1 A Classical Beginning

One of the classical gems of mathematics—and to my way of thinking, a pinnacle of hu-

man achievement—is the ancient discovery of incommensurable numbers, quantities that

cannot be expressed as the ratio of integers.

√

2

The Pythagoreans discovered in the fifth century BC that the side and diagonal of a

square have no common unit of measure; there is no smaller unit length of which they are

both integral multiples; the quantities are incommensurable. If you divide the side of a

square into ten units, then the diagonal will be a little more than fourteen of them. If you

divide the side into one hundred units, then the diagonal will be a little more than 141; if

one thousand, then a little more than 1414. It will never come out exactly. One sees those

approximation numbers as the initial digits of the decimal expansion:

√

2 = 1.41421356237309504880168872420969807856 ...

The discovery shocked the Pythagoreans. It was downright heretical, in light of their

quasi-religious number-mysticism beliefs, which aimed to comprehend all through pro-

portion and ratio, taking numbers as a foundational substance. According to legend, the

man who made the discovery was drowned at sea, perhaps punished by the gods for impi-

ously divulging the irrational.