How do you get a square root of an irrational number?

Answer 1

One way to find the square root of a number is an iterative method. This entails making a guess at the answer and then improving on it. Repeating the procedure should lead to a better estimate at each stage. One such is the Newton-Raphson method.

If you want to find the square root of q, define f(x) = x2 - q.

Then finding the square root of kkk 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. Even if your first guess is not so good:

Depending on your arithmetical skills, this method may require a calculator and, if you do have a calculator available, you may be able to use it to find the square root!

Another way is bracketing.

If you want the square root of k, find two rational numbers a and b such that

a2 < k2 < b2. Then find rational numbers between a and k and between k and b so that a similar relationship holds. You will gradually narrow the possible range for k.