Complex Numbers in Polar Form

⚡ Key Takeaways

Complex numbers in polar form express a complex number by its distance from the origin (modulus $r$ ) and its angle from the positive real axis (argument $\theta$ ), written $z = r(\cos\theta + i\sin\theta)$ . This representation turns multiplication into the simple act of multiplying lengths and adding angles. It applies wherever rotation, oscillation, or waves appear — electrical engineering, signal processing, and physics. The form was crystallised by Euler, Wessel, and Argand around the late 18th and early 19th centuries.

💡 The Core Insight

A complex number carries two independent pieces of information: how far it is from the origin and which direction it points. Cartesian form $x + iy$ hides these; polar form $r(\cos\theta + i\sin\theta)$ displays them directly. Because multiplication of complex numbers adds angles and multiplies magnitudes, polar form is the natural language for anything involving rotation.

DefinitionPolar Form of a Complex Number

Every nonzero complex number $z = x + iy$ can be written in polar (trigonometric) form:

1

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

Derivation hint: Read off $x = r\cos\theta$ and $y = r\sin\theta$ from the right triangle with legs $x$ and $y$ and hypotenuse $r$ . where the modulus (or absolute value) is

2

\quad r = |z| = \sqrt{x^2 + y^2}

Derivation hint: Apply the Pythagorean theorem to the point $(x,y)$ in the plane. and the argument $\theta = \arg z$ satisfies

3

\quad \cos\theta = \frac{x}{r}, \qquad \sin\theta = \frac{y}{r}, \qquad \tan\theta = \frac{y}{x}\ (x \neq 0)

Derivation hint: Divide the leg relations $x = r\cos\theta$ , $y = r\sin\theta$ by $r$ , and form their quotient for the tangent.

Symbols:

$z$ — the complex number
$x = \operatorname{Re}(z)$ — real part; $y = \operatorname{Im}(z)$ — imaginary part
$r$ — modulus, a non-negative real number (the distance from the origin)
$\theta$ — argument, an angle in radians (or degrees), measured anticlockwise from the positive real axis
$i$ — the imaginary unit, $i^2 = -1$

⚠️ Common mistake

The argument $\theta$ is not unique: adding any integer multiple of $2\pi$ gives the same point. The principal argument, written $\operatorname{Arg}(z)$ , is the unique value in $(-\pi, \pi]$ . When you use $\arctan(y/x)$ you must check the quadrant of $(x,y)$ — a calculator's arctangent only returns values in $(-\pi/2, \pi/2)$ .

TheoremEuler's Formula and the Exponential Form

For any real $\theta$ ,

4

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

Derivation hint: Compare the Maclaurin series of $e^{i\theta}$ with those of $\cos\theta$ and $i\sin\theta$ term by term. Hence every complex number has the compact exponential form

5

\quad z = re^{i\theta}

Derivation hint: Substitute Euler's formula [4] into the trigonometric form [1].

TheoremMultiplication 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

6

\quad z_1 z_2 = r_1 r_2\big[\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)\big]

Derivation hint: Multiply out, then apply the cosine and sine addition formulas to the real and imaginary parts.

7

\quad \frac{z_1}{z_2} = \frac{r_1}{r_2}\big[\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)\big], \quad z_2 \neq 0

Derivation hint: Multiply numerator and denominator by the conjugate of $z_2$ , or use the angle-subtraction identities in exponential form $r_1 e^{i\theta_1}/(r_2 e^{i\theta_2})$ .

ProofMultiplication adds angles, multiplies moduli

We prove equation [6].

Step 1. Write the product directly:

z_1 z_2 = r_1 r_2(\cos\theta_1 + i\sin\theta_1)(\cos\theta_2 + i\sin\theta_2).

Step 2. Expand the bracket, recalling $i^2 = -1$ :

(\cos\theta_1\cos\theta_2 - \sin\theta_1\sin\theta_2) + i(\sin\theta_1\cos\theta_2 + \cos\theta_1\sin\theta_2).

Step 3. Recognise the angle-addition identities:

\cos(\theta_1+\theta_2) = \cos\theta_1\cos\theta_2 - \sin\theta_1\sin\theta_2,

\sin(\theta_1+\theta_2) = \sin\theta_1\cos\theta_2 + \cos\theta_1\sin\theta_2.

Step 4. Substitute to obtain

z_1 z_2 = r_1 r_2\big[\cos(\theta_1+\theta_2) + i\sin(\theta_1+\theta_2)\big]. \qquad \blacksquare

The exponential form makes this instantaneous: $r_1 e^{i\theta_1}\cdot r_2 e^{i\theta_2} = r_1 r_2\, e^{i(\theta_1+\theta_2)}$ , using only the law of exponents.

∎

mathematician · Leonhard Euler | 1707–1783

Euler unified the trigonometric and exponential worlds with the formula $e^{i\theta} = \cos\theta + i\sin\theta$ (published 1748 in Introductio in analysin infinitorum). The special case $\theta = \pi$ gives the celebrated identity $e^{i\pi} + 1 = 0$ , linking five fundamental constants. Euler treated complex exponentials freely and computed with them long before their geometric meaning was fully accepted, providing the analytic backbone that polar form rests upon.

