How Imaginary Numbers Were Invented
How Imaginary Numbers Were Invented
ELI5/TLDR
For thousands of years, math was geometry. Numbers were lengths, squares, and cubes you could draw in the dirt, so negative numbers made no sense — you can’t have a square with sides of negative five. But to solve a problem that had been stuck for 4,000 years, Renaissance mathematicians had to start using the square root of negative one, a number that couldn’t possibly exist. Four hundred years later, that same impossible number turned up at the center of quantum physics, where it describes how every atom in the universe behaves. Giving up on math matching reality is what let math finally describe reality.
The Full Story
In 1494, Leonardo da Vinci’s math teacher publishes a summary of all known mathematics and concludes that one particular kind of equation — the cubic, where x is raised to the third power — simply cannot be solved. He’s wrong, but understandably so. Every civilization that had tried, from the Babylonians on, had failed.
The reason it was so hard is that math back then meant geometry. You couldn’t write equations the way we do now — symbolic algebra wouldn’t exist for another century. So when an old mathematician saw “x squared,” they pictured an actual square with sides of length x. “26x” was a rectangle. The equation was a literal jigsaw puzzle of shapes whose total area had to come out right. This worked beautifully for quadratics. The technique was called “completing the square,” and it was exactly that — you’d cut your rectangles in half, slide them around, and notice that you only needed to add one missing tile to turn the whole thing into a perfect square. Then you’d take the square root and read off the answer.
The price of this approach was that negative numbers were forbidden. A length can’t be negative. An area can’t be negative. So mathematicians simply didn’t believe in them. There wasn’t even one quadratic equation — there were six versions, rearranged so every coefficient stayed positive.
The math duels
Solving the cubic eventually fell out of an unusual feature of 1500s academic life. Holding a math professorship meant other mathematicians could show up and challenge you for your job. Each side handed the other a list of problems. Whoever solved more kept their position. The loser was publicly humiliated.
Around 1510, a Bologna professor named Scipione del Ferro figured out how to solve a stripped-down version called the depressed cubic (no x-squared term). He did the rational thing: he told nobody. For nineteen years he sat on the answer. On his deathbed he passed it to a student, Antonio Fior, who was talented enough to inherit the secret but not talented enough to keep his mouth shut. Fior bragged. Then he challenged a recently arrived rival in Venice — Niccolo Fontana, known as Tartaglia (“the stutterer”) because a French soldier had cut his face open when he was a kid.
Fior gave Tartaglia thirty depressed cubics. Tartaglia, who hadn’t solved the cubic before but now knew it was solvable, locked himself in and worked it out from scratch. He extended completing the square into three dimensions: imagine the x-cubed term as a literal cube, then pad it out with rectangular slabs until it becomes a bigger cube. The geometry, once you arrange it carefully, collapses back into a quadratic in disguise. Tartaglia solved all thirty of Fior’s problems in two hours. Fior solved zero of his.
Cardano breaks his oath, sort of
Word got out. A Milanese physician and polymath named Gerolamo Cardano spent years badgering Tartaglia for the method, alternating between flattery and insult. Eventually Tartaglia gave in, but only after Cardano swore a solemn oath never to publish it.
Cardano did something better than break the oath outright. He went to Bologna, looked through del Ferro’s old notebook, and found the same solution sitting there from decades earlier. Now, technically, he wasn’t publishing Tartaglia’s work — he was publishing del Ferro’s. He wrote a 500-page book called “Ars Magna” laying out the full method, with credit (sort of) and a workaround (sort of) for the x-squared term. Tartaglia was furious and wrote angry letters to half the mathematicians in Europe. The general method is still called Cardano’s, not Tartaglia’s.
“Written in five years, may it last for five hundred.”
The impossible step
Here’s where things get strange. While writing Ars Magna, Cardano hits a cubic — x cubed equals 15x plus four — that has a perfectly reasonable answer (x equals four, you can check by plugging it in). But when he runs his method on it, the formula spits out square roots of negative numbers along the way. Tracing back through the geometry, he finds the problem: completing the square requires him to add a piece with an area of 30, but the piece has sides of length five, which would give it an area of 25. To make the math work, he has to add negative area. There is no such thing as negative area. Cardano shrugs and moves on, calling the whole idea “as subtle as it is useless.”
Ten years later, an engineer named Rafael Bombelli decides to push through the impossibility instead of around it. He treats the square root of negative one as a new kind of number — not positive, not negative, just something else — and tries the calculation again. The square roots cancel out partway through, and the formula spits out the correct answer: four. The geometric picture is nonsense the whole time. The intermediate step requires shapes that cannot exist. But the answer at the end is real, and right.
“Schrödinger put the square root of minus one into the equation, and suddenly it made sense.”
That was the moment math broke loose from geometry. Over the next century, Descartes started using these numbers freely and gave them the dismissive name “imaginary,” which stuck. Euler labeled them with the letter i. They became respectable.
The part where they describe the atom
Fast forward four hundred years. In 1925, Erwin Schrödinger is trying to write down the equation that governs quantum particles. He needs his solutions to behave like waves. It turns out there’s a function — e to the ix — that has both a sine wave and a cosine wave hidden inside it, sitting in the imaginary direction. Geometrically, multiplying by i is just rotating by 90 degrees, and e to the ix turns those rotations into a perfect spiral, with a cosine on one face and a sine on the other.
Schrödinger plugs it in. The equation works. It describes the orbits of electrons, the structure of every atom, and almost all of chemistry. And it doesn’t work without i sitting right there in the middle of it. Schrödinger himself didn’t like this — he thought a wave function should be a real, honest, drawable thing. It isn’t. Freeman Dyson later put it plainly: nature, it turns out, runs on complex numbers, not real ones. Nobody saw that coming.
The number that got invented because a Renaissance physician needed to publish a book without breaking an oath turned out to be the language the universe was already speaking.
Claude’s Take
Veritasium is on solid ground here. The history is well-documented and the dates, names, and rivalries are accurate to what historians of mathematics generally agree on. The math-duel-as-job-security framing of Renaissance academia is real, not dramatized — that’s how it actually worked. Tartaglia really did write his solution as a poem. Cardano really did pull the bookshelf trick to wriggle out of his oath. The resentment really did last the rest of Tartaglia’s life.
The geometric intuition for completing the cube is the strongest part of the video. It’s the kind of thing that gets glossed over in textbooks because by the time you’re learning the cubic, you’re already swimming in symbolic notation, and you forget that someone had to picture this as actual blocks of wood before symbols existed. Showing it as a 3D dissection is the right move, and it makes Tartaglia’s breakthrough feel earned instead of magical.
The one place to be a little careful is the framing at the end — that nature “really” runs on complex numbers and not real ones. This is a popular and defensible take, and there’s a 2021 result (Renou et al., Nature) showing that standard quantum mechanics genuinely needs complex numbers, not just real ones, to match experiment. But the philosophical leap from “the math uses i” to “the universe is fundamentally complex-valued” is a slightly bigger step than the video lets on. You can rewrite quantum mechanics in purely real terms — it just gets uglier and requires more bookkeeping. The interesting fact is that the complex version is the one that’s clean, natural, and predictive, which is a strong argument but not quite the same as a metaphysical claim. The video blurs that line a little in service of a satisfying ending. Forgivable, given how satisfying it is.
The real moral, which the video gets right, is the one about giving up. Math made progress not by trying harder to make geometry work, but by abandoning geometry as the standard for what counts as real. That’s a recurring pattern in science, and it’s worth noticing.