And what are the methods? This was stimulated by This question could be rephrased as "use the method for finding square roots you were taught in school to find √7 to 6 decimal places". If you were taught at school. If you were not taught at school you were luckier than me, I was. It has had a negative energy for me for a long time since. I forgot by the next school year and it became something I couldn't touch. But square roots, inevitable in quadratic equations, trig, calculus, SHM and oscillations were all OK because by that time I had a slide rule, logarithms, tables (long before pocket calculators). But the arithmetical method was like a sinister room with something lurking that I've always hurried past, swept under carpet. Sure I'm not the only one? Actually the slide rule, logarithms, tables would not have been up to solving the above problem (but I would have smugly said never needed practically). The trouble was that although the method clearly worked it was never explained why. I mention all this because I am curious to know how many people were taught the same way? And how many have been taught nothing at all about it at school? In much later years I thought about it and I thought oh well you can just use for e.g. √7: if x is an approximate root let (x + a) be the exact root. Then (x + a)2 = 7 . Now (x + a)2 = x2 + 2ax + x2 Approximate this as x2 + 2ax Then the difference between x2 and 7 is approximately 2ax and you can get a (involves division ) you just continue that getting closer and closer. Actually doing it, maybe for the first time and starting with x = 2 - very crude, I would better have started with 3 - I got by third step to 2.64575 which squared is 6.99999. Sounds a bit too good actually maybe there is some fluke. It can't be the school method quite as there can be subtractIons which I don't remember. I think in any case it is not efficient and a hard slog to get to the precision required in Curieuse's question. You could program a calculator to do that in a wink and then inefficiency would not matter. - except I believe it does sometimes and there are applications like statistics and stock flow control and for all I know protein or other dynamics where they have to be calculating zillions of square roots per second. So what are the methods? I am also curious about this: I looked into calculating π recently and the most obvious method, same as Archimedes I think, involved calculating a lot of square roots. Quite time consuming. Did he do that? It also seems easier with decimals than rational fractions, or maybe it is just more natural for us.