What is the process for estimating the square root of a number that is not a perfect square?

Answer 1

The Newton-Raphson method is a pretty efficient process.

See link for details.

If you want the square root of 34, say,

define f(x) = x2 - 34

so that when x is the square root of 34, f(x) is zero.

That would imply f'(x) = 2x

So start with x0

and use the iteration xn+1 = xn - f(xn)/f'(xn) = xn - [xn2 - 34]/[2*xn]