I don't remember now why, but I wanted to take a square root in my head this morning. The first digit and a guess at the second were easy, but after that I found myself relying on a very hazy recollection of some convoluted method that seemed rather counter-intuitive, at least in contrast to the method that I had just used to obtain the first 2 digits. I saw no reason why the method that I had used couldn't be extended to obtain as many digits as desired, yet I couldn't find any reference to it.
Let's say that we want to find the square root of 62. A quick check shows that 7 is too small and 8 is too big, so we write down 7. Now (and this is the “intuitive” part that led me to this solution), we can see that 62 is pretty close to 64, the square of the next number after 7. In fact, it's a lot closer to the square of 8 than it is to the square of 7, so I guessed that the next digit was 8 (in other words, it appeared to me that 62 was about 80% of the way between 49 and 64).
Of course, in order to calculate the next digit, we'll have to do better than guess. We subtract 49 (the current square) from 62 (the target) and obtain 13. Subtracting 49 from 64 (the next square), we obtain 15. So 62 is 13/15 of the way from 49 to 64. 13 divided by 15 is 0.86, so the next number is indeed 8 (9 would be too big).
Okay, so now we need to see what percentage of the distance between 7.8 and 7.9 the next digit should be. We square 7.8 and obtain 60.84. We are 1.16 away from our target of 62. Squaring 7.9 gives 62.41. 62.41 minus 60.84 is 1.57. 1.16 divided by 1.57 is 0.73, so the next digit is 7.
We'll do one more. 7.87 squared is 61.93. Our difference between the square and the target is now 0.07. 7.88 squared is 62.09. The difference between the current square and the next square is 0.16. To see what distance the next digit covers between 7.87 and 7.88, we divide 0.07 by 0.16 and obtain 0.43. The next digit is 4.
7.874 squared is 61.999876. We're really close now, so it looks like the next digit will be a zero. In fact, the next 2 digits are zeros.
I suppose it comes down to personal preference, but this method just makes more sense to me than any other I've seen. Have any of you been taught this method, or heard of it being taught? And do you find, as I do, that this is at least as easy as any other method?