In 1545 the Italian polymath Girolamo Cardano was working on a perfectly ordinary problem: divide 10 into two parts whose product is 40. Setting one part as xx and the other as 10−x10 - x, he wrote down the equation x(10−x)=40x(10-x) = 40, expanded to x2−10x+40=0x^2 - 10x + 40 = 0, and applied the quadratic formula. The solutions came out as 5±−155 \pm \sqrt{-15}.

Cardano stared at this for a moment. The square root of a negative number didn’t make any sense — squaring any real number gives something positive. He wrote in the margin: “so subtle as to be useless.” Then he proceeded to verify that yes, multiplying (5+−15)(5−−15)(5 + \sqrt{-15})(5 - \sqrt{-15}) does indeed equal 25−(−15)=4025 - (-15) = 40, exactly as required. The arithmetic worked. The objects manipulating it didn’t exist.

This was the moment imaginary numbers entered mathematics — sideways, against the will of the mathematician using them, as a useful fiction that gave correct answers despite being apparently meaningless. It took two and a half centuries before they were understood to be neither imaginary nor useless. By the twentieth century they had become essential to quantum mechanics, electrical engineering, signal processing, and large parts of pure mathematics. The “imaginary” name is one of the worst-fitting in all of mathematics.

This article is about why that happened — what complex numbers actually are, why they ended up everywhere, and what the long delay between their invention and their acceptance tells us about how mathematics absorbs new ideas.

The setup

A real number is a point on the number line. You can add, subtract, multiply, and divide them. Squaring any real number gives something non-negative, so x2=−1x^2 = -1 has no real solution. Pre-1545, that was the end of the discussion.

Cardano’s contemporaries — Tartaglia, Bombelli, Ferrari — had a problem they couldn’t avoid. The cubic equation x3+px+q=0x^3 + px + q = 0 has a general formula, discovered around 1535. For some cubics with three obvious real roots, the formula goes through expressions involving square roots of negative numbers as intermediate steps. The square roots cancel out at the end, but you cannot evaluate the formula without passing through them. Mathematics was forcing the issue.

Bombelli, in 1572, took the leap. He defined arithmetic for what he called “plus of minus” and “minus of minus” — the symbols +−1+\sqrt{-1} and −−1-\sqrt{-1} — by the rules

(−1)2=−1,−1⋅−1=−1.(\sqrt{-1})^2 = -1, \quad \sqrt{-1} \cdot \sqrt{-1} = -1.

With these rules he could carry the cubic formula’s intermediate steps through and get correct real answers at the end. The intermediate “imaginary” quantities cancelled, but they had to be there during the calculation.

For 250 years afterwards, mathematicians used these objects without believing in them. Leibniz called them “an amphibian between being and non-being.” Euler used them constantly and produced extraordinary results. Nobody could quite say what they were.

The geometric picture

The conceptual breakthrough came in the 1790s. Caspar Wessel in Norway and Jean-Robert Argand in France, independently, suggested a simple visualisation: stop thinking of numbers as points on a line. Think of them as points on a plane.

The real number line stays put as the horizontal axis. The vertical axis is the new direction. A complex number a+bia + bi — where i=−1i = \sqrt{-1} — is just a point in this plane: aa steps to the right, bb steps up.

The four operations now have geometric meanings:

  • Addition is vector addition. To add two complex numbers, draw the parallelogram.
  • Multiplication by a real number is scaling — the same as in one dimension.
  • Multiplication by ii is rotation by 90°. Multiplying once turns east into north; twice turns east into west; four times brings you back to east. This is exactly i4=1i^4 = 1.
  • Multiplication by a general complex number is rotation combined with scaling — a single operation that captures both.

That’s the punchline. Imaginary numbers are not mysterious. They are the algebra of rotations and scalings in two dimensions, written in a notation that lets you treat them as if they were one-dimensional.

Carl Friedrich Gauss, in 1831, made this view rigorous and respectable. He coined the name “complex number” — the modern term — and showed that complex arithmetic was as legitimate as real arithmetic. Within a generation, the philosophical objections evaporated. Mathematicians stopped apologising for using ii.

Euler’s identity

The deepest payoff of the geometric view is one of the most famous equations in mathematics:

eiπ+1=0.e^{i\pi} + 1 = 0.

Read our piece on ee for why ee is fundamental and our Euler’s identity page for a step-by-step derivation. The short version: extending the exponential function exe^x to complex inputs via its Taylor series gives, for any real θ\theta,

eiθ=cos⁡θ+isin⁡θ.e^{i\theta} = \cos\theta + i\sin\theta.

Setting θ=π\theta = \pi gives eiπ=−1e^{i\pi} = -1, hence the identity. The remarkable thing is that the exponential, defined originally for compound growth, turns out to be the same function as cosine and sine — they are real and imaginary parts of the same underlying complex object.

This is why complex numbers are everywhere in physics and engineering. Whenever a system has periodic behaviour — alternating current, oscillating springs, vibrating strings, electromagnetic waves, quantum amplitudes — the equations governing it become dramatically simpler in complex form. The two operations of trigonometry (cosine, sine) collapse into one operation (complex exponential), and that single operation has clean differentiation, integration, and multiplication rules.

Where complex numbers actually live

Once mathematicians admitted complex numbers were legitimate, an extraordinary thing happened: they noticed that complex numbers are better behaved than real numbers in many ways.

