# 虛數單位

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

 ${\displaystyle \ldots }$（重複藍色區域樣式） ${\displaystyle i^{-3}=i\,\!}$ ${\displaystyle i^{-2}=-1\,\!}$ ${\displaystyle i^{-1}=-i\,\!}$ ${\displaystyle i^{0}=1\,\!}$ ${\displaystyle i^{1}=i\,\!}$ ${\displaystyle i^{2}=-1\,\!}$ ${\displaystyle i^{3}=-i\,\!}$ ${\displaystyle i^{4}=1\,\!}$ ${\displaystyle i^{5}=i\,\!}$ ${\displaystyle i^{6}=-1\,\!}$ ${\displaystyle \ldots }$（重複藍色區域樣式）

## 定義

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

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

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

## i 嘅計算

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

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

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

${\displaystyle i}$ 嘅餘弦係一個實數：

${\displaystyle \cos i=\cosh 1={{e+1/e} \over 2}={{e^{2}+1} \over 2e}\approx 1.54308064.}$

${\displaystyle i}$ 嘅正弦係一個純虛數

${\displaystyle \sin i=\,i\sinh 1={{e-1/e} \over 2}\,i={{e^{2}-1} \over 2e}\,i\approx 1.19520119i.}$