Free The King of Infinite Space by David Berlinski Page A

theorem, the square root of 2. The square root of 2 is neither a natural number nor the ratio of natural numbers. A proof simpler than Euclid’s own proceeds by contradiction. Suppose that the square root of 2 could be represented as the ratio of two integers so that √2 = a / b . Squaring both sides of this little equation: 2 = a 2 / b 2 . Cross-cutting: a 2 = 2 b 2 .
    Now the fundamental theorem of arithmetic affirms that every positive integer can be represented as a unique product of positive prime numbers. A prime number is a number divisible only by itself and the number 1. Knownto the Greeks, this theorem was known to Euclid. It was widely known; it had gotten around.
    The little equation a 2 = 2 b 2 is shortly to undergo a bad accident. Whatever the number of prime factors in a , there must be an even number of them in a 2 . There are twice as many. Ditto for b 2 . But the number 2 b 2 has an odd number of prime factors. The number 2 is, after all, prime. Either the square root of 2 is not a number, or some numbers cannot be expressed as natural numbers or as the ratio of natural numbers.
    This is the bad accident.
    The consequences are obvious. If the Euclidean line does not contain a point corresponding to the square root of 2, how can the Cantor-Dedekind axiom be true, and if it does, how can the line be Euclidean?
    T HE NATURAL NUMBERS 1, 2, 3, . . . constitute the smallest set of numbers whose existence cannot be denied without commonly being thought insane. Whatever their nature, nineteenth-century mathematicians discovered how numbers beyond them might be defined and so made useful. A single analytical tool is at work. New numbers arise as they are needed to solve equations that cannot be solved using numbers that are old. Zero is the number that results when any positive number is taken from itself and so appears as the solution to equations of the form x − x = z . The negativenumbers provide solutions to equations of the form x–y = z , where y is greater than x . The common fractions, numerator riding shotgun on top, denominator ridden below, are solutions in style to any equation of the form x ÷ y = z .
    There remained equations such as x 2 = 2. The equation is there in plain sight. What, then, is x ? The answer proved difficult to contrive. The Greeks endeavored to find a sense suitable to the square root of 2, but they did not entirely succeed, and beyond the imperative of solving this equation, mathematicians had no common currency with which they could easily pay for its solution. They had nothing in experience.
    In the late nineteenth century, Richard Dedekind defined the irrational numbers—how did numbers that are not rational come to be ir rational?—in terms of cuts , a partitioning of the integers into two classes, A and B. Every number in A, Dedekind affirmed, is less than any number in B, and what is more, there is no greatest number in A. The cuts themselves he counted as new numbers, the enigmatic square root of 2 corresponding to the cuts A and B in which all numbers less than the square root of 2 are in A, and all those greater than the square root of 2 are in B. Dedekind’s cuts are not the sort of animals one is apt to find in an ordinary zoo. Dedekind’s cuts are, it must be admitted, transgendered, their identity as numbers at odds with their appearance as classes. Dedekind demonstrated, nevertheless, that they were what they did not seem to be,and that is number-like in nature. They could be added and multiplied together; they could be divided and subtracted from one another. They took a lot of abuse. They were fine. They were, in any event, more appealing than the supposition that where there really should be a number answering to the square root of 2, there was no number at all.
    The formal introduction of the real numbers in the nineteenth century brought to a close an arithmetical saga, one in which numbers that had once inspired unease acquired