Polar Form of a Complex Number

Polar coordinates form another set of parameters characterising the vector from the origin to a point $z=x+iy$, using magnitude and direction instead of horizontal/vertical components. The polar coordinate system has a fixed point $O$ (the pole) and a horizontal half-line from $O$ (the initial line or polar axis). For a point $P$, if $r$ is the distance from $O$ to $P$ and $\theta$ is the angle (measured counter-clockwise from the initial line) to the line $OP$, then $(r,\theta)$ are the polar coordinates of $P$.

Superimposing this on the ordinary rectangular system gives

so any nonzero complex number $z=x+iy$ can be expressed as $z=r\cos\theta+ir\sin\theta$.

Definition. Let $r,\theta$ be polar coordinates of the point $P(x,y)$ corresponding to $z=x+iy\ne0$. The polar (trigonometric) form of $z$ is

Here $r$ is the modulus of $z$ (the absolute value, as before), and $\theta$ is called the argument (or amplitude) of $z$, written $\arg z$.

• If $z=0$, $\theta$ is undefined — polar coordinates always assume $z\ne0$.
• If $z=x+iy$ has polar coordinates $(r,\theta)$, its conjugate $\overline z=x-iy$ has polar coordinates $(r,-\theta)$.

Squaring and adding (1) and (2) gives $r=|z|=\sqrt{x^2+y^2}$; dividing (2) by (1) gives $\tan\theta=y/x$.

General argument vs. principal argument. $\theta$ can take infinitely many values, all differing by an integer multiple of $2\pi$ — every value of $\theta$ satisfying $\tan\theta=y/x$ in the correct quadrant for $z$ is an argument of $z$, and the full set of such values is denoted $\arg z$. There is a unique value of $\theta$ satisfying $-\pi<\theta\le\pi$; this is called the principal argument, $\operatorname{Arg}z$. In general, $\arg z=\operatorname{Arg}z+2n\pi,\ n\in\mathbb Z$.

To find $\operatorname{Arg}z$ in practice: compute $\alpha=\tan^{-1}\left|\dfrac yx\right|$ (the reference angle, always taken as the acute angle from the calculator) and then adjust for the quadrant containing $z=x+iy$:

For instance, the principal argument and (general) argument of $1,i,-1,-i$ are $0,\ \dfrac\pi2,\ \pi\ (\text{but}\ \arg=2n\pi+\pi\text{ using the branch just past }-\pi\text{ too}),\ -\dfrac\pi2$ respectively, matching the point's position on the axes.

Properties of arguments (parallel to the modulus properties):

and $\cos\theta+i\sin\theta$ has the equivalent alternate forms $\cos(2k\pi+\theta)+i\sin(2k\pi+\theta),\ k\in\mathbb Z$ (adding any whole number of full turns changes nothing).

Multiplication and division in polar form. If $z_1=r_1(\cos\theta_1+i\sin\theta_1)$ and $z_2=r_2(\cos\theta_2+i\sin\theta_2)$, then, using the cosine/sine angle-addition formulas, …