The modulus ∣z∣ and argument argz uniquely define the complex number z. That means we can represent any complex number using its modulus and argument instead of a+bi:

It is conventional to call r=∣z∣ and θ=argz. Using trigonometry, we deduce that

And we use the shorthand cisθ=cosθ+isinθ:

Example

Express z=2−2i in the form rcisθ.

First we find r=∣z∣=√22+(−2)2​=2√2.

Next, we find θ=argz=arctan(−22​)=−4π​.

Thus