In short

A complex number z = a + bi can be written as z = r(\cos\theta + i\sin\theta), where r = |z| is the modulus and \theta = \arg(z) is the argument. This is the polar form. It makes multiplication and division transparent: to multiply two complex numbers, multiply their moduli and add their arguments. There is also a compact notation z = re^{i\theta} — Euler's form — where e^{i\theta} = \cos\theta + i\sin\theta. The Cartesian form a + bi is good for addition; the polar form is good for multiplication, division, and powers.

Multiply 1 + i by itself. You get (1+i)^2 = 1 + 2i + i^2 = 2i. The result jumps from the diagonal to the imaginary axis — the number rotated 45° and stretched by \sqrt{2}. That rotation-and-stretch pattern is not a coincidence. Every complex multiplication is a rotation and a stretch, but the Cartesian form a + bi hides that geometry behind an algebraic mess.

There is a way to write complex numbers that makes the rotation visible. Instead of giving the horizontal and vertical coordinates, you give the distance from the origin and the angle from the positive real axis: "go r units in the direction \theta." This is the polar form, and in it, multiplication becomes trivially simple: multiply the distances, add the angles.

From Cartesian to polar

Take z = a + bi, plotted as the point (a, b) on the complex plane. Draw the line from the origin to this point. Its length is r = \sqrt{a^2 + b^2}, the modulus. The angle it makes with the positive real axis is \theta, the argument.

From the right triangle with legs a (horizontal) and b (vertical) and hypotenuse r, you read off:

a = r\cos\theta, \qquad b = r\sin\theta.

Substitute into z = a + bi:

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

That is the polar form. The number r tells you how far from the origin; the angle \theta tells you which direction.

Converting between Cartesian and polar form of a complex numberA coordinate plane with axes labelled Re and Im. A point z is shown at coordinates (a, b) in the first quadrant. A line from the origin to z has length r. The horizontal projection is labelled a equals r cos theta, and the vertical projection is labelled b equals r sin theta. An arc at the origin shows the angle theta. Both forms of z are written: a plus bi and r times cos theta plus i sin theta. ReIm θ a = r cos θ b = r sin θ r = |z| z = a + bi = r(cos θ + i sin θ)
The complex number $z$ has Cartesian coordinates $(a, b)$ and polar coordinates $(r, \theta)$. The horizontal leg of the right triangle is $a = r\cos\theta$; the vertical leg is $b = r\sin\theta$. The two forms describe the same point — one using a grid, the other using a compass.

From polar back to Cartesian

Going the other direction is straightforward. Given r and \theta:

a = r\cos\theta, \qquad b = r\sin\theta, \qquad z = a + bi.

For example, if r = 4 and \theta = \pi/3:

a = 4\cos\frac{\pi}{3} = 4 \cdot \frac{1}{2} = 2, \qquad b = 4\sin\frac{\pi}{3} = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3}

So z = 2 + 2\sqrt{3}\,i. Clean and direct.

Going from Cartesian to polar requires the modulus and argument calculations from the Modulus and Argument articles:

r = \sqrt{a^2 + b^2}, \qquad \theta = \operatorname{Arg}(z) \quad\text{(with quadrant adjustment)}

Polar form

A non-zero complex number z can be written uniquely as

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

where r = |z| > 0 is the modulus and \theta = \operatorname{Arg}(z) \in (-\pi, \pi] is the principal argument. The expression \cos\theta + i\sin\theta is sometimes abbreviated as \operatorname{cis}\theta.

The abbreviation \operatorname{cis}\theta (read "cis theta") is simply shorthand for \cos\theta + i\sin\theta. It appears in some textbooks but not others. You can use it to save ink, but always know what it unpacks to.

Why polar form exists: multiplication becomes beautiful

The real payoff of polar form is what happens when you multiply.

Take two complex numbers in polar form:

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

Multiply them:

z_1 z_2 = r_1 r_2 \big[(\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)\big]

The expressions in brackets are the sum formulas for cosine and sine:

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

So the product simplifies to:

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

This says: to multiply two complex numbers, multiply their moduli and add their arguments.

|z_1 z_2| = |z_1| \cdot |z_2|, \qquad \arg(z_1 z_2) = \arg(z_1) + \arg(z_2)

Multiplication scales and rotates. The modulus handles the scaling; the argument handles the rotation. This is the geometric meaning of complex multiplication — and it is invisible in Cartesian form.

