how do you find the square root of a number without using a calculator?

Question

zarkon · Answer

one could use newtons method

hero · Answer

There's a formula for it around somewhere. it's not newton's method but it also works.

anonymous · Answer

and what is it?

zarkon · Answer

one can use the iterative procedure

$$x_{n+1}=\frac{x_{n}^2+a}{2x_{n}}$$

this gives the solution to $$\sqrt{a}$$
$$x_0$$ is the initial guess for the value of the square root

anonymous · Answer

can you use it in an example pliz

hero · Answer

I could have posted a formula. It's in one of my old programming books that I recently sold :/

anonymous · Answer

can you pliz

zarkon · Answer

look at http://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method

zarkon · Answer

it is the same formula

$$x_{n+1}=\frac{x_{n}^2+a}{2x_{n}}=\frac{1}{2}\left(\frac{x_n^2}{x_n}+\frac{a}{x_n}\right)=\frac{1}{2}\left(x_n+\frac{a}{x_n}\right)$$

zarkon · Answer

this formula can be derived using newtons method.(that's how I did it)