Pythagoras was a sixth-century BCE Greek philosopher often credited with several discoveries in mathematics. If you remember high-school geometry or algebra, you may remember the theorem that bears his name. It states that the square of the length of the side opposite the right angle in a right triangle is equal to the sum of the squares of the remaining two sides. It is commonly written as \(a^2 + b^2 = c^2\), and I have written elsewhere about Pythagorean triples and how to find whole number solutions to his equation.

Pythagoras was also credited with a philosophical theory that everything in the world originated as Number, by which he meant whole numbers. He reputedly taught that all numbers could be expressed as a ratio between whole numbers. One of his followers, Hippasus of Metaponum, may have proved that the square root of two could not be expressed as a ratio between whole numbers. According to some, he was executed by other followers of Pythagoras, thrown into the sea during a voyage by ship, one in a long line of heretics killed by the orthodox for daring to espouse dissent from orthodoxy. Unlike Pythagoras, no widely taught theorem bears his name, so he is rarely mentioned in high-school mathematics classes. Mathematics, being a study of pure ideas, prefers to forget its own history and focus on the accumulation of its successes.

Hippasus is often credited with discovering the irrational numbers, so called not because they are crazy or irresponsible but because they cannot be expressed as a ratio between integers1. Irrational numbers are commonly represented either by special notation (e. g., \(\pi\) and \(e\)) or as implied calculations (e. g., \(\sqrt{2}\)). When expressed as decimals, they are commonly written with a trailing ellipsis to indicate that the decimal expansion continues indefinitely without terminating or repeating, (e. g., \(1.414213562...\)).

All the numbers we use to measure and manipulate the world are rational numbers. They can all be expressed as a ratio between integers. This is without exception. Someone may argue that we use plenty of irrational numbers, and indeed we do—to describe the world, not to measure or manipulate it. Humanity has known since ancient times that the square root of two is irrational. It can be proven using tools from high school algebra. Yet no matter how precisely we measure the diagonal of a unit square, the result is always a rational number. There is, in fact, no way to measure an irrational quantity. All measurements, with the possible exception of counting individual items2, are approximations. Only ideal squares, which exist only as ideas and not as real objects, have irrational diagonals. Real squares are not squares at all. Their sides vary imperceptibly, and none of their angles are exactly 90°. Their sides are not really straight line segments. With careful engineering we can create very precise squares, but we cannot make them more precise than can be expressed by a dozen or so digits. With relative ease we can calculate \(\sqrt{2}\) to hundreds or thousands of decimal places. Yet even the most demanding applications do not need more than 20 digits of precision. With 20 digits of precision, we could measure the distance from earth to the sun to the nearest hundredth of a micron. We could measure the diameter of the Milky Way and calculate its circumference to the nearest kilometer. With 40 digits of precision we could in principle measure and calculate the size of the universe with subatomic precision.

While it might not be wholly accurate to say that Pythagoras was right, he was certainly on to something when he—or his followers—claimed that all numbers can be expressed as a ratio of integers. In fact, we actually use rational numbers, and only rational numbers, to measure and manipulate real objects in the world. Irrational numbers cannot be represented with absolute accuracy and precision. The \(\pi\) in your calculator likely has no more than 16 digits of precision. It is a rational approximation of \(\pi\). The same is true of every other irrational number used in our calculations.

1. Integers are the set of positive and negative whole numbers along with \(0\), which includes the numbers we use to count individual items. Rational numbers are any numbers that can be expressed as a ratio between integers, e. g., \({1 \over 2}, {2 \over 3}, 0.34 = {17 \over 50}\). All terminating or repeating decimal numbers are rational numbers.

2. It's fairly easy to come up with scenarios where even counting individual items requires approximation. One need only think of the "hanging chads" and "dimpled chads" in Florida's 2000 election, where Bush ended up winning the state with just 537 votes out of six million cast. That's a margin of less than 0.01%. Democrats were dismayed and cried foul, claiming that ballots were rejected or miscounted even when the voter's intent was clear. They demanded a recount in counties thought to favor their party. Republicans sued because a recount had already occurred. The Supreme Court sided with Republicans. The law demanded only one recount regardless of its accuracy. Time to move on.