Heron's method for approximating square roots

One way to estimate $\sqrt{50}$ is to observe that $50=2\times 5^2$, and hence that $\sqrt{50}=5\sqrt{2}$.

Now, the square root of 2 is one of those constants whose first few decimal places are often useful to know—let's use 1.41 to keep things as simple as possible here. We then need only multiply by 5 to obtain a reasonable approximation. One convenient way to multiply by 5 is to multiply by 10 first and then divide by 2, readily giving 7.05. Incidentally, with this method it would, curiously enough, have been easier to use 1.414 as an approximation to $\sqrt{2}$: this would effortlessly have given 7.07, which is both more accurate and involves no carrying! Mental arithmetic sometimes calls for an eye for opportunity.

Let's return to $\sqrt{50}$ and "forget" the previous trick (which is not always available) as we look for a more general method. The first step is to find bounds for $\sqrt{50}$ using the first few perfect squares. In any case, knowing a good list of them is a prerequisite for anyone who wants to develop their mental-arithmetic skills.

This table also lays the groundwork for square roots of larger numbers, using operations involving powers of 10. For example, we can see that $\sqrt{\mathrm{150 000}}$ is slightly less than 400, since: