How you get the square root of a number that is not a perfect square?

Answer 1

If you have a non-scientific calculator you can use the Newton-Raphson method.

Suppose you wish to find the square root of 7.

Let f(x) = x2 - 7 so that f(x) = 0 when x is the square root. That is, you want to find x such that f(x) = 0.

Let f'(x) = 2x

[f'(x) is the derivative of f(x) but you do not need to know that to use the N-R method.]

Make a guess at the square root of 7, and call is x0.

Then calculate

xn+1 = xn - f(xn)/f'(xn) for n = 1, 2, 3, ...

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

It does so even if your first guess is not so good. Suppose 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.