An Imaginary Problem

Mathematicians are wonderful at a great many things, but naming things is not one of them. You might expect a profession built on precision and an almost pathological dislike of ambiguity to choose its words with care. It does not. Over the centuries, mathematicians have given us numbers called natural, rational, irrational, real and imaginary, and have managed, with real consistency, to make nearly every one of those words misleading in some way.

Things began simply enough. If you were an early farmer, you had sheep, and you could have one of them, or two, or seventeen. These are the natural numbers, the ones you count with, and for thousands of years they were all that anyone needed. The nineteenth-century German mathematician Leopold Kronecker once said that God made the whole numbers and all else is the work of man, and you can see his point. Even here, though, the profession has not managed to agree on the basics. To this day mathematicians are divided over whether zero counts as a natural number, and textbooks quietly take different sides. It is the sort of thing that would be settled in an afternoon in any other line of work.

Then someone divided a sheep. Fractions were born, and in time they were called rational numbers, not because they are sensible (though they are) but because each can be written as a ratio of two whole numbers. The followers of Pythagoras regarded these ratios with something close to religious devotion, which was natural enough, since Pythagoreanism was more or less a religion. They believed that everything (musical harmony, the movements of the planets, the very fabric of the cosmos) came down in the end to tidy ratios of whole numbers. They also, for reasons that have never been entirely clear, refused to eat beans. Both beliefs turned out to be difficult to defend, but only one of them caused a scandal.

The scandal was a square. Draw one with sides one unit long and ask how long its diagonal is. The answer is the square root of 2, which is 1.41421356… and goes on forever without ever repeating and without ever settling down into a ratio of two whole numbers, however hard you look. Somebody in the Pythagorean community worked this out, and tradition names him as Hippasus of Metapontum. What happened next depends on whose version you prefer. In one, he was expelled from the brotherhood, and a tomb was built for him as if he had died. In the more popular one, he was drowned at sea, either thrown overboard by his angry fellow believers or struck down by the gods for revealing the secret. None of these stories is very reliable, and all were written down centuries afterward. But it says something that people found it entirely believable that a mathematician might be murdered over a number.

Such numbers came to be called irrational. The Greek word was alogos, which meant "without ratio" but also, rather wonderfully, "unspeakable." The Latin translation kept only the first sense, and English has since given the word a second meaning entirely, so that the square root of 2 now sounds as if it were prone to tantrums. It is, in fact, extremely well behaved. And here is the curious thing: it turns out there are vastly more irrational numbers than rational ones. Pick a number on the number line entirely at random, and it is, in a precise mathematical sense, certain to be irrational. The Pythagoreans thought the universe was built from harmonious fractions. It is in fact built almost entirely from the thing they tried to keep quiet.

Even the humble square root has a peculiar name when you stop to think about it. Why "root"? The answer lies in ninth-century Baghdad, where the great mathematician Muhammad ibn Musa al-Khwarizmi wrote the book that gave us the word "algebra" (from al-jabr, in its title) and whose own name, mangled through Latin, gave us "algorithm." Al-Khwarizmi called the unknown side of a square its jidhr, the ordinary Arabic word for the root of a plant. When his work was translated into Latin in the twelfth century, jidhr became radix, which also gave us "radish" and "eradicate." Mathematicians grew tired of writing radix out, abbreviated it to r, and in 1525 a German named Christoph Rudolff printed a symbol, √, that most historians think began as a hastily scribbled r. Every time you press that button on a calculator, then, you are invoking a medieval vegetable.

Zero had a harder time of it. It was worked out properly in India, where by the seventh century the astronomer Brahmagupta was setting down rules for calculating with it, and it traveled west through the Islamic world. The Arabic word for it, sifr, meaning "empty," made its way into European languages twice: once, by way of the Italian zefiro, as "zero," and once as "cipher," which for a long time meant both zero and a secret code. That is a fair summary of how medieval Europeans regarded it. The Italian mathematician Fibonacci introduced Hindu-Arabic numerals, zero included, to Europe in 1202, and merchants quickly saw how useful they were. Authorities were less sure. In 1299 the city of Florence forbade its bankers from using the new numerals in their account books, apparently because they were too easy to alter: a 0 can be turned into a 6 or a 9 with a stroke of the pen, which is not something you can do with a Roman numeral. It may be the only time in history that a set of digits has been officially treated as a financial risk.

