Complex numbers: polar form, Euler and De Moivre, explained

By David Mashiah · 01.11.2026

A complex number a + bi is a point in the plane. Written in Cartesian form it is easy to add; written in polar form, as a length and an angle, it is easy to multiply, raise to a power and take roots. Switching between the two forms is most of the skill.

z = a + bi, i^{2} = - 1

∣ z ∣ = r = a^{2} + b^{2}, θ = atan2 (b, a)

z = r (cos θ + i sin θ) = r e^{i θ}

z^{n} = r^{n} e^{in θ}

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

Two forms

The Cartesian form a + bi adds component by component. The polar form r·e^(iθ) uses the modulus r, the distance from zero, and the argument θ, the angle from the positive real axis. Multiplying two complex numbers multiplies their moduli and adds their angles, which is why powers are easy in polar form.

The argument and its quadrant

The angle is not simply arctan(b/a). That formula cannot tell 1 + i from −1 − i. Use the quadrant: the two-argument atan2(b, a) does this automatically, and gives an angle between −180° and 180°.

Powers and roots

De Moivre’s formula raises a number to the nth power by raising r to the n and multiplying the angle by n. Roots go the other way, and there are always n of them: equally spaced around a circle of radius r^(1/n), one every 360°/n.

Division is easiest with the conjugate: multiply top and bottom by a − bi, since (a + bi)(a − bi) = a² + b² is real.

Worked example

For z = 1 + i: r = √2 and θ = 45°, so z = √2·e^(iπ/4).

Then z⁴ = (√2)⁴·e^(iπ) = 4·(−1) = −4. A check in Cartesian form: (1 + i)² = 2i, and (2i)² = −4. These are the calculator’s default inputs.

Polar form and power calculator

z = a + b·i, z^n

a b n

|z|–

θ (degrees)–

z^n–

Common mistakes

Taking the angle as arctan(b/a) for a number in the second or third quadrant.
Finding one root and forgetting the other n − 1.
Mixing degrees and radians inside e^(iθ), which always takes radians.