The fact that hardly anyone seems able to do mental arithmetic these days is one of my fairly standard rants, which most if not all of you have probably heard me make before, at some point. And the thing about mental arithmetic is that in a lot of cases -- especially the cases where you aren't looking for an exact answer -- the mindset and methodology is entirely different to the forma, step by step method that is taught in schools and written out on paper.

One thing that has been interesting me recently is the calculation of roots (square roots, cube roots, and so on). I'm not even sure what the "right" way to calculate these without a calculator or computer is. It's probably something to do with a series expansion, I'd suppose. Or it's dead straightforward if you have a log table, but that's just using a different method of having something else work out the result for you.

Approximations, though, are relatively straightforward. For instance I know that you can get a good estimate of the square root of 2 as follows:

7^2 = 50 (that should be an "is approximately equal to" symbol, but I'm too lazy to figure out how to get one of them)

14^2 = 2^2 * 7^2 = 4 * 50 = 200

14 = sqrt(200) = sqrt(2) * sqrt(100)

14 = 10 * sqrt(2)

sqrt(2) = 14/10 = 1.4

Which is close enough for most purposes (the actual answer is a little above 1.41).

Then just recently, I accidentally stumbled across a similar method for the cube root of 2:

1000 = 1024

10^3 = 2^10 = 2 * (2^9)

10 = cuberoot(2) * cuberoot(2^9) =cuberoot(2) * 2^3 = cuberoot(2) * 8

cuberoot(2) = 10/8 = 5/4 = 1.25

Again, this is good enough for most purposes (the actual value is just slightly below 1.26).

There's nothing particularly difficult about any of the maths involved (at least, not for people who deal with maths frequently; I appreciate that most of this will have gone over the head of the non-mathematical out there, but I doubt that any of you ever need to know what the square root of 2 is). Mental arithmetic of that sort is primarily difficult because people just don't learn of the tricks and techniques involved, and so don't have any idea how to tackle such things.

Admittedly, these particular tricks aren't of any particular use, since you pretty much have to know them in order to use them. They do rather nicely illustrate the mindset involves though, I think. I'm meaning to get around to hunting for other such methods of approximation for other roots at some point, though I haven't got there yet.

- Arithmetical tricks

jenmcd