How do you find the square root of 2?

Answer 1

With great difficulty because the square root of 2 is an irrational number

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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 2, define f(x) = x2 - 2.

Then finding the square root of 2 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:

Suppose you start with x0 = 2 (a pretty poor choice since 22 is 4, which is nowhere near 2).

Even so, x3 = 1.414215686, which is less than 1-in-a billion from the true value. Finally, remember that the negative value is also a square root.