Засварлах: ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ (хэсэгчилэн)

== ᠵᠠᠷᠢᠮ ᠦᠢᠯᠡᠳᠦᠯ ==
<br>
=== ᠦᠷᠡᠵᠢᠭᠦᠯᠬᠦ ᠬᠤᠪᠢᠶᠠᠬᠤ ===
[[ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭ᠎ᠠ]] ‍ᠢᠢ ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠪᠤᠶᠤ <span style="writing-mode: horizontal-tb;"><i>i</i></span> ᠲᠣᠭᠠᠨ ᠳ᠋ᠦ ᠦᠷᠡᠵᠢᠬᠦᠯᠦᠭᠰᠡᠨ ‍ᠢᠢᠡᠷ᠄  
<br><br>
<span style="writing-mode: horizontal-tb;"> 
<br><math>i\,(a + bi)</math>
<br><math> = ai + bi^2</math>
<br><math> = -b + ai</math>
</span>


ᠭᠠᠷᠤᠮᠤᠢ᠃

ᠡᠨᠡ ᠨᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠡᠭᠡᠷᠡᠬᠢ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ‍ᠢ ᠲᠣᠭᠤᠷᠢᠭᠤᠯᠤᠨ ᠸᠧᠺᠲ᠋ᠣᠷ ‍ᠢ<sup>vector</sup> ᠨᠠᠷᠠ ᠪᠤᠷᠤᠭᠤ 90°ᠡᠷᠭᠢᠭᠦᠯᠦᠭᠰᠡᠨ ᠲᠡᠢ ᠠᠭᠠᠷ ᠨᠢᠭᠡᠨ ᠪᠤᠶ ‍ᠠ᠋᠃ 
 
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>
<span style="writing-mode: horizontal-tb;">
<br><math>\frac{1}{i} = \frac{1}{i} \cdot \frac{i}{i}</math>
<br><math> = \frac{i}{i^2} = \frac{i}{-1}</math>
<br><math> = -i</math>
</span>


ᠡᠭᠦᠨ ‍ᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠲᠣᠭᠠᠨ ᠳ᠋ᠤ ᠬᠡᠷᠡᠭᠯᠡᠪᠡᠰᠦ:
<br><br>
<span style="writing-mode: horizontal-tb;">
<br><math>\frac{a + bi}{i} = -i\,(a + bi)</math>
<br><math> = -a i - bi^2 = b - a i~.</math>
</span>


ᠡᠨᠡ ᠨᠢ ᠺᠣᠮᠫ᠊ᠯᠧᠺᠰ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳᠡᠭᠡᠷᠡᠬᠢ ᠡᠬᠢᠯᠡᠯ ᠴᠡᠭ ‍ᠢ ᠲᠣᠭᠤᠷᠢᠭᠤᠯᠤᠨ ᠸᠧᠺᠲ᠋ᠣᠷ ‍ᠢ<sup>vector</sup> ᠨᠠᠷᠠ ᠵᠥᠪ 90°ᠡᠷᠭᠢᠭᠦᠯᠦᠭᠰᠡᠨ ᠲᠡᠢ ᠠᠭᠠᠷ ᠨᠢᠭᠡᠨ ᠪᠤᠶ ‍ᠠ᠋᠃ 


===ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ===
<math>i</math> ᠲᠣᠭ᠎ᠠ ‍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>
 


{| class="wikitable"
|+
|<span style="writing-mode: horizontal-tb;"> i<sup>0</sup> = 1
|<span style="writing-mode: horizontal-tb;"> i<sup>1</sup> = i
|<span style="writing-mode: horizontal-tb;"> i<sup>2</sup> = -1
|<span style="writing-mode: horizontal-tb;"> i<sup>3</sup> = i<sup>2</sup> • i = (-1) • i = -i
|-
|<span style="writing-mode: horizontal-tb;"> i<sup>4</sup> = i<sup>3</sup> • i = (-i) • i = -i<sup>2</sup> = 1
|<span style="writing-mode: horizontal-tb;"> i<sup>5</sup> = i<sup>4</sup> • i = 1 • (i) = i
|<span style="writing-mode: horizontal-tb;"> i<sup>6</sup> = i<sup>4</sup> • i<sup>2</sup> = 1 • (-1) =-1
|<span style="writing-mode: horizontal-tb;"> i<sup>7</sup>= i<sup>4</sup> • i<sup>3</sup> = 1 • (-i) = -i
|-
|<span style="writing-mode: horizontal-tb;"> i<sup>8</sup> = i<sup>4</sup> • i<sup>4</sup> = 1 • 1 = 1
|<span style="writing-mode: horizontal-tb;"> i<sup>9</sup>= i<sup>4</sup> • i<sup>4</sup> • i = 1 • 1• i = i
|<span style="writing-mode: horizontal-tb;"> i<sup>10</sup> = (i<sup>4</sup>)<sup>2</sup> • i<sup>2</sup> = 1 • (-1) = -1
|<span style="writing-mode: horizontal-tb;"> i<sup>11</sup> = (i<sup>4</sup>)<sup>2</sup> • i<sup>3</sup> = 1 • (-i) = -i
|}


ᠡᠭᠦᠨ ‍ᠢ ᠶᠡᠷᠦᠩᠬᠡᠢᠢᠯᠡᠭᠰᠡᠨ ᠬᠡᠯᠪᠡᠷᠢ ‍ᠪᠡᠷ ᠪᠢᠴᠢᠪᠡᠯ᠄


<span style="writing-mode: horizontal-tb;">
<br><math>i^{4n} = 1</math>
<br><math>i^{4n+1} = i</math>
<br><math>i^{4n+2} = -1</math>
<br><math>i^{4n+3} = -i</math>
<br><br><span style="writing-mode: vertical-lr;">
ᠡ᠊ᠨᠳᠡ ᠡᠴᠡ᠄ 
<math>i^n = i^{(n \bmod 4)}</math>
</span></span>