Multiplying two complex numbers in polar form: moduli multiply, arguments addThe complex plane showing three points: z1 at angle theta1 and distance r1, z2 at angle theta2 and distance r2, and their product z1 z2 at angle theta1 plus theta2 and distance r1 times r2. Lines from the origin to each point are drawn, with the angle between the real axis and each line marked. The product point is farther from the origin and at a larger angle. ReIm θ₁ θ₂ θ₁ + θ₂ z₁ (r₁, θ₁) z₂ (r₂, θ₂) z₁z₂ (r₁r₂, θ₁ + θ₂)
Multiplying $z_1$ and $z_2$ in polar form. The product $z_1 z_2$ sits at distance $r_1 r_2$ from the origin (farther out, since both moduli are greater than $1$ here) and at angle $\theta_1 + \theta_2$ from the positive real axis. The moduli multiply; the arguments add.

Division in polar form

The same logic works for division, but in reverse.

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

To divide, divide the moduli and subtract the arguments. Compare this with division in Cartesian form, where you multiply numerator and denominator by the conjugate and expand — polar form is dramatically simpler.

Euler's form: re^{i\theta}

There is an even more compact way to write the polar form. The expression \cos\theta + i\sin\theta turns out to be equal to e^{i\theta}, where e \approx 2.718 is the base of the natural logarithm. This identity,

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

is called Euler's formula. It connects the exponential function with trigonometry in a single equation.

With Euler's formula, the polar form becomes:

z = re^{i\theta}

Multiplication now looks like ordinary algebra of exponents:

z_1 z_2 = r_1 e^{i\theta_1} \cdot r_2 e^{i\theta_2} = r_1 r_2\, e^{i(\theta_1 + \theta_2)}

The rule "multiply moduli, add arguments" falls out automatically from the law of exponents e^A \cdot e^B = e^{A+B}. Division is equally clean:

\frac{z_1}{z_2} = \frac{r_1}{r_2}\, e^{i(\theta_1 - \theta_2)}

And powers follow instantly:

z^n = r^n\, e^{in\theta}

This is why mathematicians and physicists overwhelmingly prefer the exponential form — it reduces complex arithmetic to exponent rules.

A special case of Euler's formula, when \theta = \pi:

e^{i\pi} = \cos\pi + i\sin\pi = -1 + 0 = -1

which gives e^{i\pi} + 1 = 0. This single equation ties together five fundamental constants — e, i, \pi, 1, and 0 — using the three basic operations of addition, multiplication, and exponentiation. It is widely considered the most beautiful equation in mathematics.

The unit circle showing Euler's formula e to the i theta equals cos theta plus i sin thetaA unit circle centred at the origin on the complex plane. A radius at angle theta from the positive real axis meets the circle at a point labelled e to the i theta. The horizontal projection is cos theta and the vertical projection is sin theta. Key positions are marked: 1 at angle 0, i at angle pi over 2, negative 1 at angle pi, negative i at angle three pi over 2. ReIm θ cos θ sin θ e^(iθ) 1 i −1 −i
Euler's formula on the unit circle. The point $e^{i\theta}$ sits on the unit circle at angle $\theta$ from the positive real axis. Its real part is $\cos\theta$ and its imaginary part is $\sin\theta$. At $\theta = 0$ the point is at $1$; at $\theta = \pi/2$ it is at $i$; at $\theta = \pi$ it is at $-1$; at $\theta = 3\pi/2$ (or $-\pi/2$) it is at $-i$.

Converting between all three forms

There are now three ways to write a complex number:

Form Notation Best for
Cartesian a + bi Addition, subtraction
Polar (trigonometric) r(\cos\theta + i\sin\theta) Seeing the geometry explicitly
Polar (exponential / Euler) re^{i\theta} Multiplication, division, powers

The conversion rules:

Cartesian \to Polar:

r = \sqrt{a^2 + b^2}, \qquad \theta = \operatorname{Arg}(a + bi) \text{ (with quadrant adjustment)}

Polar \to Cartesian:

a = r\cos\theta, \qquad b = r\sin\theta

Choose the form that makes the computation simplest. If you need to add two complex numbers, stay in Cartesian. If you need to multiply, divide, or raise to a power, switch to polar.

