Fundamental Mathematics/Arithmetic/Arithmetic Number/Imaginary Number

Imaginary Number

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

Mathematical Operations

Operation on 2 different complex numbers

 Addition ${\displaystyle i+i=2i}$ Subtraction ${\displaystyle i-i=0}$ Multiplication ${\displaystyle i\times i={\sqrt {-1}}{\sqrt {-1}}=1}$ Division ${\displaystyle {\frac {i}{i}}=1}$

Operation on complex numbers and its conjugate

 Addition ${\displaystyle i+(-i)=0}$ Subtraction ${\displaystyle i-(-i)=2}$ Multiplication ${\displaystyle i\times (-i)=-1}$ Division ${\displaystyle {\frac {i}{(-i)}}=-1}$

Power of imaginary numbers

${\displaystyle i={\sqrt {-1}}}$
${\displaystyle i^{2}={\sqrt {-1}}{\sqrt {-1}}=1}$
${\displaystyle i^{3}={\sqrt {-1}}{\sqrt {-1}}{\sqrt {-1}}=-1{\sqrt {-1}}=-i}$
${\displaystyle i^{4}={\sqrt {-1}}{\sqrt {-1}}{\sqrt {-1}}{\sqrt {-1}}=1}$

We can condense these down to:

${\displaystyle i^{4n}=1}$

${\displaystyle i^{4n+1}=i}$

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

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

where n = 0,1,2,3...

Hence,

${\displaystyle i^{n}=\pm 1}$
${\displaystyle i^{n}=1}$ . Even power
${\displaystyle i^{n}=\pm i}$ . Odd power