If you’ve been following the Flying Colours Maths blog for any length of time, you’ll have noticed that every Monday, there’s a new Secret of the Mathematical Ninja, a quick tip or trick for coming up with a more-or-less accurate estimate of some crazy sum you might see in your A-level class.

And you’d be quite justified to take a look at those articles, look at me sardonically and say “I have a calculator.”

There’s a bit in one of my favourite books, Numerical Recipes in C, where they’re talking about the error bounds in a fourth-order Runge-Kutta expansion (racy stuff, I tell you). They point out that you can use your error term to get a better estimate — but if you do that, you’ve no idea how wrong your estimate is. After all, you just used the error term.

You should learn the ninja tricks — or at least to take pride in your mental arithmetic — for much the same reason. Your calculator gives you a number… but is it a sensible number? You get an exact expression for a volume — but is it a reasonable answer? Without the ninja tricks — or at least some way of checking — you have no idea.

Let me be clear: I’m not expecting you to do your paper in your head. Just that if you can come up with a ballpark number, even on paper, you can say “that looks just about right”, or “hang on — that’s absurd”, which tells you to go back and check.

I make mistakes all the time in class. Probably as many as you do. The difference is, I catch them.

* Edited 2014-10-04 to fix typo.