🧠 Intuition

Think of a complex number as an arrow in the plane. Its length is $r$ ; the direction it points is $\theta$ . To add two complex numbers you place arrows tip-to-tail (the parallelogram rule) — Cartesian form handles that easily. But to multiply, you stretch and turn: scale one arrow's length by the other's length, and rotate it by the other's angle. Multiplying by $i$ (which has $r=1$ , $\theta = 90°$ ) rotates any arrow a quarter-turn anticlockwise without changing its length. Multiplying by a positive real just stretches. Polar form separates these two effects cleanly so that products, powers, and roots become geometry rather than algebra.

Figure

The complex number

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

in the Argand plane

ExampleConverting Cartesian to polar form

Given: $z = -1 + i\sqrt{3}$ . Find its polar and exponential forms.

Step 1 — modulus.

r = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2.

Step 2 — reference angle.

\tan\theta = \frac{y}{x} = \frac{\sqrt{3}}{-1} = -\sqrt{3} \;\Rightarrow\; \text{reference angle} = \frac{\pi}{3}.

Step 3 — fix the quadrant. Here $x < 0$ , $y > 0$ , so $z$ lies in the second quadrant. The principal argument is

\theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3}.

Step 4 — assemble.

z = 2\left(\cos\tfrac{2\pi}{3} + i\sin\tfrac{2\pi}{3}\right) = 2e^{i\,2\pi/3}.

\boxed{z = 2\left(\cos\tfrac{2\pi}{3} + i\sin\tfrac{2\pi}{3}\right) = 2e^{i2\pi/3}, \quad r = 2,\ \theta = 120°}

ExampleMultiplying using polar form

Given: $z_1 = 3\big(\cos 40° + i\sin 40°\big)$ and $z_2 = 2\big(\cos 70° + i\sin 70°\big)$ . Find $z_1 z_2$ and $z_1/z_2$ .

Step 1 — product (multiply moduli, add angles), using [6].

r = 3 \times 2 = 6, \qquad \theta = 40° + 70° = 110°.

z_1 z_2 = 6\big(\cos 110° + i\sin 110°\big).

Step 2 — quotient (divide moduli, subtract angles), using [7].

r = \frac{3}{2} = 1.5, \qquad \theta = 40° - 70° = -30°.

\frac{z_1}{z_2} = 1.5\big(\cos(-30°) + i\sin(-30°)\big) = 1.5\left(\tfrac{\sqrt3}{2} - \tfrac{1}{2}i\right).

\boxed{z_1 z_2 = 6\,\mathrm{cis}\,110°, \qquad \frac{z_1}{z_2} = 1.5\,\mathrm{cis}(-30°) \approx 1.30 - 0.75i}

(Here $\operatorname{cis}\theta$ is shorthand for $\cos\theta + i\sin\theta$ .)

⚠️ Common mistake

The naive arctangent trap. For $z = -1 - i$ a calculator gives $\arctan\!\big(\frac{-1}{-1}\big) = \arctan(1) = 45°$ , but $z$ is in the third quadrant, so the true principal argument is $45° - 180° = -135°$ . Always plot the point or check the signs of $x$ and $y$ before trusting $\arctan$ .

CounterexampleModulus is never negative

One might think a polar form like $-2(\cos 30° + i\sin 30°)$ is acceptable because the algebra "works." But by definition $r = |z| \geq 0$ . A negative leading coefficient must be absorbed into the angle: since $-1 = \cos 180° + i\sin 180°$ ,

-2(\cos 30° + i\sin 30°) = 2(\cos 210° + i\sin 210°).

The standard polar form has a non-negative modulus; the sign lives in $\theta$ .

connection

Polar form is the launchpad for De Moivre's Theorem, which raises $r\,\mathrm{cis}\,\theta$ to the $n$ -th power by $r^n\,\mathrm{cis}(n\theta)$ , and for the Roots of Complex Numbers, which spaces $n$ solutions evenly around a circle. It rests on the Argand Diagram for its geometry and on Euler's Formula for its exponential shorthand. The angle-addition machinery ties it directly to Trigonometric Identities, and rotation/scaling links it to Linear Transformations of the Plane.

🔬 Where this appears

⏳ History

- 1545 — Cardano encounters square roots of negative numbers while solving cubics, dismissing them as "sophistic."

- 1637 — Descartes coins the dismissive term imaginary for these quantities.

- 1707–1748 — Euler develops the analytic theory; in 1748 he publishes

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

.

- 1722 — De Moivre states the power formula

(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta

implicitly through trigonometric work.

- 1799 — Caspar Wessel (a Norwegian–Danish surveyor) gives the first clear geometric representation of complex numbers as directed line segments.

- 1806 — Jean-Robert Argand independently publishes the plane diagram now bearing his name.

- 1831 — Gauss popularises the geometric interpretation and the term complex number, lending it full mathematical legitimacy.

- 1837 — Hamilton recasts complex numbers as ordered pairs

(x,y)

with formal multiplication rules, removing the mystique of

i

.

open-problem

Polar form is beautifully clean for multiplication but multivalued by nature: the argument is defined only modulo $2\pi$ , and the complex logarithm $\log z = \ln r + i(\theta + 2\pi k)$ has infinitely many values. Choosing a single "branch" forces a discontinuity (a branch cut), and at $z = 0$ the argument is undefined entirely. These features are not flaws but the seeds of an entire field — complex analysis with its Riemann surfaces — where functions like $\sqrt{z}$ and $\log z$ live naturally on multi-sheeted domains. The student who masters polar form has already met, in miniature, the deepest subtlety of analytic function theory.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also