Polar Representation of a Complex Number

Polar Representation of a Complex Number

From the right triangle used to define the modulus and argument (§4), the real and imaginary parts of $z=a+ib$ can themselves be written in terms of $r=|z|$ and $\theta=\arg(z)$:

Substituting these into $z=a+ib$ gives the polar form:

Here $r=|z|\ge0$ is the modulus and $\theta=\arg(z)$ is the argument (usually taken as the principal value in $(-\pi,\pi]$ unless stated otherwise). The ordered pair $(r,\theta)$ is called the polar coordinates of the point $P$ representing $z$, and by tradition the origin $O$ is called the pole in this description.

Converting Cartesian $\to$ polar means computing $r=\sqrt{a^2+b^2}$ and finding $\theta$ from the quadrant-corrected inverse tangent (§4), then writing $z=r(\cos\theta+i\sin\theta)$.

$z=1+i\sqrt3$: $r=\sqrt{1+3}=2$; since $a=1>0,b=\sqrt3>0$ (Quadrant I), $\theta=\tan^{-1}(\sqrt3/1)=\pi/3$. So $z=2\left(\cos\dfrac{\pi}{3}+i\sin\dfrac{\pi}{3}\right)$.

Converting polar $\to$ Cartesian means simply evaluating $\cos\theta$ and $\sin\theta$ for the given angle and multiplying through by $r$.

For $r=2,\ \theta=\pi/3$: $\cos(\pi/3)=\tfrac12$, $\sin(\pi/3)=\tfrac{\sqrt3}2$, so $z=2\left(\tfrac12+i\tfrac{\sqrt3}2\right)=1+\sqrt3\,i$. …