## Approximating Square Root of Numbers by Averaging

There are several algorithms developed to manually calculate the square root of a number. My favorite is the method of averaging, which is probably the least complicated.  In this method, the number is divided by the nearest square root estimate, and then the quotient and the square root estimate are averaged. The result is the new estimate.

Sounds complicated? Not really.

Example: Approximate $\sqrt{24}$.

Note: Using a calculator, $\sqrt{24} \approx 4.8989794855$.

Solution

We know that $\sqrt{24}$ is between $4^2$ and $5^2$. Between the two, it is nearer to $5^2$. So, the first estimate is $5$. Next, we divide $24$ by the first estimate and “average” it with the quotient . » Read more