Florilegium Physicae

Hippasus of Metapontum is a little-known enigmatic historical figure who lived in Magna Graecia (modern-day southern Italy) during the 6th – 5th century BC. Not much is known of this person, other than that he was a Greek philosopher and one of the early followers of the philosophic tradition called Pythagoreanism.

Pythagoreanism was the brainchild of the Greek mathematician Pythagoras—of Pythagorean theorem fame—who lived the latter part of his life in Magna Graecia and influenced many people with his teachings, beliefs, and charisma. The followers of this tradition, appropriately called Pythagoreans, held several beliefs that would perhaps be perceived as strange, esoteric, or even cult-like in our times. These included taking vows of secrecy, a belief in reincarnation, avoiding all meat and beans, reverence for particular numbers (like ten), and the idea that all numbers can be expressed as ratios of integers. The latter belief is what we will be focusing on in the rest of this blog post.

In modern terms, we would say that the Pythagoreans rejected the existence of irrational numbers. According to legend, this belief would be shattered by Hippasus—himself a Pythagorean—who stumbled across unequivocal mathematical proof that irrational numbers do, in fact, exist.

Let us now go through one of the most popular proofs of the existence of irrational numbers. Our starting point will be a right triangle whose perpendicular sides are both equal to one (in arbitrary units). According to the Pythagorean theorem, the length of the largest side—called the hypotenuse—is equal to a number which, when multiplied with itself, is equal to 2. In modern mathematical notation, we denote this number by and refer to it as the “square root of two.”

As we will demonstrate shortly, is an example of an irrational number. In other words, there is no way of expressing it as a ratio of two integers. In order to prove this, we will employ a mathematical technique referred to as reductio ad absurdum (or proof of contradiction): we will assume the opposite of what we want to prove is true and show that this leads to absurdity or contradiction. We will thus begin by assuming that

(1)

where are integers. By squaring both sides of equation (1) and rearranging, we find

(2)

According to equation (2), is divisible by 2. By definition, this means that is an even number. But this implies that itself must be an even number, since, as can be demonstrated, (a) can only be either even or odd and, (b) the square of a number even if and only if the number itself is even. Equation (2) therefore implies that

(3)

where is some other integer. Substituting equation (3) into (2) and solving for , we get

(4)

which, in turn, implies that

(5)

where is some other non-zero integer. Substituting equations (3) and (5) into equation (1) gives

(6)

Equation (6) tells us that if can be written as a ratio of two given integers, then it can be reduced to a simplified ratio of two different integers. However, if we repeat the same process that led us from equation (1) to equation (6) by replacing and , we would deduce that equation (6) implies can be simplified even further. This process will continue ad infinitum (forever), meaning that if equation (1) is true, then can be expressed as a ratio of integers that can be simplified indefinitely. And since every rational number can always be expressed as a ratio of integers that cannot be simplified further (e.g., 2/4 = 1/2), we deduce—by means of proof of contradiction—that cannot be rational.

It should be noted at this point that the above proof is not the kind that an ancient Greek mathematician would have come up with. For one, they did not use Arabic numerals or algebra. Nevertheless, as the story goes, Hippasus’ proof of the existence of irrational numbers was convincing enough to have caused his fellow Pythagoreans so much distress that they sentenced him to death by drowning.

According to Wikipedia, the historical evidence linking the discovery of irrational numbers to Hippasus is actually unclear. However, there is a certain “catchy” element to the legend of him losing his life by going against established dogma, which resonates to this day. And so, regardless of its validity, it functions as a useful springboard to discuss not only the existence of irrational numbers but also to highlight how dangerous and threatening new ideas can be to rigid dogmatic beliefs.