Zero - Charles Seife [14]
In fact, no matter how tiny you make the bits, it is impossible to choose a common yardstick that will measure both the side and the diagonal perfectly: the diagonal is incommensurable with the side. However, without a common yardstick, it is impossible to express the two lines in a ratio. For a square of size one, this means that we cannot choose counting numbers a and b such that the diagonal of the square can be expressed as a/b. In other words, the diagonal of that square is irrational—and nowadays we recognize that number as the square root of two.
This meant trouble for the Pythagorean doctrine. How could nature be governed by ratios and proportions when something as simple as a square can confound the language of ratios? This idea was hard for the Pythagoreans to believe, but it was incontrovertible—a consequence of the mathematical laws that they held so dear. One of the first mathematical proofs in history was about the incommensurability/irrationality of the square’s diagonal.
Irrationality was dangerous to Pythagoras, as it threatened the basis of his ratio-universe. To add insult to injury, the Pythagoreans soon discovered that the golden ratio, the ultimate Pythagorean symbol of beauty and rationality, was an irrational number. To keep these horrible numbers from ruining the Pythagorean doctrine, the irrationals were kept secret. Everyone in the Pythagorean brotherhood was already tight-lipped—nobody was allowed even to take written notes—and the incommensurability of the square root of two became the deepest, darkest secret of the Pythagorean order.
However, irrational numbers, unlike zero, could not easily be ignored by the Greeks. The irrationals occurred and reoccurred in all sorts of geometrical constructions. It would be hard to keep the secret of the irrational hidden from a people so obsessed with geometry and ratios. One day someone was going to let the secret out. This someone was Hippasus of Metapontum, a mathematician and member of the Pythagorean brotherhood. The secret of the irrationals would cause him great misfortune.
The legends are very hazy and contain contradictory stories about the betrayal and ultimate fate of Hippasus. Mathematicians to this day tell of the hapless man who revealed the secret of the irrational to the world. Some say that the Pythagoreans tossed Hippasus overboard, drowning him, a just punishment for ruining a beautiful theory with harsh facts. Ancient sources talk about his perishing at sea for his impiety, or alternatively, say that the brotherhood banished him and constructed a tomb for him, expelling him from the world of human beings. But whatever Hippasus’s true fate was, there is little doubt that he was reviled by his brothers. The secret he revealed shook the very foundations of the Pythagorean doctrine, but by considering the irrational an anomaly, the Pythagoreans could keep the irrationals from contaminating their view of the universe. Indeed, over time the Greeks reluctantly admitted the irrationals to the realm of numbers. The irrationals didn’t kill Pythagoras. Beans did.
Just as the legends of Hippasus’s murder are hazy, so too are the legends of Pythagoras’s end. Nevertheless, they all imply that the master died in a bizarre way. Some say that Pythagoras starved himself, but the most common versions all say that beans were his undoing. One day, according to a version of the legend, his