Interactive: Cartesian and polar side by side

Drag the red point on the complex plane below. The readouts show both the Cartesian form (a + bi) and the polar data (r and \theta) updating simultaneously. Watch how moving radially changes r while moving along a circle changes \theta.

Interactive polar form explorerA coordinate plane from negative five to five on both axes. A red draggable point represents z. Readouts show the Cartesian coordinates a and b, the modulus r, and the argument theta, all updating as the reader drags the point. ReIm 1234 −1−2−3−4 drag the red point
Drag the red point anywhere on the plane. The top readouts show the Cartesian parts $a$ and $b$; the bottom readouts show the polar parts $r$ and $\theta$. Move the point along a circle centred at the origin and $r$ stays constant while $\theta$ changes. Move it along a radial line and $\theta$ stays constant while $r$ changes.

Worked examples

Example 1: Convert $z = 1 + i$ to polar form and Euler's form

A first-quadrant complex number with equal real and imaginary parts — the simplest non-trivial case.

Step 1. Find the modulus.

r = \sqrt{1^2 + 1^2} = \sqrt{2}

Why: the point (1, 1) sits at distance \sqrt{2} from the origin along the diagonal.

Step 2. Find the principal argument.

\theta = \arctan\!\left(\frac{1}{1}\right) = \arctan(1) = \frac{\pi}{4}

Why: both components are positive (first quadrant), so \arctan(b/a) gives the correct angle directly. The line to (1,1) makes a 45° angle with the real axis.

Step 3. Write the trigonometric polar form.

z = \sqrt{2}\left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right)

Why: substitute r and \theta into the template r(\cos\theta + i\sin\theta).

Step 4. Write Euler's form.

z = \sqrt{2}\, e^{i\pi/4}

Why: replace \cos\theta + i\sin\theta with e^{i\theta}. Both forms describe the same number; the exponential form is just more compact.

Result: 1 + i = \sqrt{2}\left(\cos\dfrac{\pi}{4} + i\sin\dfrac{\pi}{4}\right) = \sqrt{2}\, e^{i\pi/4}.

The complex number 1 plus i shown in both Cartesian and polar formA coordinate plane with the point (1, 1) marked. A line from the origin to the point has length root 2. The angle from the positive real axis is pi over 4. Dashed lines project the point onto the real and imaginary axes at 1 each. Both forms of the number are written beside the point. ReIm 012 1 π/4 r = √2 1 + i = √2 · e^(iπ/4)
The number $1 + i$ sits at $(1, 1)$ on the complex plane, at distance $\sqrt{2}$ from the origin and angle $\pi/4$ from the real axis. The dashed lines show the Cartesian projections; the solid line and arc show the polar description. The two representations carry identical information.

You can verify by expanding: \sqrt{2}(\cos(\pi/4) + i\sin(\pi/4)) = \sqrt{2} \cdot \frac{\sqrt{2}}{2} + i\sqrt{2} \cdot \frac{\sqrt{2}}{2} = 1 + i. The polar form reproduces the Cartesian form exactly.

Example 2: Multiply $z_1 = 2\left(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\right)$ and $z_2 = 3\left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right)$

Both numbers are already in polar form. The multiplication rule says: multiply moduli, add arguments.

Step 1. Multiply the moduli.

r = r_1 \cdot r_2 = 2 \times 3 = 6

Why: the product sits at distance r_1 r_2 from the origin — the distances multiply, just as with positive reals.

Step 2. Add the arguments.

\theta = \theta_1 + \theta_2 = \frac{\pi}{6} + \frac{\pi}{4} = \frac{2\pi}{12} + \frac{3\pi}{12} = \frac{5\pi}{12}

Why: multiplying complex numbers rotates by the second number's angle. The combined angle is the sum. Since 5\pi/12 < \pi, it is already a valid principal argument — no adjustment needed.

Step 3. Write the product in polar form.

z_1 z_2 = 6\left(\cos\frac{5\pi}{12} + i\sin\frac{5\pi}{12}\right)

Why: assemble the product from the new modulus and argument. The angle 5\pi/12 = 75° sits in the first quadrant, between \pi/4 = 45° and \pi/2 = 90°.

Step 4. Convert to Cartesian form (optional, for verification).

