虛數單位

定義

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

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

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

${\displaystyle i={\sqrt {-1}}\,}$ 往往被誤認為是錯的，他們的證明的方法是：

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

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

i和−i

${\displaystyle -i={\frac {-ii}{i}}={\frac {1}{i}}}$

正當的使用

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

i的運算

i平方根為：

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

 ${\displaystyle \left[\pm {\frac {1}{\sqrt {2}}}(1+i)\right]^{2}\ }$ ${\displaystyle =\left(\pm {\frac {1}{\sqrt {2}}}\right)^{2}(1+i)^{2}\ }$ ${\displaystyle =(\pm 1)^{2}{\frac {1}{2}}(1+i)(1+i)\ }$ ${\displaystyle ={\frac {1}{2}}(1+2i+i^{2})\quad \quad \quad (i^{2}=-1)\ }$ ${\displaystyle ={\frac {1}{2}}+i-{\frac {1}{2}}\ }$ ${\displaystyle =i\ }$

${\displaystyle \!\ x^{ni}=\cos \ln x^{n}+i\sin \ln x^{n}.}$

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

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

${\displaystyle \mathbb {Z} }$代表整數集，代入不同的k值，可計算出無限多的解。

${\displaystyle \log _{i}x={{2\ln x} \over i\pi }.}$

${\displaystyle i}$餘弦是一個實數

${\displaystyle \cos i=\cosh 1={{e+{\frac {1}{e}}} \over 2}={{e^{2}+1} \over 2e}}$

${\displaystyle i}$正弦純虛數

${\displaystyle \sin i=\,i\sinh 1={{e-{\frac {1}{e}}} \over 2}\,i={{e^{2}-1} \over 2e}\,i}$

在程式語言

• Matlab虛數單位的表示方法為ij，但ijfor迴圈可以有其他用途。
• Maple，必須啟用虛數功能，並選擇用i還是j表示虛數單位

註解

1. ^ Maple中，${\displaystyle {\sqrt {i}}={\frac {1}{\sqrt {2}}}(1+i).}$
2. ^ (University of Toronto Mathematics Network: What is the square root of i? URL retrieved March 26, 2007.)
3. ^ "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, Page 26.

參考文獻

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