In the 16th and 17th centuries, in the age of calculus with its penchant for infinity, mathematicians came up with a number of expressions for \(\pi\) using infinite sums and products. Modern approaches to estimating \(\pi\) use methods that are surprisingly un-geometrical. But Zu Chongzhi’s effort also illustrates a limitation of this approach: you might have to make an enormous number of calculations to get a result that’s only accurate to a few decimal places. Although we can never pin \(\pi\) down exactly, it’s possible, at least in theory, to calculate it to any degree of accuracy by increasing the number of sides of the polygon you inscribe in a circle. Archimedes was out-performed, at least in calculational stamina, by the Chinese mathematician and astronomer Zu Chongzhi who in the fifth century AD worked out that \ Zu Chongzhi’s original writings are lost, but if he used Archimedes’ method then he must have inscribed a regular polygon with \(24,576\) sides in a circle and made a huge number of calculations to admirable accuracy.Īrchimedes’ method for approximating \(\pi\) is reassuring. It’s an assertion that makes most mathematicians scream: writing \(\pi\) as a bog standard fraction is denying its irrationality and thereby much of its mystery. The larger of these two numbers, \(22/7\), might be familiar-even today some people think that \(\pi\) is exactly equal to it. The areas and perimeters of these two shapes sandwich those of the circle, and Archimedes arrived at the estimate \ Some smaller polygons sandwiching the circle Then he drew a polygon with the same number of sides around the circle, so that the centres of its sides just touched the circle. He fitted a regular polygon with \(96\) sides inside a circle, so that its vertices were sitting on the circle. The first person to approach the problem of estimating \(\pi\) systematically seems to have been Archimedes working in the third century BC. It measured \(30\) cubits in circumference and \(10\) in diameter, giving the crude approximation of \(30/10 =3\). The number \(\pi\) even comes up in the Bible, implicitly, in a passage of 1 Kings 7 which sees Huram of Tyre build a circular “sea of cast metal” for King Solomon. The Egyptians produced a closer estimate of \(3.16\). The Babylonians, whose culture flourished around 3000 years ago, seem to have mostly made do with a value of \(3\), though one clay tablet gives the slightly better value of \(3.125\) (written in modern notation). Since it’s irrational, we can’t write it as a simple fraction, its decimal expansion is infinite and doesn’t end in a recurring block of numbers, so any attempt to write it down will be nothing more than an approximation. What they didn’t know, and what we still don’t know, is the exact value of \(\pi\). People seem to have have known about \(\pi\) for a very long time-after all, it’s quite natural to want to compute the area or perimeter of a circle.
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