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
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 ?