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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. Suppose you’re trying to compute the square root of 2 and you have a guess, say. If your guess is exactly right thenīut if your guess isn’t right, won’t be quite equal to. So it makes sense to take the average of and, and use that as a new guess. In fact it converges very rapidly: at each step, the number of correct digits in your guess will approximately double! If your original guess wasn’t too bad, and you keep using this procedure, you’ll get a sequence of guesses that converges to. Now 1 sure isn’t equal to 2/1, but we can average them and get a better guess: We start with an obvious dumb guess, namely 1. We’re doing the calculation in painstaking detail for two reasons. Now we’ll average 17/12 and 2/(17/12):ĭo you remember what 17 times 17 is? No? That’s bad.įirst, we want to prove that we’re just as good at arithmetic as the ancient Babylonians: we don’t need a calculator for this stuff! Second, a cute pattern will show up if you pay attention. Do you remember what 12 times 24 is? Well, maybe you remember that 12 times 12 is 144. Which is what the Babylonians seem to have used!ĭo you see the cute pattern? No? Yes? Even if you do, it’s good to try another round of this game, to see if this pattern persists. Besides, it’ll be fun to beat the Babylonians at their own game and get a better approximation to.

So, let’s average 577/408 and 2/(577/408):ĭo you remember what 577 times 577 is? Heh, neither do we. In fact, right now a calculator is starting to look really good. So that’s our new approximation of, which is even better than the best known in 1700 BC! Let’s see how good it is: Just one less! And that’s the pattern we were hinting at: it’s been working like that every time.