Imaginary Numbers

The name “imaginary number” sounds almost like a joke — how can a number be imaginary? The name reflects a history of suspicion. These numbers were useful before anyone was comfortable saying they were real.

The need for them appears when we try to solve equations with no answer among the real numbers. The simplest example:

x² + 1 = 0

No real number squared gives −1. The real numbers have no solution. The first serious mathematician to work with square roots of negative numbers in calculation was Girolamo Cardano. In his 1545 book Ars Magna, he encountered expressions like √−15 while solving cubic equations. He called such expressions “fictitious” and found them confusing, but noted they appeared to work in calculations if you followed the rules. He did not explain why.

The term “imaginary” came from René Descartes in 1637, used dismissively. Gottfried Wilhelm Leibniz in the late 1600s called them “an elegant and wonderful recourse of the divine spirit” while also describing them as “a sort of amphibian between being and not-being.”

Leonhard Euler in the 18th century gave the imaginary unit its modern symbol i and established the relation i² = −1. Together with his work connecting complex numbers to trigonometry, this produced the formula:

e^(ix) = cos(x) + i sin(x)

Setting x = π gives the special case e^(iπ) + 1 = 0, now called Euler’s identity, which connects five fundamental mathematical constants in a single equation. Carl Friedrich Gauss gave complex numbers their geometric interpretation in the early 19th century: the complex plane, where real numbers appear on a horizontal axis and imaginary numbers on a vertical one. Multiplication by i is geometrically a 90-degree rotation in this plane, which turns out to explain a great deal about why complex numbers are useful.

Why they are not imaginary

By the early 19th century, complex numbers had become indispensable in several areas of mathematics. Augustin-Louis Cauchy developed complex analysis — the study of functions of complex variables — which turned out to be one of the most powerful and beautiful areas of mathematics ever developed.

The applications are not abstract. In electrical engineering, impedance (the complex resistance to alternating current) is expressed as Z = R + jX, where j is the engineering notation for i, R is resistance, and X is reactance. The mathematics of alternating current circuits only works cleanly in the complex number system. In signal processing, the Fourier transform — which decomposes a signal into its frequency components — is expressed most compactly using complex exponentials. In quantum mechanics, the Schrödinger equation, which governs how quantum states evolve, requires complex amplitudes; the probabilities we observe are the squared magnitudes of those complex numbers.

I like the history of imaginary numbers because it illustrates a pattern that appears throughout mathematics. Sometimes a problem cannot be solved inside the system we already have. The right response is not to try harder within the existing rules, but to expand the rules — to enlarge the number system to include what was previously impossible. The expanded system then turns out to have more structure and more applications than the original. A number that once looked impossible becomes, a few generations later, a fundamental tool that engineers use every day.