Fundamental Mathematics/Arithmetic/Arithmetic Number/Imaginary Number

Imaginary Number

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

Mathematical Operations

Operation on 2 different complex numbers

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

Operation on complex numbers and its conjugate

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

Power of imaginary numbers

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

$i^{2}={\sqrt {-1}}{\sqrt {-1}}=1$

$i^{3}={\sqrt {-1}}{\sqrt {-1}}{\sqrt {-1}}=-1{\sqrt {-1}}=-i$

$i^{4}={\sqrt {-1}}{\sqrt {-1}}{\sqrt {-1}}{\sqrt {-1}}=1$

We can condense these down to:

$i^{4n}=1$

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

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

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

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

Hence,

$i^{n}=\pm 1$

$i^{n}=1$ . Even power

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

Reference

• Imaginary Number