Complex Numbers Are Real

We call them "imaginary numbers" and use the symbol i for √-1, making them sound like mathematical fiction. But complex numbers are as real as any other number — and arguably more fundamental. Let me show you why.

The “Imaginary” Problem

In school, they tell you that √-1 doesn’t exist because “you can’t take the square root of a negative number.”

Then they introduce i = √-1 and call it “imaginary.”

This naming is terrible. It makes complex numbers sound like a hack, a trick, something not quite legitimate.

But here’s the truth: complex numbers are just as “real” as regular numbers. The problem isn’t the numbers — it’s how we think about them.

Numbers Are Tools for Solving Problems

Let’s step back. What even is a number?

Counting numbers (1, 2, 3, …) let you count sheep.

Integers (…, -2, -1, 0, 1, 2, …) let you represent debt and temperature below zero.

Rational numbers (fractions) let you divide things fairly.

Real numbers (including π, √2, e) let you measure continuous quantities.

Each expansion of “number” seemed weird at first. Romans didn’t have zero. Negative numbers were controversial for centuries. Irrational numbers freaked out the Pythagoreans.

But each time, we expanded the concept because it was useful.There’s a famous story that the Pythagoreans threw Hippasus overboard for proving √2 is irrational. Probably apocryphal, but shows how shocking the idea was!

Complex numbers are the next step. Not a trick — a natural extension.

The Geometric View

Here’s the key insight that makes everything clear: stop thinking of complex numbers as weird algebra, and start thinking of them as points in a plane.

Every complex number is a point in 2D space:

• The real part is the x-coordinate
• The imaginary part is the y-coordinate

So 3 + 4i is just the point (3, 4).

Complex numbers aren't about √-1 being magical. They're about working in two dimensions instead of one.

Regular numbers live on a line. Complex numbers live on a plane. That’s it.

What Is i, Really?

In this geometric view, what’s i?

i is a 90-degree rotation.

Multiplying by i rotates a point counterclockwise by 90°:

• Start with 1 (pointing right along the real axis)
• Multiply by i → you get i (pointing up along the imaginary axis)
• Multiply by i again → you get i² = -1 (pointing left)
• Multiply by i again → you get i³ = -i (pointing down)
• Multiply by i again → you get i⁴ = 1 (back where you started)

That’s why i² = -1. Not because of mystical square roots, but because rotating twice by 90° flips you 180°, which is multiplication by -1.

Suddenly, it’s not mysterious. It’s geometry.

Why Complex Numbers Matter

Once you see complex numbers geometrically, you realize they’re not optional — they’re essential for describing rotation, oscillation, and waves.

Electrical Engineering

AC circuits are impossible to analyze without complex numbers. Impedance, reactance, phase shifts — all naturally described using complex arithmetic.Engineers sometimes use j instead of i because i already means current. Same concept, different letter.

Quantum Mechanics

The wave function — the fundamental description of quantum states — is inherently complex. Schrödinger’s equation doesn’t work with real numbers alone.

The universe, at its most fundamental level, seems to run on complex numbers.

Signal Processing

Remember Fourier transforms? They convert signals into complex exponentials. The entire framework of modern signal processing depends on complex numbers.

Fractals

The Mandelbrot set — that beautiful infinite fractal — is defined using complex numbers. It literally can’t exist without them.

Euler’s Formula: The Most Beautiful Equation

Once you understand complex numbers geometrically, you can appreciate why this is stunning:

e^(iθ) = cos(θ) + i·sin(θ)

This says: raising e to an imaginary power gives you rotation.

An exponential (usually about growth) becomes circular motion. This connects:

• Exponentials (e^x)
• Trigonometry (sine and cosine)
• Rotation (the complex plane)
• π (when θ = π, you get e^(iπ) = -1)

"e^(iπ) + 1 = 0" — Euler's identity, connecting five fundamental constants

This equation is geometric truth written algebraically. e^(iπ) means “rotate by π radians (180°)” which gets you to -1.

Complex Numbers Complete Algebra

Here’s another reason they’re fundamental: complex numbers complete algebra.

The Fundamental Theorem of Algebra says: every polynomial has a solution in the complex numbers.

• x² + 1 = 0 has no real solutions, but has solutions x = i and x = -i
• x³ - 1 = 0 has one real solution (x = 1) and two complex solutions
• Every n-degree polynomial has exactly n complex roots (counting multiplicity)

With complex numbers, algebra is complete. No more exceptions, no more “this equation has no solution.” Every polynomial factorizes perfectly.

This isn’t a coincidence. Complex numbers are where algebra wants to live.

Building Intuition

The key to understanding complex numbers:

Once you see it this way, the “imaginary” label falls away. These numbers are describing real geometric operations.

Why the Name Stuck

René Descartes coined “imaginary” in 1637 as a dismissive term. He meant it mockingly — these numbers don’t exist, they’re just imagination.

But mathematicians kept using them because they worked. They solved problems. They revealed patterns. They made calculations cleaner.

By the 1800s, Gauss and others showed the geometric interpretation, proving complex numbers are perfectly rigorous.

But we’re stuck with the name. “Imaginary” is historical baggage, not mathematical truth.

The Bigger Picture

Complex numbers teach us something important: mathematical truth isn’t about physical tangibility.

You can’t hold the number three. You can’t point to π. You can’t touch an integral.

Mathematics describes patterns, structures, relationships. Complex numbers are a structure that happens to describe rotation, oscillation, and wave phenomena perfectly.

That makes them real — real in the way that matters for mathematics.

My Takeaway

The moment I stopped thinking of complex numbers as “weird algebra with √-1” and started seeing them as “points in a plane where multiplication rotates,” everything clicked.

They’re not imaginary. They’re not mysterious. They’re just 2D numbers with beautiful geometric properties.

And once you see it, you realize: the universe runs on oscillation, rotation, and waves.

No wonder complex numbers are everywhere.

Resources: “Visual Complex Analysis” by Tristan Needham is the most beautiful book on this topic — entirely geometric, deeply insightful.