How do you calculate the square root of 1009?

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.

Since you want to find the square root of 1009, define f(x) = x^2 - 1009.

Then finding the square root of 1009 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.

30*30 = 900 which is reasonably close to 1009 so let x0 = 30, then by x3, the absolute error is less than a quarter in 1 billion!

The solution, using this method, is 31.76476035