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᠃ 


ᠦᠯᠡᠳᠡᠭᠳᠡᠯ ᠢᠨᠦ <span style="writing-mode: horizontal-tb;">0</span> ᠲᠡᠢ ᠲᠡᠩᠴᠡᠬᠦ ᠠᠪᠠᠰᠤ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠦᠭᠰᠡᠨ ‍ᠦ ᠬᠠᠷᠢᠭᠤ ᠠᠨᠤ <math>1</math>᠂

ᠦᠯᠡᠳᠡᠭᠳᠡᠯ ᠢᠨᠦ <span style="writing-mode: horizontal-tb;">1</span> ᠲᠡᠢ ᠲᠡᠩᠴᠡᠬᠦ ᠠᠪᠠᠰᠤ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠦᠭᠰᠡᠨ ‍ᠦ ᠬᠠᠷᠢᠭᠤ ᠠᠨᠤ <math>i</math>᠂
 
ᠦᠯᠡᠳᠡᠭᠳᠡᠯ ᠢᠨᠦ <span style="writing-mode: horizontal-tb;">2</span> ᠲᠡᠢ ᠲᠡᠩᠴᠡᠬᠦ ᠠᠪᠠᠰᠤ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠦᠭᠰᠡᠨ ‍ᠦ ᠬᠠᠷᠢᠭᠤ ᠠᠨᠤ <math>-1</math>᠂

ᠦᠯᠡᠳᠡᠭᠳᠡᠯ ᠢᠨᠦ <span style="writing-mode: horizontal-tb;">3</span> ᠲᠡᠢ ᠲᠡᠩᠴᠡᠬᠦ ᠠᠪᠠᠰᠤ ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠦᠭᠰᠡᠨ ‍ᠦ ᠬᠠᠷᠢᠭᠤ ᠠᠨᠤ <math>-i</math>

ᠪᠠᠢᠢᠨ᠎ᠠ ᠬᠡᠮᠡᠨ ᠲᠣᠭᠲᠠᠭᠠᠵᠤ ᠪᠣᠯᠤᠨ᠎ᠠ᠃

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᠃


===<math>i</math> ᠲᠣᠭ᠎ᠠ ‍ᠢᠢ <math>i</math> ᠵᠡᠷᠭᠡ ᠳ᠋ᠦ ᠳᠡᠪᠰᠢᠭᠦᠯᠬᠦ===
<br>
<span style="writing-mode: horizontal-tb;">
<math>i^i = \left( e^{i (\pi/2 + 2k \pi)} \right)^i </math>
<br>
<math>= e^{i^2 (\pi/2 + 2k \pi)} </math>
<br>
<math>= e^{- (\pi/2 + 2k \pi)}</math>
</span>

ᠡᠭᠦᠨ ᠳ᠋ᠦ <span style="writing-mode: horizontal-tb;">''k'' ∈ ℤ</span> ᠪᠤᠶᠤ ᠪᠦᠬᠦᠯᠢ ᠲᠣᠭ᠎ ‍ᠠ᠋ ‍ᠢᠢᠨ ᠣᠯᠠᠨᠯᠢᠭ᠃

<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>


===ᠬᠠᠭᠤᠷᠮᠠᠭ ᠨᠢᠭᠡᠴᠡ ᠡᠴᠡ ᠢᠵᠠᠭᠤᠷ ᠭᠠᠷᠭᠠᠬᠤ===

<br>
{| class="wikitable"
|+
|[[File:3-root of imaginary unit,.svg|thumb]]
|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>
<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᠃ 

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>
<span style="writing-mode: horizontal-tb;">
<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><small><math>\quad k=0,1,...,n-1</math> </small>
<br><br><span style="writing-mode: vertical-lr;"><math>i</math> ᠲᠣᠭᠠᠨ ‍ᠤ <br>ᠺᠸᠠᠲᠷᠠᠲ <br>ᠢᠵᠠᠭᠤᠷ᠄</span>
<br><small><math>\{\sqrt i\} = \left\{\frac{1+i}\sqrt2; ~\frac{-1-i}\sqrt2 \right\}</math> </small>
<br><br><span style="writing-mode: vertical-lr;"><math>i</math> ᠲᠣᠭᠠᠨ ‍ᠤ <br>ᠺᠦᠪ <br>ᠢᠵᠠᠭᠤᠷ᠄</span>
<br><small><math>\{\sqrt[3]i\} = \left\{-i;~\frac{i+\sqrt3}2;~ \frac{i-\sqrt3}2\right\}.</math></small>
</span>


===<math>i</math> ᠲᠣᠭᠠᠨ ᠡᠴᠡ <math>i</math> ᠵᠡᠷᠭᠡ ‍ᠢᠢᠨ ᠢᠵᠠᠭᠤᠷ ᠭᠠᠷᠭᠠᠬᠤ===

<br>
<span style="writing-mode: horizontal-tb;">
<br><math>\sqrt[i]{i}
= i^{\frac{1}{i}\frac{i}{i}}
= i^{-i}</math>
<br><math>= \biggl(e^{{i}\frac{\pi}{2}}\biggr)^{-i}</math>
<br><math>= e^{-i^2\frac{\pi}{2}}
= e^{\frac{\pi}{2}}</math>
</span>