Algebra closes up. The fundamental theorem of algebra — proved by Gauss — says every polynomial of degree nn with complex coefficients has exactly nn complex roots, counted with multiplicity. Over the reals, x2+1x^2 + 1 has no roots. Over the complexes, every polynomial factors completely. The complex numbers are algebraically closed; the reals are not.

Calculus becomes more rigid. A function of one real variable can be smooth (infinitely differentiable) without being analytic (equal to its Taylor series). For complex-differentiable functions, the two coincide: any complex-differentiable function is automatically analytic, and its values on a small disk determine its values everywhere. This rigidity is the basis of complex analysis, one of the most beautiful subjects in pure mathematics.

Number theory simplifies. The Riemann zeta function, originally defined as an infinite series in real variables, only reveals its connection to the prime numbers when extended to the complex plane. The Riemann hypothesis — perhaps the most famous open problem in mathematics — is a statement about where the complex zeros of this function lie.

Geometry generalises. A complex number with magnitude 1 represents a rotation. The set of such numbers forms a circle, S1S^1, with multiplication as the group operation. Generalise this to higher dimensions and you get the unitary groups U(n)U(n), which describe quantum mechanics. The whole apparatus of Lie groups underlying modern physics has complex numbers built into its foundations.

Quantum mechanics needs complex numbers

The most striking demonstration that complex numbers are not just a calculational convenience comes from quantum mechanics.

Classical mechanics describes the world with real-valued positions, velocities, and forces. Quantum mechanics replaces these with state vectors in a complex Hilbert space. A particle’s wave function ψ(x,t)\psi(x, t) is a complex-valued function. The fundamental equation governing its evolution — the Schrödinger equation — has an explicit factor of ii:

iℏ∂ψ∂t=H^ψ.i\hbar \frac{\partial \psi}{\partial t} = \hat H \psi.

You cannot write quantum mechanics without complex numbers. Real-valued formulations have been attempted; they require ugly mathematical contortions and produce no new predictions. The complex-valued framework, by contrast, predicts everything from the spectrum of hydrogen to the behaviour of transistors.

Why is reality complex-valued? Nobody fully knows. The leading interpretation is that complex amplitudes encode interference, and interference is a basic feature of quantum reality that has no simple real-number analogue. Whatever the deep reason, the mathematical structure has been independently confirmed by every quantum experiment ever performed.

Engineering applications

Outside fundamental physics, complex numbers are bread-and-butter tools for engineers.

AC circuits. Voltage and current oscillating sinusoidally are most naturally represented by complex amplitudes. The relationship between them through capacitors and inductors becomes a simple algebraic equation in the complex domain — one that would require differential equations in the real domain.

Signal processing. Every digital filter, radio modem, MP3 codec, and image-compression algorithm relies on the fast Fourier transform — which decomposes signals into complex exponentials.

Control theory. The stability of feedback systems is determined by the location of complex poles in a transfer function. Whether your aircraft autopilot is stable, your power grid stays synchronised, or your audio amplifier oscillates depends on numbers in the complex plane.

Antenna design. The radiation patterns of antennas are computed via complex impedance functions. The Smith chart, a graphical tool every microwave engineer learns, lives in the complex plane.

If complex numbers were “imaginary” in any meaningful sense, none of this technology would work.

What the history teaches

The story of complex numbers is a story about mathematical maturity. For 250 years, mathematicians used objects they didn’t understand because the objects gave correct answers. The acceptance came not when philosophers approved of them, but when geometric and algebraic frameworks — Wessel’s plane, Gauss’s algebraic closure, Riemann’s complex analysis — made the formal manipulations transparent.

A pattern that recurs throughout mathematics: a useful tool is invented to solve a specific problem, used uneasily for generations, and eventually understood to be a fundamental object whose deeper structure was waiting to be noticed. The same arc applies to negative numbers (resisted into the 1700s), to limits (only rigorously defined in the 1820s), and to infinitesimals (formalised only in the 1960s).

The lesson, perhaps, is that the difference between “real” and “imaginary” mathematics is mostly a matter of how long the community has had to get used to the idea. Cardano’s “useless” square roots became, four centuries later, the foundation of quantum mechanics. The next generation’s “useless” idea is probably already in print, being grumbled about, and waiting its turn.

Frequently asked

Why are they called 'imaginary'?

Because Descartes, in 1637, used the word 'imaginaires' dismissively — to suggest these numbers were unreal compared to ordinary 'real' ones. The terminology stuck even after mathematicians realised they were perfectly legitimate. The names 'real' and 'imaginary' are historical accidents, not statements about reality.

Are complex numbers actually used in physics?

Constantly. Quantum mechanics is fundamentally complex: state vectors live in complex Hilbert spaces, and the Schrödinger equation has an explicit i in it. Electrical engineering uses complex impedance for AC circuits. Signal processing uses the Fourier transform, which only makes sense over the complex numbers.

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Formula

Euler's Identity

The fundamental relation linking the five essential constants of mathematics through the exponential map on the complex plane — and a gateway into modern analysis.

Read →Formula

The Fourier Transform

The integral transform that decomposes a function into its frequency spectrum — foundation of harmonic analysis, quantum mechanics, and modern signal processing.

Read →Formula

Schrödinger Equation

The partial differential equation that governs the evolution of a quantum system's wavefunction — the mathematical core of quantum mechanics.

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