Polar Form and De Moivre’s Theorem

A New Way to Describe Complex Numbers

Every complex number z = a + bi can be visualised as the point (a, b) on the Argand diagram. The Cartesian form tells us the horizontal and vertical coordinates. But there is another equally natural way to locate a point in a plane: give its distance from the origin and the angle it makes with the positive real axis. This is the polar (or modulus-argument) form, and it turns out to be far more powerful for multiplication, division, and finding powers of complex numbers.

Modulus and Argument

The modulus of z = a + bi is r = |z| = √(a² + b²). This is simply the distance from the origin to the point (a, b), calculated by Pythagoras’ theorem. The argument θ = arg(z) is the angle (measured anticlockwise from the positive real axis) to the line segment joining the origin to z. By convention we use the principal argument where θ ∈ (−π, π].

To find the argument: compute the reference angle α = arctan(|b/a|) and then adjust for the quadrant. If z is in Q1: θ = α. Q2: θ = π − α. Q3: θ = −(π − α) = α − π. Q4: θ = −α.

The cis Notation

We write cisθ as shorthand for cosθ + i sinθ. So any complex number with modulus r and argument θ can be written as z = r cisθ. Converting back to Cartesian form: a = r cosθ, b = r sinθ. This connection between polar coordinates and trigonometry is why this form is so useful.

Multiplication and Division in Polar Form

This is where polar form shines. If z&sub1; = r&sub1; cisθ&sub1; and z&sub2; = r&sub2; cisθ&sub2;, then:

z&sub1;z&sub2; = r&sub1;r&sub2; cis(θ&sub1; + θ&sub2;)  —  multiply the moduli, add the arguments

z&sub1;/z&sub2; = (r&sub1;/r&sub2;) cis(θ&sub1; − θ&sub2;)  —  divide the moduli, subtract the arguments

This is far simpler than expanding in Cartesian form. Geometrically, multiplication by a complex number simultaneously scales and rotates the original number.

De Moivre’s Theorem

Applying the multiplication rule repeatedly: (r cisθ)² = r² cis(2θ), (r cisθ)³ = r³ cis(3θ), and in general:

[r cisθ]ⁿ = rⁿ cis(nθ) for any integer n.

This theorem is extraordinary: it says that to raise a complex number to the n-th power, you simply raise the modulus to the n-th power and multiply the argument by n. What would be a lengthy algebraic expansion collapses to two arithmetic steps. De Moivre’s theorem also extends to rational exponents, making it the standard tool for finding square roots and cube roots of complex numbers in polar form.

Finding Roots of Complex Numbers

To find the n-th roots of z = r cisθ, use: z^1/n = r^1/n cis((θ + 2kπ)/n) for k = 0, 1, 2, …, n−1. This gives exactly n distinct roots, equally spaced around a circle of radius r^1/n in the Argand plane. For example, the three cube roots of 8 are found by writing 8 = 8 cis(0) and computing: 2 cis(0) = 2, 2 cis(2π/3) = −1 + √3 i, 2 cis(4π/3) = −1 − √3 i.

Why This Matters

De Moivre’s theorem underpins trigonometric identities (you can derive cos(3θ) and sin(3θ) from (cisθ)³), signal processing, electrical engineering (phasors), and quantum mechanics. In this course, it is the gateway to solving polynomial equations completely over the complex numbers.