Secrets of the Mathematical Ninja: Estimating powers of e

Euler’s constant, $e$ (about 2.718 281 828) is one of the most important numbers in maths – both pure and applied. (Thinking about my final year university courses, the only one I’m pretty sure had no use for $e$ was History of Maths, and frankly that was an oversight.)

As a budding mathematical ninja, you’re doubtless keen to learn how to estimate powers of $e$ for pleasure and profit. Unfortunately, only pleasure is likely to come of it, unless you become a maths tutor.

Now: it’s easy enough to estimate $e$ – it has a nice, memorable decimal expansion, and depending on how roughly you want to play, you can look at it as 3 (-10%), 2.7(+0.7%) or $\frac{30}{11}$ (-0.3%).

However, your estimates for powers of $e$ are likely to be off by more than you’re used to in ninja maths – this is one situation where small errors add up fast. If you’re about as good as me, you’ll be happy to get things within about 5%.

If you want to know $e^3$, you might pick the first one, and say ‘it’s 27 less 30% – take away about 8, so 19. It’s actually 20.08 – not brilliant, but ok for a ballpark figure. It’s not really obvious how to do ${2.7}^3$ – unless you know that ${27}^3$ is about 19,700, so ${2.7}^3$, plus 2.1%, would be 19.7 plus about 0.4, or 20.1. That’s bang on.

Alternatively, you can do $\frac{{30}^3}{{11}^3}$, which gives $\frac{27,000}{1,331}$. Multiplying top and bottom by 3 gives $\frac{81,000}{3,993}$; that’s a shade over $\frac{81}{4}$, or 20.25 – less about 1% to make up for the original estimate, making 20ish. The only limit is your number handling!

If you’re hot on your natural logs, you can reverse-engineer powers of $e$ from there, too – if you want to know $e^{3.5}$, you can ask ‘$\ln$ of what is 3.5? Well, that’s $0.7\times 5$, so it must be about $2^5$ or 32. (It’s 33.11 – not bad).

When you’re working with negative powers of $e$, the fractional version ($e\approx \frac{30}{11}$) comes into its own - because $e^{-1}$ is just $\frac{11}{30}$, and it’s easy to take powers of both of those.

For instance, $e^{-2}$ is just $\frac{{11}^2}{{30}^2}$, or $\frac{121}{900}$. That’s a bit more than $\frac{40}{300}$, or 0.133. (In fact, it’s 0.135). And $e^{-1}$ itself comes up a lot: $\frac{11}{30}$ is 0.367, while $e^{-1}$ is 0.368. I could live with that as an estimate!