Mathematics

Complex Numbers

Solve x² = 4 and you get ±2. Solve x² = −1 and no real number works. Rather than declaring it impossible, define a number that does: i² = −1. Drag the vector below to see what i actually does.

Before you read on: on the real number line, multiplying by −1 is a 180° rotation. What single operation, applied twice, gives a 180° rotation?

A 90° rotation. If i means "rotate 90° anticlockwise," then i × i means "rotate 90°, then another 90°" — a full 180° turn, identical to multiplying by −1. So i² = −1 is not an arbitrary rule. It follows from treating i as a rotation.

Common mistake

"i is imaginary, so it isn't a real thing." The word "imaginary" was a historical insult, not a mathematical description. Complex numbers are exactly as real as negative numbers once were — unfamiliar at first, indispensable once understood.

Multiplication is rotation

On the Argand plane, every complex number is a vector from the origin. Multiplying by i rotates that vector 90° anticlockwise — try it below.

z = 0.83 + 0.56i · |z| = 1 · arg(z) = 34.4°

Drag the vector tip to rotate it · click ×i to rotate 90° anticlockwise

In ℝ, multiplying by 3 scales. Multiplying by −1 reflects. In ℂ, multiplying by z = re^(iθ) does both at once: it scales by r and rotates by θ. This is why polar form exists — and why complex exponentials make the Fourier transform, AC circuit phasors, and quantum wavefunctions all work.

z₁ · z₂ = r₁r₂ · e^(i(θ₁+θ₂))

Magnitudes multiply. Arguments add. Two rotations applied in sequence equal a single rotation by the sum of their angles.

Euler's formula

If multiplying by i is a 90° rotation, multiplying by e^(iθ) should rotate by θ. Expand e^(iθ) as a Taylor series and group real and imaginary terms using i² = −1, i³ = −i, i⁴ = 1:

e^(iθ) = 1 + iθ − θ²/2! − iθ³/3! + θ⁴/4! + ⋯ = (1 − θ²/2! + θ⁴/4! − ⋯) + i(θ − θ³/3! + θ⁵/5! − ⋯) = cos θ + i sin θ

At θ = π, rotating from the point 1 by 180° lands on −1:

e^(iπ) + 1 = 0

This is not arranged to look beautiful — it is beautiful because rotation by π radians sends the point 1 to the point −1. Five of mathematics' most important constants, connected by one geometric fact.

Where i is not optional

AC circuits

Impedance: Z = R + j(ωL − 1/ωC)

The imaginary part is real reactance — genuine opposition from capacitors and inductors. Remove j and the equation falls apart.

Signal processing

Fourier transform: f̂(ξ) = ∫ f(t)·e^(−2πiξt) dt

The e^(iθ) term lets one equation decompose any signal into frequencies. Your phone uses this continuously.

Quantum mechanics

Schrödinger equation: iℏ ∂ψ/∂t = Ĥψ

i is not a convenience here. The wavefunction ψ is complex-valued by physical necessity.

Quiz

5 questions · Rotation, Euler's formula, applications.

Question 1

Why does i² = −1?

Question 2

Multiplying a complex number z = re^(iθ) by i:

Question 3

Euler's formula e^(iπ) + 1 = 0 follows from:

Question 4

In ℂ, multiplying z₁ · z₂ where z₁ = r₁e^(iθ₁) and z₂ = r₂e^(iθ₂) gives a result with:

Question 5

In AC circuit analysis, the impedance Z = R + j(ωL − 1/ωC). The imaginary part represents: