Let $z$ be a complex number $z\neq 0$, and let $z=x+iy$ be its usual Cartesian form. A polar form is an expression

where $r>0$ is the modulus and $\theta$ an argument of $z$, measured in radians.

The value of the argument such that $-\pi<\theta\leq\pi$ is called the principal value. Write $r=|z|$ and $\theta=\arg z$. (Argument means variable or angle, and not a dispute.)

To find a polar form, we plot the points on the Argand diagram and solve the equations

Clearly $r=\sqrt{x^{2}+y^{2}}$; then we can find $\theta$ by trigonometry.