If you don’t remember much else about high school geometry, you probably remember the Pythagorean theorem: the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. Or, to put it in a formula:

(Credit: pedica018/Shutterstock)

The formula is handy in construction, surveying and navigation, among other down-to-earth activities such as deciding the size of the monitor or television to buy. The theorem also led to mathematical breakthroughs, such as calculus.

But who was Pythagoras, the person behind the famous and useful theorem?

## Mathematics and Mystics

Beyond the fact that he was born on the Greek island of Samos around 569 BC. AD and died around 475 BC. J.-C., we do not know much about him. Pythagoras left no writings, but he founded a sect (or, what some would consider it a cult): the Divine Brotherhood of Pythagoras. Its followers are often referred to simply as the Pythagoreans, a secret society dedicated to a combination of scientific and mystical precepts.

They were an eccentric bunch, and the legends surrounding Pythagoras and his followers are delightful. It is said that Pythagoras refused to eat beans and did not allow his followers to do so, as he thought beans had a soul. The Pythagoreans were also said to have been very suspicious of sex, and according to Jordan Ellenberg in his book How Not to Be Wrong: The Power of Mathematical Thinkingthey may have believed that the Earth had a twin planet on the other side of the sun.

Some of Pythagoras’ followers, it is said, believed he was a god. Yet it is also said that they threw him overboard and left him to die because the Pythagorean theorem was also, as one might say today, disruptive.

Although this last myth is as unlikely as the others – and the poor turf thrown overboard is generally identified as Hippase of Metapontum, another Pythagorean, rather than Pythagoras himself – this is certainly appropriate, given the impact of the theorem. It was dangerous, at least for the Pythagorean worldview. By elaborating these specific properties of a triangle, Pythagoras had come across the existence of irrational numbers. And that was a big deal.

## “All is number”

The basic tenet of Pythagoreanism was that numbers are the essence of everything, as their motto states: “All is number”. They had good reason to see things that way. They were the ones who discovered that musical intervals corresponded to the length of the strings of a stringed instrument, and they deduced the happy mediumor golden ratio, by examining patterns in nature, such as nautilus shells and flower petals.

As David Foster Wallace explains in Everything and more: a compact history of infinity, the “attempts of the Pythagoreans to articulate the connections between mathematical reality and the physical world were part of the larger project of pre-Socratic philosophy, which was essentially to give a rational, non-mythopoetic account of what was real and from where it was coming.” They wanted to understand how things worked – missing gods with lightning bolts – and everywhere they looked they found numbers.

Before the discovery of the Pythagorean theorem, the Greeks believed that all numbers could be expressed as a whole number or as a fraction – a ratio of two whole numbers. But the Pythagorean theorem blew a big hole in that notion. Hiding in plain sight in the theorem was something very disturbing. If, for example, you are looking at a right triangle with two sides of 1 inch (or foot, or something like that), the hypotenuse is a number whose square is 2.

So what is the square root of 2? Something the Pythagoreans couldn’t handle: an irrational number; that is, a number that cannot be written as a fraction (or a ratio). For us, that just means the math is a little harder (OK, maybe a lot harder). For the Pythagoreans, it was a challenge to their whole worldview, which relied on the supremacy, even divinity, of numbers that didn’t do such weird things.

Of course, we now have all kinds of weird numbers: imaginary numbers, transcendent numbers, the really disturbing zero, plus quantum mechanics. And we manage more or less finely, even managing to fly planes and invent computers.

## Old version of the theorem

The writings of other members of the society were often attributed to Pythagoras himself, rather than the writer himself, probably out of respect (or perhaps as a form of self-defense). So it’s really not fair to give all the weirdness or all the credit to Pythagoras.

Much of what we attribute to Pythagoras may well be the work of another or more Pythagoreans. But real or composite, Pythagoras was not the first to understand the famous relationship. clay tablets found in what is now central Iraq show that Babylonian mathematicians knew the basics of the famous theorem at least 1,000 years before it freaked out the Pythagoreans. We don’t know how the Babylonians reacted, but we can hope they resisted pushing anyone out of a boat.