How can you find the decimal approximation of the square roots that are irrational?

Answer 1

One way to estimate the square root of a number is by iteration. 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 k, define f(x) = x^2. Then finding the square root of k 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 x(0) as the first guess. Then let x(n+1) = x(n) - f[x(n)]/f'[x(n)] 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. It works even if your first guess is not so good:
Suppose, for calculating sqrt(7) you start with x(0) = 5 (a pretty poor choice since 5^2 is 25, which is nowhere near 7).
Even so, x(3) = 2.2362512515, which is less than 0.01% from the true value. Finally, remember that the negative value is also a square root.

Answer 2

One way to estimate the square root of a number is by iteration. 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 k, define f(x) = x^2 – k.

Then finding the square root of k 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.

It works even if your first guess is not so good:

Suppose you want the square root of 7 and you start with x0 = 5 (a pretty poor choice since 5^2 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.