My original article can also be found in Babylonian system is credited as being the first known positional numeral system, in which the value of a particular digit depends both on the digit itself and its position within the number. Finally, to know more about the Newton-Raphson formula (and other mathematicians that were also involved) you may be interested in Historical Development of the Newton-Raphson Method, by Tjalling J.It’s worth mentioning that also mathematicians of other nations seemed to be aware of this iteration. The Indians also knew this method: Methods of computing square roots - the Bakhshali method, from Wikipedia.(By the way, this is the same Heron who developed a formula for the area of a triangle, given its sides.) The Greeks were also aware, and the method is linked to Heron of Alexandria: A History of Greek Mathematics, pages 323–326, by Thomas Little Heath, Clarendon Press,1921.A modern description for the algorithm: The Babylonian method for finding square roots by hand, by Rick Wicklin, on The DO Loop.Some archaelogical considerations: Square Root Approximations in Old Babylonian Mathematics, by David Fowler and Eleanor Robson, in Historia Mathematica, Volume 25, Issue 4, November 1998.These are not new results, for sure, but the simple way we were able to use simple algebra in order to derive them, that’s the interesting part… even if the Babylonians didn’t work that way! References We have taken a look at the Babylonian method for calculating square roots, and we have been able to prove its convergence (for any initial value) and we also found an estimate for the error after any number of steps - all without resorting to derivatives or any other methods of calculus. Also, the error in each line (again, excepting the initial case) is less than half the error in the previous line at the seventh iteration, we got a precision over 12 decimals.Īs to the quadratic convergence, we can see that the number of correct decimals starts (more than) doubling: x₄ (4.69…) has two x₅ (4.6904…) has four, and x₆ (4.6904157598…) has 10. The table above verifies all our assertions: except for the initial error, all errors are positive. Let’s now prove that the average of x and p/ x (in other terms, the next value in the succession of approximations) is greater than √p.ħ iterations are enough to calculate √22 with 12 decimals of precision If you wanted to find √2 but started with a negative value (say, x=-1) then you get exactly the same approximations (-1.5, -1.41666…, etc., as we saw above) just with changed sign. We will only consider positive values for x. Of course, the initial guess does matter in a sense if it’s closer to √p, then convergence will be speedier what we mean is that we can pick any initial value, and the successive approximations all will be greater than √p. Let’s start by proving that, no matter what the initial guess, all the successive approximations will be greater than the root that we seek, so it doesn’t matter where we start. Let’s get started! The initial guess doesn’t matter The algorithm is said to have “quadratic convergence”. When the approximations get close enough to √p, the error at one step is less than the square of the error at the previous step - convergence gets really fast.This means that the error tends to zero, so the iteration will actually converge, and the algorithm will calculate the root we seek. Successive approximations will get closer to √p, and the error of a new approximation will be less than half the error of the previous one.Of course, that alone doesn’t mean anything the values could grow away from the root! But… This implies that no matter where you start, the approximations will (hopefully!) approach the root we seek from above. No matter what the initial guess, all successive approximations will be greater than √p.Each new value in the succession is closer to √2 (1.4142135624…) and it seems that the method is converging to the desired root, but is it (necessarily) so?

Subtracting babylonian numerals