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I need to know what calculation needs to be done to get "1.4142135623730950488016887242097" from √2 (square root 2)

I've seen wiki (below link), but didn't yet understand how it's get done. http://en.wikipedia.org/wiki/Square_root_of_2

anyone who know's this, please help me to understand. Thanks

kamal pal
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  • Try the method(s) described under the "Computation algorithms" section – Hagen von Eitzen May 04 '14 at 13:46
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    If you're looking for a method to extract them with pencil and paper, you might want to look at this for a description of the method with example. – Edward May 04 '14 at 13:48
  • Thanks Edward. Your posted link contains all, what I wanted to know. – kamal pal May 04 '14 at 13:54
  • There are many numerical methods, but even the simplest bisection (aka trial and error) works just fine (although it's slow). You could just increment a digit up until the square of the number becomes larger than 2, then keep it at the last one that's below the root, add another digit and proceed. That's not how it's done seriously but if you only have pen and paper and want to do something that you understand how it works, it's a nice start, at least for elementary school level. – orion May 04 '14 at 14:14

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Let us say that you want to solve for $x$ the equation $$f(x)=x^2-2=0$$ A general method for solving nonlinear equations such as $f(x)=0$ is Newton iterative scheme which, starting from a reasonable guess $x_0$, gives $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$ So, in your case, $f'(x)=2x$ and then the previous equation can be rewritten in several form starting with $$x_{n+1}=x_n-\frac{x_n^2-2}{2 x_n}=\frac{x}{2}+\frac{1}{x}$$ Suppose we start iterating with $x_0=1$, then applying the above formula gives sucessively $1.5$, $1.41667$,$1.41422$, $1.41421$.

Starting from $x_0=3$, the iterates would heve been $1.83333$, $1.46212$, $1.41500$,$1.41421$.

Is this making things clearer ?

Claude Leibovici
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