How can you do square root?

Answer 1

There is no simple way.

There is one way which resembles long division but with a divisor which changes at each step and I regret that I cannot explain it here.

Another method is based on iteration and using the Newton-Raphson method seems the simplest. The method is as follows:

If you want to find the square root of a number n then define f(x) = x^2 – n.

Then finding the square root of n is equivalent to solving f(x) = 0.

Let f’(x) = 2x. This is the derivative of f(x) but you do not need to know that to use the N-R method.

Start with x0 as the first guess.

Then let xn+1 = xn - f(xn)/f’(xn) for n = 0, 1, 2, …

Provided you made a reasonable choice for the starting point, the iteration will very quickly converge to the true answer.

It works even if your first guess is not so good:

Suppose you want the square root of 7 and you start with x0 = 5 (a pretty poor choice since 52 is 25, which is nowhere near 7).

Even so, x3 = 2.2362512515, which is less than 0.01% from the true value. Finally, remember that the negative value is also a square root.