Approximate Square Root

Unlike most of my friends who simply dove into the JEE preparation after announcement of their tenth results, I spent my summers reading Asimov and playing with numbers. In fact more than half the time of my class XI was spent just playing with Mathematics, rather than solving problems of the correspondence courses and joining coaching classes. I remember the thrill and excitement that came from exploring the woods of numbers and the freedom to choose my own direction. In fact my interest in mathematics could largely be attributed to that period alone.

I discovered this formula way back in the starting of class XI. The formula is startlingly simple and gives good results in most cases. Furthermore, I barely knew enough calculus at that time to actually prove it and I am always amazed at how useful it has been.

Given a number x, choose any a such that a^2 is the closest to x (and you remember the 'a and a^2' pair; that usually happens with integers). The approximate square root of x is simply (x+a^2)/2a. One could alternatively express it as a+(x-a^2)/2a if that sounds easier. The proof is a simple class XI exercise in calculus.

Around that time I read another approximate formula for square root in Mathematics Today. Comparing the two, I found that their formula was much less accurate and had a smaller domain of applicability. I remember that I was very encouraged by this fact and even wrote to them, though they did not publish my work.