# 虛數單位

${\displaystyle \mathbb {N} \subseteq \mathbb {Z} \subseteq \mathbb {Q} \subseteq \mathbb {R} \subseteq \mathbb {C} }$

2i
-1+i i 1+i
-2 -1 0 1 2
-1-i -i 1-i
-2i

## 定義

 ${\displaystyle \ldots }$ ${\displaystyle {{i}^{-{3}}}=i}$ ${\displaystyle {{i}^{-{2}}}=-1}$ ${\displaystyle {{i}^{-{1}}}=-i}$ ${\displaystyle {{i}^{0}}=1}$ ${\displaystyle {{i}^{1}}=i}$ ${\displaystyle {{i}^{2}}=-1}$ ${\displaystyle {{i}^{3}}=-i}$ ${\displaystyle {{i}^{4}}=1}$ ${\displaystyle {{i}^{5}}=i}$ ${\displaystyle {{i}^{6}}=-1}$ ${\displaystyle \ldots }$

${\displaystyle {{x}^{2}}=-1}$

${\displaystyle i={\sqrt {-{1}}}}$

${\displaystyle i^{3}=i^{2}i=(-1)i=-i}$
${\displaystyle i^{4}=i^{3}i=(-i)i=-(i^{2})=-(-1)=1}$
${\displaystyle i^{5}=i^{4}i=(1)i=i}$

${\displaystyle i^{4n}=1}$
${\displaystyle i^{4n+1}=i}$
${\displaystyle i^{4n+2}=-1}$
${\displaystyle i^{4n+3}=-i}$
${\displaystyle i^{n}=i^{n{\bmod {4}}}}$

## i和-i

${\displaystyle -i^{2}=1}$
${\displaystyle -i=i^{-1}={\frac {1}{i}}}$

## 正当的使用

${\displaystyle -1=i\cdot i={\sqrt {-1}}\cdot {\sqrt {-1}}={\sqrt {(-1)\cdot (-1)}}={\sqrt {1}}=1}$ （不正确）
${\displaystyle -1=i\cdot i=\pm {\sqrt {-1}}\cdot \pm {\sqrt {-1}}=\pm {\sqrt {(-1)\cdot (-1)}}=\pm {\sqrt {1}}=\pm 1}$ （不正确）
${\displaystyle {\frac {1}{i}}={\frac {\sqrt {1}}{\sqrt {-1}}}={\sqrt {\frac {1}{-1}}}={\sqrt {-1}}=i}$ （不正确）

## i的运算

• ${\displaystyle i}$ 平方根为：
${\displaystyle \pm \left({\frac {\sqrt {2}}{2}}+{\frac {\sqrt {2}}{2}}i\right)=\pm {\frac {\sqrt {2}}{2}}(1+i)}$

{\displaystyle {\begin{aligned}\left[\pm {\frac {\sqrt {2}}{2}}(1+i)\right]^{2}&=\left(\pm {\frac {\sqrt {2}}{2}}\right)^{2}(1+i)^{2}\ \\&={\frac {1}{2}}(1+2i+i^{2})\\&={\frac {1}{2}}(1+2i-1)\\&=i\end{aligned}}}

${\displaystyle {\sqrt {i}}={\frac {\sqrt {2}}{2}}(1+i)}$

• 一个数的${\displaystyle ni}$ 次幂为：
${\displaystyle x^{ni}=\cos \ln x^{n}+i\sin \ln x^{n}}$

${\displaystyle {\sqrt[{ni}]{x}}=\cos \ln {\sqrt[{n}]{x}}-i\sin \ln {\sqrt[{n}]{x}}}$

${\displaystyle i^{i}=\left[e^{i({\frac {\pi }{2}}+2k\pi )}\right]^{i}=e^{i^{2}({\frac {\pi }{2}}+2k\pi )}=e^{-({\frac {\pi }{2}}+2k\pi )}}$ ${\displaystyle k\in \mathbb {Z} }$

• ${\displaystyle i}$ 为底的对数为：
${\displaystyle \log _{i}x={{2\ln x} \over i\pi }}$
• ${\displaystyle i}$ 余弦是一个实数
${\displaystyle \cos i=\cosh 1={{e+{\frac {1}{e}}} \over 2}={{e^{2}+1} \over 2e}\approx }$ 1.5430806348152...
• ${\displaystyle i}$ 正弦纯虚数
${\displaystyle \sin i=i\sinh 1={{e-{\frac {1}{e}}} \over 2}i={{e^{2}-1} \over 2e}i\approx }$ 1.1752011936438...${\displaystyle i}$

## 註解

1. ^ University of Toronto Mathematics Network: What is the square root of i?页面存档备份，存于互联网档案馆） URL retrieved March 26, 2007.
2. ^ "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, Page 26.
3. ^ Rob Pike. Constants. The Go Blog. 2014-08-25 [2022-05-27]. （原始内容存档于2022-06-28）.

## 参考文献

• Paul J. Nahin, An Imaginary Tale, The Story of √-1, Princeton University Press, 1998