Every once in a while, my mind gets snagged by numbers, nowhere as often as when I was seven or eight and used to work out sums in my head (that was before I discovered masturbation).<\/p>\n
A Financial Times<\/em> recent piece by Tim Harford entitled “We must face facts — even the ones we don’t like<\/em>” provides a good example.<\/p>\n The universe is not constructed in terms of whole numbers. A\u00a0hypotenuse\u00a0across a square \u00a0is prima facie evidence, a dilemma for true-believing Pythagoreans, who thought whole numbers were the basis of everything. <\/span>Hence the fun in using the Pythagorean theorem to show they are not. <\/span><\/p>\n The square of the hypotenuse is the sum of the square of the other two sides. Formulaically: \u00a0c2<\/sup> = a2<\/sup> + b<\/strong>2<\/strong>.\u00a0<\/sup>\u00a0<\/span><\/p>\n Assume a whole number fraction, a\/b<\/strong>, does equal\u00a0\u221a2<\/b>, that is\u00a0<\/span>a\/b \u00a0= \u00a0\u221a2<\/strong><\/b>\u00a0. <\/span>Let’s also assume that a\/b<\/strong> is the simplest possible fraction, with a<\/strong> and b<\/strong> sharing no common factors.<\/p>\n Rearranging a\/b \u00a0= \u00a0\u221a2<\/strong> \u00a0gives us 2b2 = a2<\/strong>.<\/p>\n That means that a2<\/strong> is an even number, which implies four thngs: a<\/strong> is also even. and therefore a2\/2<\/strong> is also even, therefore b2<\/strong> is even and therefore b<\/strong> is even.<\/p>\n Alas, we began by assuming that a\/b<\/strong> was the simplest possible whole number fraction, but we’ve just proved that a\/b<\/strong> is the ratio of two even numbers and it follows that this fraction could be simplified by dividing both of them by 2<\/strong>.<\/p>\n This contradion shows that our original assumption — that a<\/strong> and b<\/strong> exist at all — must be wrong.<\/p>\n From Tim Harford, Undercover Economist, Financial Times, 10\/11 July, 2021, p 18.<\/em><\/p>\n","protected":false},"excerpt":{"rendered":" Every once in a while, my mind gets snagged by numbers, nowhere as often as when I was seven or eight and used to work out sums in my head (that was before I discovered masturbation). A Financial Times recent piece by Tim Harford entitled “We must face facts — even the ones we don’t … <\/p>\n