${\displaystyle z=a+bi=r\left(\cos \phi \ +i\sin \phi \right)}$

where

• i is the imaginary number ${\displaystyle \left(i\ ={\sqrt {-1}}\right)}$
• the modulus ${\displaystyle r=\operatorname {mod} (z)=|z|={\sqrt {a^{2}+b^{2}}}}$
• the argument ${\displaystyle \phi =\arg(z)}$ is the angle formed by the complex number on a polar graph with one real axis and one imaginary axis. This can be found using the right angle trigonometry for the trigonometric functions.

This is sometimes abbreviated as ${\displaystyle r\left(\cos \phi \ +i\sin \phi \right)=r\operatorname {cis} \phi }$ and it is also the case that ${\displaystyle r\operatorname {cis} \phi =re^{i\phi }}$ (provided that ${\displaystyle \phi }$ is in radians). The latter identity is called Euler's formula.

Euler's formula can be used to prove DeMoivre's formula: ${\displaystyle (\cos \phi \ +i\sin \phi )^{n}=\cos(n\phi )+i\sin(n\phi ).}$ This formula is valid for all values of n, real or complex.