But the strangest naming story of all begins in sixteenth-century Italy, with a feud so bitter it makes the Bernoullis look like a well-adjusted family.

At the time, mathematicians made their reputations, and often their livings, by winning public problem-solving contests, in which rivals set each other lists of questions and the loser was publicly humiliated. Knowing how to solve something that nobody else could was therefore a valuable trade secret, and the most valuable secret of all was how to solve cubic equations, a problem that had defeated everyone since antiquity. Around 1515, a professor at Bologna named Scipione del Ferro worked out how to solve a large class of them, and told almost no one. Shortly before his death he passed the method on to a student, Antonio Fior, who was, unfortunately, not a very good mathematician. Fior, armed with his one great secret, challenged a Venetian teacher named Niccolò Tartaglia to a contest in 1535.

Tartaglia had had a hard life. As a boy of about twelve, he had been caught up in the sack of his home town of Brescia by French soldiers, and was slashed across the face and jaw with a sabre. He survived, but with a stammer, and "Tartaglia," which means "stammerer," was a nickname he took on as his surname. He was also, as Fior was about to discover, very, very good. Fior set him thirty problems, every one a cubic equation of the type he alone knew how to solve. A few days before the deadline, Tartaglia worked out the method for himself and solved all thirty in a couple of hours. Fior managed to solve hardly any of Tartaglia's questions, and more or less vanishes from history after that.

Enter Gerolamo Cardano, one of the most remarkable and least restful people of the Renaissance. Cardano was a physician of European fame, a compulsive gambler, an astrologer and the author of something like two hundred works on everything from medicine to the afterlife. He also wrote the first serious mathematical analysis of probability, which he based, naturally, on dice. He wrote one of the earliest true autobiographies, in which he reports among other things that he could not bear to be without some pain, so that if he was free of it he would bite his lips until they bled. He was later jailed by the Inquisition, apparently in part for having cast the horoscope of Jesus. According to a story that is too good to check and too dubious to repeat with confidence, he had predicted the date of his own death and, when it came, made sure of it by starving himself.

Cardano wanted Tartaglia's formula, and for years Tartaglia would not give it up. Eventually, in 1539, Cardano lured him to Milan, flattered him, and extracted the method in exchange for a solemn oath that he would never publish it. Tartaglia handed it over in the form of a poem, which tells you something about Italian mathematics in the 1530s. Then, some years later, Cardano and his brilliant student Lodovico Ferrari went to Bologna and saw del Ferro's original notes, and they decided that since Tartaglia had not been first, the oath no longer applied. In 1545 Cardano published the solution in his great book, the Ars Magna. He gave full credit to both del Ferro and Tartaglia, which might have been expected to count for something. It did not. Tartaglia was beside himself, and there followed years of furious pamphlets, accusations and counter-accusations, much of it conducted by the hot-tempered Ferrari, who eventually beat Tartaglia in a public debate in Milan in 1548. Tartaglia, it is generally agreed, left town after the first day.

All of this matters to our story because in the course of working with these equations, Cardano kept running into something he couldn't explain: square roots of negative numbers. Every number multiplied by itself gives a positive result, so a square root of −15, say, should not exist at all. Yet in the Ars Magna, Cardano asked how to split 10 into two parts whose product is 40, and found that the answer, if you were prepared to swallow the absurdity, was 5 + √−15 and 5 − √−15. Multiply them together and you do indeed get 40. Cardano called the whole business "as subtle as it is useless" and moved briskly on. Worse, his own formula for cubics sometimes demanded these square roots of negatives as a stepping stone, even for equations whose answers were perfectly ordinary numbers. To get from a real question to a real answer, you had to pass through something that apparently did not exist.

