Polar representation of complex numbers

The polar representation of a complex number describes it by its distance from the origin (magnitude) and its angle from the positive real axis (argument), instead of by its real and imaginary components. Equivalent to Cartesian form but more useful for multiplication, division, and powers.

Image: A complex number in modulus–argument form, CC BY-SA 3.0

For a nonzero $z=x+iy\in \mathbb{C}$, define:

• Magnitude (modulus): $r=|z|=\sqrt{x^2+y^2}$.
• Argument (angle): $\theta$ = angle from positive real axis to the line from $0$ to $z$.

The pair $(r,\theta )$ is the polar coordinates of $z$.

Cartesian ↔ polar

Going from Cartesian to polar:

$r=\sqrt{x^2+y^2},\theta =\arctan \frac{y}{x}$

Quadrant warning (this is not a side note — it bites every time): the function $\arctan (y/x)$ returns values only in $(-\pi /2,\pi /2)$, so it can’t distinguish a point in the second quadrant ($x<0,y>0$) from one in the fourth quadrant ($x>0,y<0$); both can give the same ratio $y/x$. To get the right $\theta$, you need to inspect the signs of $x$ and $y$ and add $\pi$ (or subtract $\pi$) when $x<0$. In code, this is exactly what the two-argument atan2(y, x) function does — always use it instead of atan(y/x) for complex-number arguments.

Going from polar to Cartesian:

$x=r\cos \theta ,y=r\sin \theta$

Substituting:

$z=r(\cos \theta +i\sin \theta )$

This is the trigonometric form of the polar representation.

Euler’s formula

The most beautiful identity in mathematics:

$e^{i\theta }=\cos \theta +i\sin \theta$

Combining with the trig form:

$z=r{e}^{i\theta }$

This is the exponential form — the most compact way to write a complex number.

Special cases:

• $e^{i\pi }=-1$ (the famous Euler identity, $e^{i\pi }+1=0$).
• $e^{i\pi /2}=i$.
• $e^{i\cdot 0}=1$.
• $e^{2\pi i}=1$ (full rotation back to start).

Multiplication and division become trivial

For $z_1=r_1{e}^{i{\theta }_1}$ and $z_2=r_2{e}^{i{\theta }_2}$:

$z_1\cdot z_2=r_1{r}_2{e}^{i({\theta }_1+{\theta }_2)}$

Magnitudes multiply, arguments add. Geometrically: scale by $r_1{r}_2$, rotate by ${\theta }_1+{\theta }_2$.

$\frac{z_1}{z_2}=\frac{r_1}{r_2}{e}^{i({\theta }_1-{\theta }_2)}$

Magnitudes divide, arguments subtract.

These are much easier than the Cartesian formulas $(x_1{x}_2-y_1{y}_2)+i(x_1{y}_2+x_2{y}_1)$ for multiplication, etc.

Powers and roots

De Moivre’s theorem:

$z^n=r^n{e}^{in\theta }=r^n(\cos (n\theta )+i\sin (n\theta ))$

Powers in polar form are trivial — just exponentiate the magnitude and multiply the angle.

For roots:

$z^{1/n}=r^{1/n}{e}^{i(\theta +2\pi k)/n},k=0,1,\dots ,n-1$

There are $n$ distinct $n$-th roots, evenly spaced around a circle of radius $r^{1/n}$.

Argument is multivalued

The argument $\theta$ is only defined modulo $2\pi$. The principal value $\mathrm{Arg}(z)$ is conventionally taken in $(-\pi ,\pi ]$ (or sometimes $[0,2\pi )$). For the full set of arguments:

$\arg (z)=\mathrm{Arg}(z)+2\pi k,k\in \mathbb{Z}$

This multivaluedness can complicate calculations involving logarithms or fractional powers — be careful which branch you’re on.

Why polar dominates EE

Phasors in AC circuit analysis are complex numbers. Their magnitude is voltage/current amplitude; their argument is phase angle. Multiplying impedances in series, dividing currents through parallel paths — all way easier in polar form. See Phasor and Phasor relationships for circuit elements.

In signals and systems, the Laplace transform and Fourier transforms produce complex-valued frequency-domain representations, almost always handled in polar form.

For the algebraic side and the complex plane geometry, see Algebra and Geometry in the complex plane.