Math & Money

Making sense of irrational numbers

Like many heroes of Greek myths, the philosopher Hippasus was rumored to have been mortally punished by the gods. But what was his crime? Did he murder guests, or disrupt a sacred ritual? No, Hippasus’s transgression was a mathematical proof: the discovery of irrational numbers. Hippasus belonged to a group called the Pythagorean mathematicians who had a religious reverence for numbers. Their dictum of, “All is number,” suggested that numbers were the building blocks of the Universe and part of this belief was that everything from cosmology and metaphysics to music and morals followed eternal rules describable as ratios of numbers.

Thus, any number could be written as such a ratio. 5 as 5/1, 0.5 as 1/2 and so on. Even an infinitely extending decimal like this could be expressed exactly as 34/45. All of these are what we now call rational numbers. But Hippasus found one number that violated this harmonious rule, one that was not supposed to exist. The problem began with a simple shape, a square with each side measuring one unit. According to Pythagoras Theorem, the diagonal length would be square root of two, but try as he might, Hippasus could not express this as a ratio of two integers.

And instead of giving up, he decided to prove it couldn’t be done. Hippasus began by assuming that the Pythagorean worldview was true, that root 2 could be expressed as a ratio of two integers. He labeled these hypothetical integers p and q. Assuming the ratio was reduced to its simplest form, p and q could not have any common factors. To prove that root 2 was not rational, Hippasus just had to prove that p/q cannot exist. So he multiplied both sides of the equation by q and squared both sides. which gave him this equation. Multiplying any number by 2 results in an even number, so p^2 had to be even.

That couldn’t be true if p was odd because an odd number times itself is always odd, so p was even as well. Thus, p could be expressed as 2a, where a is an integer. Substituting this into the equation and simplifying gave q^2 = 2a^2 Once again, two times any number produces an even number, so q^2 must have been even, and q must have been even as well, making both p and q even. But if that was true, then they had a common factor of two, which contradicted the initial statement, and that’s how Hippasus concluded that no such ratio exists.

That’s called a proof by contradiction, and according to the legend, the gods did not appreciate being contradicted. Interestingly, even though we can’t express irrational numbers as ratios of integers, it is possible to precisely plot some of them on the number line. Take root 2. All we need to do is form a right triangle with two sides each measuring one unit. The hypotenuse has a length of root 2, which can be extended along the line. We can then form another right triangle with a base of that length and a one unit height, and its hypotenuse would equal root three, which can be extended along the line, as well.

The key here is that decimals and ratios are only ways to express numbers. Root 2 simply is the hypotenuse of a right triangle with sides of a length one. Similarly, the famous irrational number pi is always equal to exactly what it represents, the ratio of a circle’s circumference to its diameter. Approximations like 22/7, or 355/113 will never precisely equal pi. We’ll never know what really happened to Hippasus, but what we do know is that his discovery revolutionized mathematics. So whatever the myths may say, don’t be afraid to explore the impossible.

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