Every now and then mathematics has been convulsed by a row, not over where numbers come from-but over what should be allowed to count as one. Two millennia ago, inspired by such discoveries as the relationship between musical pitch and the lengths of vibrating strings {double the length of the string and the note falls by an octave), the followers of Pythagoras decided that all of Nature must be expressible as ratios of whole numbers. Their discovery that one very simple geometric ratio-that of the length of a square's diagonal to the length of its side-could not was, according to legend, so shocking that it was kept a secret on pain of death.
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