Math formulas and cheat sheets for Complex numbers

## Complex numbers

### Definitions:

A complex number is written as a+bi$a + b\,i$ where a$a$ and b$b$ are real numbers an i$i$, called the imaginary unit, has the property that i2=1$i^2 = -1$.

The complex numbers z=a+bi$z = a + b\,i$ and z¯=abi$\overline{z} = a - b\,i$ are called complex conjugate of each other.

### Formulas:

Equality of complex numbers

 a+bi=c+di⟺a=c  and  b=d

 (a+bi)+(c+di)=(a+c)+(b+d)i

Subtraction of complex numbers

 (a+bi)−(c+di)=(a−c)+(b−d)i

Multiplication of complex numbers

Division of complex numbers

Polar form of complex numbers

 a+bi=r⋅(cosθ+isinθ)

Multiplication and division of complex numbers in polar form

 [r1(cosθ1+i⋅sinθ1)]⋅[r2(cosθ2+i⋅sinθ2)]=r1⋅r2[cos(θ1+θ2)+i⋅sin(θ1+θ2)]
 r1(cosθ1+isinθ1)r2(cosθ2+isinθ2)=r1r2[cos(θ1−θ2)+i⋅sin(θ1−θ2)]

De Moivre's theorem

 [r(cosθ+isinθ)]n=rn(cos(nθ)+isin(nθ))

Roots of complex numbers

 [r(cosθ+isinθ)]1/n=r1/n(cosθ+2kπn+isinθ+2kπn)  k=0,1,…,n−1