The first person to look at this calmly was Rafael Bombelli, an engineer from Bologna who spent part of his career draining marshes and, during a pause in the work, wrote a book on algebra. Published in 1572, it set out plain rules for calculating with these odd quantities and showed that, if you simply followed them, Cardano's formula delivered the right answer every time. For the equation x³ = 15x + 4, for instance, the formula plunged into the square root of −121 and climbed back out with the answer 4, which is correct. Bombelli admitted that the whole thing seemed "more sophistic than true," but he could see that it worked, and that was enough for him. It was not enough for anyone else for a very long time.

The name came in 1637, from René Descartes. In a book on geometry, Descartes divided the solutions of equations into those that were "real" and those he called imaginary, meaning, roughly, that you could imagine they were there but nothing actually corresponded to them. Whether he meant it as an insult is debated, but it was certainly not intended as a compliment, and it has dogged these numbers ever since. The same stroke of the pen also gave us "real numbers," which were named only in contrast to the imaginary ones, rather as nobody had needed the word "acoustic" before electric guitars came along.

It took Leonhard Euler, the most productive mathematician who ever lived, to take imaginary numbers seriously. Late in his life he started writing the square root of −1 simply as i, and the habit caught on. Much earlier, in 1748, he had published a formula linking imaginary numbers to the exponential function and to trigonometry, and from it falls out, almost as a by-product, the following:

This is Euler's identity, and in a single short line it ties together the five most important numbers in mathematics: 0, 1, π, e and i. Whenever mathematicians are asked to name the most beautiful equation ever written, they choose this one, and they do so with such regularity that the poll has become slightly pointless. It should be said that Euler himself never seems to have written it down in quite this form, which is the sort of detail that only a mathematician would care about, and which mathematicians care about very much. But it is his, and it is built around a number that had been named, in effect, as a put-down.

The person who finally made imaginary numbers respectable was Carl Friedrich Gauss, who had little patience with the old name. The key idea was to stop thinking of these numbers as sitting on the ordinary number line and to give them a dimension of their own. Ordinary numbers run left and right; imaginary ones run up and down; and any combination of the two can be plotted as a point on a flat plane. Gauss popularised this picture and gave such combinations the name complex numbers, which is at least not insulting, even if it does make them sound harder than they are. He also complained in print that if +1, −1 and √−1 had been called "direct," "inverse" and "lateral" units, rather than "positive," "negative" and "imaginary," nobody would ever have found them mysterious in the first place. He was entirely right and was entirely ignored.

As it happens, Gauss was not the first to think of the plane. A Norwegian-Danish surveyor named Caspar Wessel had described it in 1797, in a paper written in Danish that nobody outside Denmark read for a hundred years. A Swiss bookkeeper living in Paris named Jean-Robert Argand came up with the same idea independently in 1806 and published it at his own expense, without bothering to put his name on it. For a long time afterward the plane was often called the "Argand diagram," which is at least an improvement on naming it after someone who called it imaginary.

In time, complex numbers turned out to be not merely respectable but indispensable. Electrical engineers use them to describe alternating current, which means that they are quietly involved in every light switch you have ever flicked. They are woven into the mathematics of waves, of fluid flow and of signal processing. The fundamental equation of quantum mechanics has an i sitting in it, right at the front, as if the universe itself were insisting on the point. Your phone does complex arithmetic an enormous number of times every second, just to let you make a call, and neither of you gives it a moment's thought.

So here, after two and a half thousand years of effort, is where the naming stands. The "natural" numbers are the only ones you could plausibly find in nature, though mathematicians still can't agree on whether they include zero. The "irrational" numbers are perfectly well behaved. The "real" numbers include such things as an endless, never-repeating decimal, which no one has ever seen or measured in the physical world. And the "imaginary" numbers are woven so deeply into the workings of electronics and physics that it is fair to say the modern world could not run without them.

It is a scheme that would embarrass a zoologist. But it does suggest a pattern. Again and again, a new kind of number has turned up, somebody important has declared it absurd, or useless, or dangerous, and then a generation or two later somebody else has quietly used it to build something essential. Hippasus may have been drowned for irrational numbers. Florence banned zero and its new companions from its ledgers. Cardano called imaginary numbers useless, and Descartes gave them a name that still makes schoolchildren suspicious. If history is any guide, then, whatever today's mathematicians are dismissing as nonsense will probably turn up one day in your phone, still wearing whatever unfortunate name it was given at birth.