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 $$n$$.