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Q.E.D.

38 39 the nature oF numbers another proof by contradiction On the number line below, every point represents one of the real numbers that we use for measuring distances, areas, and volumes. By dividing the intervals between the integers into two parts, three parts, four parts, and so on, we single out the fractions, or rational numbers. Even the tiniest patch of the number line contains infinitely many rational numbers and it may therefore seem reasonable to expect that every real number is rational. The Pythagoreans reputedly sacrificed a hecatomb, or one hundred oxen, to celebrate the discovery of a proof that 2, the length of the diagonal of a unit square, is irrational, so not a rational number. Our proof opposite is an example of a proof by contradiction. We start by assuming that 2 is rational. This first implies the existence of an integer square a square with integer diagonals and sides and eventually a contradiction, that is, a statement that is not true. We conclude that our assumption is false. Therefore, 2 is irrational. In general, it can also be shown that if a natural number is not a square, then its square root is an irrational number. This means that infinitely many of the radii of the root spiral lower, opposite are irrational. Also, it turns out that, in a sense, there are many more irrational numbers than there are rational ones.
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