Thread: √2 as a fraction View Single Post
 2021-11-14, 09:28 #2 Batalov     "Serge" Mar 2008 Phi(4,2^7658614+1)/2 3×29×113 Posts @Matt - Here's an easy construction for square roots approximations of any arbitrary numbers. No need for matrices. Use Newton's method for solving f(x)=x2-a=0. You know f'(x). It is 2x. xnew = x - f(x)/f'(x) = x - (x^2-a)/(2x) = (x^2+a)/2x ...or (x+a/x)/2 as frequently taught in schools For $$\sqrt 2$$: use a=2 and apply this repeatedly: Code: a=2; x=1; x=(x+a/x)/2 3/2 x=(x+a/x)/2 17/12 x=(x+a/x)/2 577/408 x=(x+a/x)/2 665857/470832 x=(x+a/x)/2 886731088897/627013566048 x=(x+a/x)/2 1572584048032918633353217/1111984844349868137938112 For $$\sqrt 10$$: use a=10 and apply this repeatedly: Code: a=10; x=3; x=(x+a/x)/2 19/6 x=(x+a/x)/2 721/228 x=(x+a/x)/2 1039681/328776 x=(x+a/x)/2 2161873163521/683644320912 x=(x+a/x)/2 9347391150304592810234881/2955904621546382351702304 ... Now, try the same to get fast approximation of a cubic root of 2: xnew = x - f(x)/f'(x) = x - (x3-a)/(3x2) = (2x^3+a)/(3x^2) ...