Pythagoras could have pushed up the invention of the real number system by many centuries, had he done a simple thing: given the diagonal a name, say, *d*, or even better, [symbol for the square-root of 2]. Had he done that, he might have pre-empted Descartes’s coordinate revolution, for, absent a numerical representation, the need to describe this new number begged for the invention of the number line. Instead, Pythagoras retreated from his promising practice of associating geometric figures with numbers, and proclaimed that some lengths cannot be expressed as a number. The Pythagoreans called such lengths *alogon*, “not a ratio”, which we today translate as “irrational”. The word *alogon* had a double meaning, though: it also meant “not to be spoken”. Pythagoras had solved his dilemma with a doctrine that would have been hard to defend, so, in keeping with his overall doctrine of secrecy, he banned his followers from revealing the embarrassing paradox. Not obeyed. According to legend, one of his followers, Hippasus, did reveal the paradox. Today people are murdered for many reasons — love, politics, money, religion — but not because somebody squealed about the square root of 2. To the Pythagoreans, though, mathematics was a religion, so when Hippasus broke the oath of silence, he was assassinated. Resistance to irrationals continued for thousands of years. In the late nineteenth century, when the gifted German mathematician Georg Cantor did groundbreaking work to put them on firmer footing, his former teacher, a crab named Leopold Kronecker who “opposed” the irrationals, violently disagreed with Cantor and sabotaged his career at every turn. Cantor, unable to tolerate this, had a breakdown and spent his last days in a mental institution.

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