How do you get the square root of not perfect square?

Answer 1

One way is to solve the equation x^2 - number = 0 with Newton's method.

Start with a guess for the answer (say your number is 20, guess 5 or something - it hardly matters how bad your guess is). Say your guess is G. Then compute :

0.5 * (G + 20/G)

This will give you a new guess. Rinse and repeat. The number 20 needs to be replaced with whatever number you're trying to find the square root of.

Example with √30 :

Take a guess : 6.

Compute 0.5 * (6 + 30/6) = 5.5.

Compute 0.5 * (5.5 + 30/5.5) = 5.47727272728.

Compute 0.5 * (5.47727272728 + 30/5.47727272728) = 5.47722557525.

The true answer is 5.47722557505. You already had four correct decimals after the second step. The third step yields nine correct decimals. It keeps growing like this really fast. This is way better than the longhand method your grandparents learned in school. ^_^

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Excellent answer except for one minor (pedantic) point. The true answer is NOT 5.47722557505. The square root of 20 is an irrational and therefore has an infinitely long, non-recurring decimal representation. 5.47722557505 is a rational - tediously long, but even so.