How can you approximate a square root?

Answer 1

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

Suppose you want to find the square root of 7.

Let f(x) = x2 - 7, and f'(x) = 2x

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

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

Then calculate

xn+1 = xn - f(xn)/f'(xn)

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