Irrational Numbers

Irrational numbers are numbers that are not rational; that is, they cannot be represented in the form ^a/_b where a and b are integers. Around 500 B.C., Hippasus of Metapontum showed that 2 is irrational. Pythagoras likely discovered this fact earlier, but kept it secret. Hippasus, then, revealed this secret knowledge of the Pythagoreans to the Greek world.

There are many stories about what happened to Hippasus after that. One story is that, in an act of divine retribution, he died in a shipwreck. Another is that the Pythagoreans, or even Pythagoras himself (although because of the date, this would have been unlikely) killed him. It was also written that the Pythagoreans sacrificed a hecatomb of oxen because of this discovery. At any rate, these people were very enthused about this discovery. Due to political pressure, the sect was disbanded shortly after, but the Pythagoreans then went to Tarentum and elsewhere in Magna Graecia (southern Italy).

The discovery of irrational numbers caused a break in Greek mathematics between arithmetic and geometry. The Greeks could not accept the fact that some lengths were incommensurable with rational numbers. Therefore, they decided that numbers could not be associated with lengths. Unfortunately, this decision led to a division between arithmetic and geometry that was not reconciled until the time of Descartes.

Getting back to irrational numbers themselves, we find that, since all repeating decimals can be expressed as fractions, that the decimal expansion of an irrational number never repeats. For example, the first 100 decimals of the square root of 2 never start to repeat. This property holds for the entire decimal expansion of the square root of 2, and for any other irrational number. This means that we can look at the decimals of an irrational number as random, by some meaning of that word.