I recently came across the following computational trick: to estimate , use
where is the closest perfect square to . For example,
I had not seen this trick before — at least stated in these terms — and I’m definitely not a fan of computational tricks without an explanation. In this case, the approximation is a straightforward consequence of a technique we teach in calculus. If , then , so that . Since , the equation of the tangent line to at is
The key observation is that, for , the graph of will be very close indeed to the graph of . In Calculus I, this is sometimes called the linearization of at . In Calculus II, we observe that these are the first two terms in the Taylor series expansion of about .
For the problem at hand, if , then
if is close to zero. Therefore, if is a perfect square close to so that the relative difference is small, then
One more thought: All of the above might be a bit much to swallow for a talented but young student who has not yet learned calculus. So here’s another heuristic explanation that does not require calculus: if , then the geometric mean will be approximately equal to the arithmetic mean . That is,