A Modified Babylonian Algorithm

Abstract

One may infer (Mainzer, p. 44) from nature of their approximations that Babylonians had discovered following algorithm for a sequence x1, x2,... of successively improving approximations to v/;. The sequence is defined inductively as x1 = x and Xk = (xk1 + x/xkl1)/2 for k ? 2. In case of C2, their best approximation was fourth term of this sequence rounded to two sexagesimal places. It was accurate to within 10-6. The choice of x1 is flexible, but for purposes of our exposition x1 = x is best. For Babylonians it was important that Xk was rational if x was, and so could easily be approximated in decimal notation. Modern calculus texts efficiently treat this algorithm as a special case of Newton-Raphson method, but one could object to this treatment on grounds that it obscures fact that convergence of Xk to Cx is an easier consequence of algebra than of calculus, and it does not justify fact that convergence is quadratic (one usually states this to students as the number of significant digits is better than doubled by proceeding from Xk to Xk+1,' but seldom is it proven to freshmen). Finally it does not address issue of non-uniformity of convergence of sequence x1, x2, ... to Cx/. That fact justifies normal strategy of conditioning x by multiplying or dividing it by a suitably chosen perfect integer square to obtain a number larger than 1 and less than or equal to (say) 10, and then applying algorithm to result. We propose following modification of Babylonian algorithm for finding square roots as a good means of illucidating above points.