: | ||
= 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" /> | ||
| 50-р мөр: | 50-р мөр: | ||
= 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> | |||
| 59-р мөр: | 59-р мөр: | ||
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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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| 66-р мөр: | 66-р мөр: | ||
= 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᠃ | |||
| 75-р мөр: | 75-р мөр: | ||
==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> | |||
| 82-р мөр: | 82-р мөр: | ||
==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 | ||