Let’s demonstrate the method with an example. Consider computing the square root of n=33. We start by finding a number that forms a perfect square that is close to 33. Here, let’s pick g=6, since 62=36. Then we compute a second number, b=n/g. In practice, computing b in your head may require an approximation. Here, we can compute it exactly as 33/6=5.5. Then our final guess is the average of these two numbers or


which in our example is


That is pretty good. The relative error is less than 0.1%! And furthermore, this is pretty straightforward to do in your head when n isn’t too large.

Estimating Square Roots in Your Head
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