What Is a Complex Number Actually Good For?

"When will I ever use imaginary numbers?" It's one of the most common questions in math class. The honest answer is that you use them every time you flip on a light, make a phone call, stream a song or board a plane. You just don't see them.

Complex numbers are the natural language for anything that rotates or oscillates, and the modern world is full of things that do.

Complex Numbers Make Real Problems Easier

Adding an "imaginary" dimension sounds like it would make calculations harder. In practice it often does the opposite.

A line often attributed to French mathematician Jacques Hadamard (who credited the idea to Paul Painlevé) sums it up: "The shortest path between two truths in the real domain passes through the complex domain." Engineers take that detour because a problem full of sines, cosines and derivatives can turn into simple algebra once you step into the complex plane.

1. The Power Grid: AC Circuits

The electricity in your wall is alternating current (AC): voltage swinging back and forth as a sine wave, 60 times a second in North America and 50 in much of the world.

In AC circuits, resistors, capacitors and inductors don't just reduce current. Capacitors and inductors shift its timing relative to voltage. Tracking both size and timing with trigonometry quickly gets messy.

Complex numbers bundle both into one quantity called impedance:

Z = R + iX

R is the resistance and X is the reactance from capacitors and inductors. Take a circuit with R = 30 Ω and X = 40 Ω:

|Z| = √(30² + 40²) = 50 Ω
phase angle = arctan(40 / 30) ≈ 53.13°

So the current is limited as if by a 50 Ω resistor and lags the voltage by about 53°. Check these with the right triangle calculator and arc tangent calculator. (Electrical engineers write j instead of i, since i already means current.)

An Insider Reference: Charles Proteus Steinmetz

In 1893, Charles Proteus Steinmetz, a German-born engineer who emigrated to the U.S., presented a paper on "Complex Quantities and Their Use in Electrical Engineering" at the International Electrical Congress in Chicago.

Before Steinmetz, AC analysis required solving differential equations. His method, now called phasor analysis, turned it into complex-number arithmetic that engineers could do by hand. He became General Electric's leading engineer and was nicknamed "the Wizard of Schenectady." Practically every AC power system designed since has used his approach.

2. Signals, Sound and Radio

Every sound, radio transmission and Wi-Fi signal can be broken into a sum of waves of different frequencies. The Fourier transform does that decomposition, and it's written most naturally with complex exponentials:

e^(iωt) = cos(ωt) + i sin(ωt)

A single complex number captures both the strength (modulus) and timing (angle) of each frequency. That's how:

• Music apps display equalizers and spectrograms
• MP3 and AAC compress audio
• Phones and Wi-Fi pack data into radio waves, using schemes like QAM, where each symbol is literally a point on the complex plane

3. Quantum Mechanics

In 1926, Erwin Schrödinger published the equation that governs quantum systems:

iħ ∂ψ/∂t = Ĥψ

The i is right there at the front. The wave function ψ is complex-valued, and the probability of finding a particle somewhere is |ψ|², the squared modulus.

For a long time, some physicists wondered whether complex numbers were truly necessary or just a convenience. Experiments reported in 2022 by independent research teams (published in Physical Review Letters) supported the conclusion that standard quantum theory genuinely needs them: versions using only real numbers make predictions that conflict with experiment.

4. Airplane Wings

Around 1910, Russian scientist Nikolai Zhukovsky (Joukowsky) used a complex function to transform a simple circle into the shape of an airfoil:

w = z + 1/z

Airflow around a spinning cylinder is easy to calculate. Mapping it through the transformation gives the airflow around a wing, and from that, the lift. Conformal mapping, which preserves angles, became a foundational tool in early aerodynamics.

5. Stability and Control

Will a bridge sway out of control? Will a drone's autopilot oscillate wildly? Control engineers find the roots of a system's characteristic equation, and those roots are often complex:

• If every root has a negative real part, the system settles down (stable)
• If any root has a positive real part, oscillations grow (unstable)
• The imaginary part sets how fast it oscillates

Plotting roots on the complex plane shows at a glance whether a design is safe.

6. Graphics and Fractals

Multiplying by a complex number rotates and scales in one step, which makes complex numbers handy for 2D graphics. And repeatedly applying z → z² + c produces the Mandelbrot set, whose boundary is infinitely detailed. Explore related mathematical beauty in the interactive Lissajous curve and cardioid. The cardioid is the shape of the Mandelbrot set's main body.

Two Concepts Worth Knowing

Phasors

A phasor is a complex number representing a sine wave's amplitude and phase. Adding waves of the same frequency becomes simple complex addition.

Polar Form

Writing z = r·e^(iθ) makes multiplication easy: multiply the r values and add the angles. Practice with the polar to rectangular converter.

Quick Answer: What Are Complex Numbers Used For?

Complex numbers are used to analyze AC electrical circuits, process audio and radio signals with Fourier transforms, describe quantum mechanics, model airflow around wings, test the stability of control systems, and create fractals. They're ideal for anything involving rotation or oscillation, since one complex number captures both size and phase.

Try Them Yourself

• Polar to Rectangular Converter: the two faces of a complex number
• Right Triangle Calculator: impedance magnitude
• Arc Tangent Calculator: phase angle
• Sine Calculator: AC waveforms
• Cardioid: the heart of the Mandelbrot set
• The Strange World of Imaginary Numbers: how i works

Look at a device's power adapter. Its rating assumes a sine-wave supply at 50 or 60 Hz, and the engineer who designed it almost certainly analyzed it with complex numbers.