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And by “numbers,” they did not mean what we mean: they did not include what we now call negative numbers, imaginary numbers, irrational numbers, infinitesimals, and more. Instead, it seems that they included only what we call the natural integers and ratios of them. This leads directly to a major legend about the Pythagoreans, namely that they were so fanatical in believing that everything is numerical that when Hippasus, a member of the cult, discovered that 18 A N I R R AT I O N A L M U R D E R AT S E A 3 3 5 4 ?

43 Aristoxenus was a student of some Pythagoreans and of Aristotle, so his words seem to carry authority. 44 Next, about two centuries after Pythagoras died, a few writers, nonmathematicians, apparently claimed very briefly that he studied mathematics, among other things. ”45 It’s a doubtful claim, considering that the ancient Egyptians did not believe in transmigration. Still, some of Diodorus’s sources for his Bibliotheca Historica were copies of the writings of Hecataeus of Abdera (ca. 360–290 BCE), and hence some historians arbitrarily have ascribed the brief line in question to that early Table 2.

But it can’t be! That’s a contradiction, because we found above that it had to be odd. It is impossible that b is both even and odd. But since we 26 A N I R R AT I O N A L M U R D E R AT S E A reached that conclusion by assuming that √2=a/b, then we conclude that this assumption is false. We conclude that √2 cannot be a ratio of integers. So this is the kind of argument that Aristotle and others gave to prove that the diagonal of a square is incommensurable with its side. So what? Why does it matter that there doesn’t exist one unit of length that divides both the side and the diagonal of a square?