Originally Posted by **hpe**

The square root of a number x can be computed approximately to any precision as follows. I suspect that that's close to the way it's done in a calculator:

Start with a guess a (a number near $\displaystyle \sqrt{x}$). Then compute $\displaystyle b =\frac{1}{2}(a + \frac{x}{a})$, $\displaystyle c =\frac{1}{2}(b + \frac{x}{b})$ , $\displaystyle c =\frac{1}{2}(b + \frac{x}{b})$ and so on. After a few steps, you will be near $\displaystyle \sqrt{x}$. Note that you only have to be able to add and divide.

There is a bit of theory that allows you to tell when to stop. It can be easly implemented in hardware. This is a special instance of the Newton-Raphson algorithm

Example: To find $\displaystyle \sqrt{23}$, start with a = 5. Then b = 4.80000, c = 4.79583333333333, d = 4.79583152331306, e = 4.79583152331272 which is correct to all 15 digits. Works for any x and can be done also for all other roots.