The Invention Of Imaginary Numbers: Separating Math From Reality

Mathematics began very close to the physical world. People counted objects, measured land, followed the movement of stars, and solved practical problems.

For a long time, numbers were expected to represent something we could imagine directly. Then algebra started asking questions that did not fit so comfortably inside physical reality.

The cubic equation and the problem that forced the issue

One famous example came from the search for general solutions to cubic equations. In Renaissance Italy, mathematicians competed intensely over these problems — solutions were valuable in public contests and kept secret for advantage. Luca Pacioli had written in 1494 that a general solution to the cubic seemed impossible. Not long after, Scipione del Ferro found a method for one class of cubics and shared it with his student before he died.

The story became dramatic when Niccolò Tartaglia rediscovered methods and shared them with Gerolamo Cardano under conditions of secrecy and a sworn oath. Cardano eventually published a general treatment in Ars Magna in 1545, acknowledging del Ferro’s earlier priority but breaking his promise to Tartaglia — a dispute that continued bitterly for years.

But the formulas created a strange problem. Even when a cubic equation had three perfectly real positive solutions, the formula required passing through expressions like √-1 in intermediate steps. Cardano himself called these “sophistic quantities” — formally useful but without meaning. The square root of a negative number appeared in the middle of a calculation that ended with a real, sensible answer. This was deeply uncomfortable for everyone who encountered it.

Bombelli takes the strange expressions seriously

Rafael Bombelli made the decisive step in his 1572 work L’Algebra. He did not dismiss the square roots of negative numbers as meaningless. Instead, he developed consistent rules for manipulating them: a positive times a positive imaginary gives a negative, and so on. He showed that if you followed these rules carefully, the strange intermediate expressions could cancel out and leave real solutions. The imaginary quantities were not obstacles to be apologised for — they were part of the calculation that led to the correct answer.

This was not a proof that √-1 existed in any physical sense. It was a demonstration that treating it as a symbol governed by consistent rules produced reliable, correct results. Mathematics did not need every symbol to point to a visible object.

The word “imaginary” and what comes after

René Descartes coined the term “imaginary” in 1637, intending it as a criticism — these were numbers that could not exist. The name stuck long after the criticism lost its force. By the eighteenth century, mathematicians including Euler and Gauss had developed the full system of complex numbers. Gauss proved the fundamental theorem of algebra in his 1799 doctoral dissertation using complex numbers: every polynomial equation of degree n has exactly n roots in the complex numbers. The theorem cannot be stated or proved staying inside the real numbers alone.

The geometric interpretation arrived in 1806 when Jean-Robert Argand published a way to represent complex numbers as points in a plane — real part on the horizontal axis, imaginary part on the vertical. This picture made multiplication by i visible as a 90-degree rotation, and it connected complex numbers to geometry in a way that made their behaviour intuitive.

Euler’s formula e^(ix) = cos(x) + i sin(x) — and the special case e^(iπ) + 1 = 0, sometimes called the most beautiful equation in mathematics — showed that imaginary exponents described rotation in the complex plane. These were not isolated tricks. They were a deep connection between algebra, geometry, and analysis.

Where they appear today

Engineering uses complex numbers constantly. Alternating current circuits are analysed using impedance Z = R + jX, where the imaginary component represents the phase shift introduced by capacitors and inductors. Signal processing and control systems use the Fourier transform, which decomposes a signal into complex exponentials. Quantum mechanics describes quantum states as vectors in a complex Hilbert space; the Schrödinger equation itself is complex.

I like this history because it shows how knowledge sometimes grows. A strange idea appears, people resist it because it does not fit the old picture, and then someone asks a better question: not “Is this real in the way I already understand reality?” but “Is this idea consistent, useful, and mathematically meaningful?” That question, taken seriously, can change everything.