Square Roots by Hand

If you're good with long division, here's a quick way to find pretty accurate square roots without the aid of a calculator. Let's try 24.6.

1. Make a guess. It can be a very bad guess. It doesn't matter. You can even guess 1. Let's try 5 since $5^{2}$ is 25, which is pretty close to 24.6.
2. Divide 24.6 by 5. $\frac{24.6}{5} = 4.92$.
3. Now, comes the trick: Pick a new guess between 5 and 4.92 and divide it into 24.6 again. Let's try 4.95. $\frac{24.6}{4.95} = 4.96$. 4.96 is pretty close to 4.9598 which is the actual square root of 24.6.
4. Repeat steps 2 and 3 to any desired level of accuracy. The further you go, the harder the long division becomes. But the first few cycles yield a pretty close answer.

The reason this works is because $n^{2}$ = 24.6 and $n = \frac{24.6}{n}$. Therefore, the real square root will always be somewhere between $\frac{24.6}{n}$ and $n$.