So it makes sense to take the average of and, and use that as a new guess. If your guess is exactly right thenīut if your guess isn’t right, won’t be quite equal to. Suppose you’re trying to compute the square root of 2 and you have a guess, say. So, Indian mathematicians may have known the same algorithm.īut what is this algorithm, exactly? Joseph describes it, but Sridhar Ramesh told us about an easier way to think about it. The number 577/408 also shows up as an approximation to in the Shulba Sutras, a collection of Indian texts compiled between 800 and 200 BC. In his book The Crest of the Peacock, George Gheverghese Joseph points out that a number very much like this shows up at the fourth stage of a fairly obvious recursive algorithm for approximating square roots! The first three approximations areīut if you work it out to 3 places in base 60, as the Babylonians seem to have done, you’ll get the number on this tablet! … even if it were only due to our incomplete knowledge of the sources that we assume that the Babylonians did not know that had no solution in integer numbers and, even then the fact remains that the consequences of this result were never realized.īut there is evidence that the Babylonians knew their figure was just an approximation. One of the great experts on Babylonian mathematics, Otto Neugebauer, wrote: There seems to be no evidence that they knew about irrational numbers. This is an impressively good approximation toīut how did they get this approximation? Did they know it was just an approximation? And did they know is irrational? But let’s get back to our original question: what did they know about ? So, we know from tables of reciprocals that Babylonians wrote 1/2 as 30. This is cool, because modern algebra also sees reciprocals as logically preceding division, even if most non-mathematicians disagree! To calculate they would break up into factors, look up the reciprocal of each, and take the product of these together with.
They even checked their answers the obvious way: by taking the reciprocal of the reciprocal! They put together tables of reciprocals and used these to tackle more general division problems. According to Jöran Friberg’s book A Remarkable Collection of Babylonian Mathematical Texts, there are tablets where a teacher has set some unfortunate student the task of inverting some truly gigantic numbers such as 3 25 How do we know the Babylonians wrote 1/2 as 30? One reason is that they really liked reciprocals. Once you start worrying about these things, there’s no end to it. And for a beginner, or indeed any mathematician, it makes a lot of sense to take 1/2 and multiply it by to get. But this tablet was probably written by a beginner, since the writing is large. So maybe the square’s side has length 1/2… but maybe it has length 30.
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