Complex Numbers on the Plane

For centuries, mathematicians were troubled by the equation x^2 = -1. No real number satisfies it. The solution was to invent one: the imaginary unit i, defined so that i^2 = -1. Far from being a mathematical curiosity, complex numbers turned out to be essential for describing electricity, quantum physics, signal processing, and fractal geometry.

The Complex Plane

Every complex number z = a + bi can be plotted as a point on a plane, where:

• The horizontal axis (real axis) represents the real part a.
• The vertical axis (imaginary axis) represents the imaginary part b.

This is the complex plane (also called the Argand diagram).

We can visualize the complex number as a point and its distance from the origin:

The purple line goes from the origin to the point z = (a, b). The red horizontal line shows the imaginary part. The complex number sits at the intersection.

Magnitude (Modulus) of a Complex Number

The magnitude (or modulus) of z = a + bi is its distance from the origin:

|z| = sqrt(a^2 + b^2)

This is just the Pythagorean theorem applied to the complex plane.

All complex numbers with the same magnitude lie on a circle centered at the origin:

The yellow circle shows all complex numbers with the same magnitude as z. The point z sits on this circle. When a = 3 and b = 4, |z| = 5 — the classic 3-4-5 right triangle.

Addition: Parallelogram Rule

Adding complex numbers works component-wise, just like vector addition:

(a + bi) + (c + di) = (a + c) + (b + d)i

The green line shows the sum z1 + z2. Just like vector addition, the result follows the parallelogram rule: place the two arrows tip-to-tail, and the diagonal of the parallelogram is the sum.

Multiplication: Rotation + Scaling

Here is where complex numbers become magical. When you multiply two complex numbers, you multiply their magnitudes and add their angles:

|z1 * z2| = |z1| * |z2| arg(z1 * z2) = arg(z1) + arg(z2)

Multiplication by a complex number is a rotation and a scaling in one operation.

When we multiply any point w on the unit circle by z, the point rotates by arg(z) and scales by |z|. Let’s see the unit circle and its image after multiplication by z:

The gray circle is the original unit circle. The purple circle has radius |z|, showing how multiplication scales everything by the magnitude. The rotation by arg(z) moves every point along the circle by that angle.

Euler’s Formula: The Crown Jewel

The most beautiful equation in mathematics connects exponentials, trig functions, and complex numbers:

e^(itheta) = cos(theta) + isin(theta)

This means every point on the unit circle can be written as e^(i*theta). And the famous special case at theta = pi gives:

e^(i*pi) + 1 = 0

— connecting e, i, pi, 1, and 0 in a single equation.

The teal circle is the unit circle. The red line shows sin(theta) (the imaginary part of e^(itheta)) and the blue line shows cos(theta) (the real part). As theta sweeps from 0 to 2pi, the point e^(i*theta) traces the entire unit circle.

Powers of i

Since i represents a 90-degree rotation, repeating the multiplication cycles through four values:

Every fourth power returns to 1. This cycle — 1, i, -1, -i, 1, i, -1, -i, … — is the heartbeat of complex arithmetic.

The purple curve shows the real part cycling through 1, 0, -1, 0, 1, … The red curve shows the imaginary part cycling through 0, 1, 0, -1, 0, … They are just cosine and sine sampled at quarter turns.