\begin{definition}[Definition:Complex Number/Polar Form]
For any complex number $z = x + i y \ne 0$, let:
{{begin-eqn}}
{{eqn | l = r
      | r = \cmod z = \sqrt {x^2 + y^2}
      | c = the modulus of $z$, and
}}
{{eqn | l = \theta
      | r = \arg z
      | c = the argument of $z$ (the angle which $z$ yields with the real line)
}}
{{end-eqn}}
where $x, y \in \R$.
From the definition of $\arg z$:
:$(1): \quad \dfrac x r = \cos \theta$
:$(2): \quad \dfrac y r = \sin \theta$
which implies that:
:$x = r \cos \theta$
:$y = r \sin \theta$
which in turn means that any number $z = x + i y \ne 0$ can be written as:
:$z = x + i y = r \paren {\cos \theta + i \sin \theta}$
The pair $\polar {r, \theta}$ is called the '''polar form''' of the complex number $z \ne 0$. 
The number $z = 0 + 0 i$ is defined as $\polar {0, 0}$.
\end{definition}