Did Greek Philosopher Hippasus Discover Irrational Numbers?

Hippasus of Metapontum was a Greek philosopher who followed Pythagoras. He is said to have discovered the existence of irrational numbers.

He was probably the best student who ever attended the school of Pythagoras, the Greek philosopher and mathematician famous for the Pythagorean theorem. At the same time, however, it is also believed he managed to confute the Pythagoreans, introducing a new, crucial chapter in mathematics.

Hippasus was likely born in the late 5th century BC, a century after Pythagoras. There is no certainty about his birthplace either, though there are various theories about where he may have been born.

The irrational numbers that Hippasus discovered are real numbers. These cannot be expressed as a ratio of integers, however. For instance, √2 is an irrational number. We cannot express any irrational number in the form of a ratio, such as p/q, where p and q are integers, q≠0.

Examples of irrational numbers are the ratio π of a circle’s circumference to its diameter, Euler’s number e, the golden ratio φ, and the square root of two. Indeed, the square roots of all natural numbers are irrational. Only with perfect squares would this not be the case.

The famous School of Pythagoras and “divine” numbers

The Pythagoreans were not a purely scientific group. In fact, their main concerns revolved around philosophy and religion. Mathematics, however, was of the utmost important in terms of their ideology. For Pythagoras and his students, numbers were sacred and even divine and were not mere symbols that make it easier for man to count and calculate. They were  something superior to the material world in which they found their application. In fact, they belonged to the realm of the ideal and were accessible only by the highest intellect.

According to the Pythagoreans, the entire universe was composed of numbers and geometry. However, for the ancient Greek mathematician and those who studied at the Omoakoion, the building where the Pythagoreans were taught in the Greek colony of Crotone, Italy, most of the numbers did not exist.

In mathematical thought of the time, all numbers could be expressed as fractions of two integers. For the Pythagoreans, everything in the world was equivalent to a corresponding rational number. But if this were the case, what role did the irrational numbers play, which in fact are incomparably more numerous? And what about the numbers that another Greek philosopher and mathematician, Hippasus, discovered?

Greek philosopher Hippasus confuted everything Pythagoreans believed

The mathematics department of the Pythagorean school was one of the most advanced of its time. Although historians have occasionally expressed doubts about Pythagoras being the “father” of the theorem, it has been proven that it was discovered at the time when the School of Pythagoras was at its peak. In fact, writers such as Euclid and Cicero confidently attribute the achievement to the great mathematician.

It is said that the founder of the mathematics department in Omoakoion was the Greek philosopher Hippasus, one of the most brilliant students of Pythagoras and the one who was to completely demolish all the ideas that had been expressed within the distinguished school. The tool he used was the theorem that made Pythagoras famous.

In a right triangle, the square of the two perpendicular sides is equal to the square of the hypotenuse. If the two vertical sides are equal to 1, however, then the hypotenuse has a length equal to the root of 2. This number had puzzled the Pythagoreans but did not shake their irrevocable belief that there is some equivalent number equal to the root of 2. Besides this, the Pythagoreans realized that there are too many rational numbers and, therefore, the root they were looking for might be hidden behind some very large numbers.

Hippasus attempted to prove it was equal to some number, but he managed to show that this number was not rational. Pythagorean philosophy received an incurable blow. A number that is not rational could not fit into the thinking of the Pythagoreans. Their entire universe was made of fractions, leaving no room for irrational numbers whose decimals have no end. Pythagoreans were outraged by the discovery and accused Hippasus of treason.

The unlucky end of the Greek philosopher Hippasus

The discovery of irrational numbers is said to have been shocking to the Pythagoreans, and Hippasus was believed to have been punished by the gods for divulging this. The fact that Hippasus was drowned at sea afterwards was seen as the appropriate punishment from the gods.

However, the few ancient sources which describe this story either do not mention Hippasus by name—like Greek mathematician Pappus of Alexandria, for example—or alternatively claim that Hippasus drowned because he revealed how to construct a dodecahedron within a sphere.

Later historians debated about the Greek philosopher and whether he should take credit for the discovery of irrational numbers. Pappus merely says that the knowledge of irrational numbers originated in the Pythagorean school and that the member who first divulged the secret died by drowning. Iamblichus is inconsistent. In one story, he explains how a Pythagorean was expelled for divulging the nature of the irrational numbers, but, in another, he cites the legend of the Pythagorean who drowned at sea for making the construction of the regular dodecahedron in the sphere known.

Nonetheless, some 20th century scholars credited Hippasus with the discovery of the irrationality of √2.