Polar Form of Complex Numbers

Every complex number can be described in two fundamentally different but equivalent ways. The familiar rectangular form z = a + bi locates a number by its real and imaginary parts on the complex plane, much like grid coordinates. The polar form, by contrast, describes the same point using its distance from the origin and the angle it makes with the positive real axis — a representation that often makes multiplication, division, and powers dramatically simpler. The distance from the origin to the point (a, b) in the complex plane is called the modulus of z, written |z| or r, and is computed by the Pythagorean theorem as r = √(a² + b²). The angle measured anticlockwise from the positive real axis to the line segment connecting the origin to the point is called the argument, written arg(z) or θ. Together, r and θ completely determine the complex number, and we write z = r(cos θ + i sin θ), often abbreviated using Euler's formula as z = re^(iθ). Polar form is not merely a notational convenience. It reveals the geometric meaning of complex multiplication — multiplying two complex numbers multiplies their moduli and adds their arguments — and it is the natural language for De Moivre's theorem, finding roots of complex numbers, and analysing alternating-current circuits in physics and engineering. Mastering polar form is therefore a gateway to many of the most powerful applications of complex numbers.

Key takeaways

Plot the complex number on the Argand diagram
Compute the modulus r
Find the principal argument θ
Write the polar form
Convert back to rectangular form

Plot the complex number on the Argand diagram

Write z = a + bi and identify the point (a, b) in the complex plane. The horizontal axis represents the real part and the vertical axis represents the imaginary part. Sketching the point first prevents sign errors in later steps.

Compute the modulus r

Apply the Pythagorean theorem to the right triangle formed by a, b, and the line segment to the origin: r = |z| = √(a² + b²). The modulus is always non-negative. For example, if z = 3 + 4i then r = √(9 + 16) = √25 = 5.

Find the principal argument θ

Use the relation tan θ = b/a to find the reference angle α = arctan(|b|/|a|). Then adjust for the correct quadrant: if (a, b) is in Q1 then θ = α; Q2 then θ = π − α; Q3 then θ = −(π − α) or equivalently π + α measured differently; Q4 then θ = −α. The principal argument is restricted to (−π, π]. Many calculators provide atan2(b, a) which handles this automatically.

Write the polar form

Express the result as z = r(cos θ + i sin θ). This is sometimes written z = r cis θ as shorthand. Using Euler's formula (derived from the Taylor series of e^x, sin x, cos x), this is equivalently z = re^(iθ). Both notations are standard at this level.

Convert back to rectangular form

Given r and θ, recover a and b by a = r cos θ and b = r sin θ. For example, z = 4(cos(π/3) + i sin(π/3)) gives a = 4 cos(π/3) = 4 · (1/2) = 2 and b = 4 sin(π/3) = 4 · (√3/2) = 2√3, so z = 2 + 2√3 i.

The Key Equation

z = r(\cos\theta + i\sin\theta) = re^{i\theta}

z — Complex number (dimensionless): The complex number being represented, equal to a + bi in rectangular form.
r — Modulus (dimensionless): The distance from the origin to z in the Argand plane; r = √(a² + b²) ≥ 0.
\theta — Argument (radians): The angle measured anticlockwise from the positive real axis to the position vector of z; the principal argument lies in (−π, π].
e^{i\theta} — Euler exponential (dimensionless): Shorthand combining cos θ + i sin θ via Euler's formula; allows polar form to be written compactly as re^(iθ).

Worked Example

Problem. Express z = −1 + √3 i in polar form, and hence write z³ in rectangular form using De Moivre's theorem.

z = 2\!\left(\cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3}\right),\quad z^3 = 8

Answer. In polar form z = 2(cos(2π/3) + i sin(2π/3)); and z³ = 8.

Figure

Diagram

Origins · 1 minAbraham de Moivre and how this was discovered

Abraham de Moivre

1667–1754 · French–British

De Moivre formulated the theorem bearing his name — (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ) — which makes raising a complex number in polar form to an integer power trivially simple. He stated it around 1707 and it was proved rigorously using Euler's formula later in the century. De Moivre also made foundational contributions to probability theory, including deriving the normal distribution as an approximation to the binomial. De Moivre reportedly predicted the date of his own death by noticing he was sleeping 15 minutes longer each night and calculating when the total would reach 24 hours — and he was correct to within a day.

Rectangular Form of Complex Numbers — Polar form is an alternative representation to rectangular form a + bi; fluency in converting between both is essential.
Euler's Formula — Euler's formula e^(iθ) = cos θ + i sin θ is the bridge that allows polar form to be written compactly as re^(iθ).
De Moivre's Theorem — De Moivre's theorem uses polar form to raise complex numbers to powers and find nth roots efficiently.
Modulus and Argument — The modulus |z| and argument arg(z) are the two quantities that define polar form, so understanding them separately is a prerequisite.