\cos\frac{5\pi}{12} = \cos 75° = \frac{\sqrt{6} - \sqrt{2}}{4}, \qquad \sin\frac{5\pi}{12} = \sin 75° = \frac{\sqrt{6} + \sqrt{2}}{4}
z_1 z_2 = 6 \cdot \frac{\sqrt{6} - \sqrt{2}}{4} + 6 \cdot \frac{\sqrt{6} + \sqrt{2}}{4}\,i = \frac{3(\sqrt{6} - \sqrt{2})}{2} + \frac{3(\sqrt{6} + \sqrt{2})}{2}\,i

Why: the Cartesian form is messier — nested surds — which is exactly the point. Polar form kept the multiplication to two trivial steps; Cartesian form buries the same answer under algebra.

Result: z_1 z_2 = 6\left(\cos\dfrac{5\pi}{12} + i\sin\dfrac{5\pi}{12}\right) = 6\,e^{i \cdot 5\pi/12}.

Multiplication of two complex numbers in polar form showing the angles addingThe complex plane showing three line segments from the origin. The first goes to z1 at angle pi over 6 and length 2. The second goes to z2 at angle pi over 4 and length 3. The third, longest line goes to the product z1 z2 at angle 5 pi over 12 and length 6. Arcs at the origin show the three angles. The product line is drawn in accent colour. ReIm z₁ r₁ = 2, θ₁ = π/6 z₂ r₂ = 3, θ₂ = π/4 π/6 5π/12 z₁z₂ r = 6, θ = 5π/12
The product $z_1 z_2$ (accent line) sits at distance $6$ from the origin and at angle $5\pi/12 = 75°$. The moduli multiplied ($2 \times 3 = 6$) and the arguments added ($30° + 45° = 75°$). Polar form reduces complex multiplication to one multiplication and one addition.

The contrast is stark: in polar form, the multiplication was essentially mental arithmetic — 2 \times 3 and \pi/6 + \pi/4. In Cartesian form, the same product involves four multiplications, a subtraction, an addition, and two surd simplifications.

Common confusions

Going deeper

If you can convert between Cartesian, trigonometric polar, and Euler's form, and you know how multiplication and division simplify in polar form, you have the core of this article. The following material is for readers who want to see the deeper connections.

Why Euler's formula works

Define the function f(\theta) = \cos\theta + i\sin\theta. Compute the derivative with respect to \theta:

f'(\theta) = -\sin\theta + i\cos\theta = i(\cos\theta + i\sin\theta) = i \cdot f(\theta)

So f satisfies the differential equation f'(\theta) = i \cdot f(\theta) with f(0) = 1. The unique solution to g'(\theta) = i \cdot g(\theta), g(0) = 1 among smooth functions is g(\theta) = e^{i\theta}. Therefore \cos\theta + i\sin\theta = e^{i\theta}.

This argument uses calculus, which is why the formula often appears "without proof" at the JEE level. The algebraic approach is to expand e^{i\theta}, \cos\theta, and \sin\theta as power series and verify the identity term by term — a beautiful computation that uses only the pattern i^2 = -1, i^3 = -i, i^4 = 1.

The unit circle group

The set of all complex numbers of modulus 1 — that is, the set \{e^{i\theta} : \theta \in \mathbb{R}\} — forms a group under multiplication. Multiplying two unit-modulus numbers gives another unit-modulus number (because 1 \times 1 = 1). The identity is e^{i \cdot 0} = 1. The inverse of e^{i\theta} is e^{-i\theta}. This is the circle group, and it is one of the simplest non-trivial examples in group theory. The argument function \arg is a group homomorphism from this circle group to (\mathbb{R}, +) modulo 2\pi — it converts multiplication into addition.

Connections to physics

Alternating current in electrical engineering is modelled as a rotating complex number: V(t) = V_0 e^{i\omega t}, where \omega is the angular frequency. The polar form makes impedance calculations — which involve multiplication and division of complex amplitudes — straightforward. Quantum mechanics uses e^{i\theta} even more fundamentally: the time evolution of a quantum state is |\psi(t)\rangle = e^{-iHt/\hbar}|\psi(0)\rangle. In both fields, the polar form is not a mathematical curiosity but the working notation of the discipline.

Where this leads next

The polar form is the platform for several powerful results.