Secrets of the mathematical ninja: some numbers worth knowing
Learning the rough value of a few key numbers worth knowing can make ninja maths a lot more impressive later on - especially if you know how roughly how rough the rough values are.
A little bit about the tables below: I’m giving you decimals to two sig figs, the best simple fraction I know, and the approximate error for each (a + means you have to adjust the estimate upwards by the given percentage, and a - means you adjust it downwards. An = means it’s pretty much bang on). You’re going to whine about the fractions, aren’t you? It’s actually a lot easier to divide by a fraction than a decimal, so there.
Basic numbers worth knowing: Square roots
Knowing a handful of square roots always looks impressive. (My first exposure to ninja maths was Mr Rowley looking at $\frac{3}{\sqrt{7}}$, frowning slightly, and saying “which is about 1.13” before anyone in the class had even turned their calculators on.)
So, here are some square roots that are numbers worth knowing - the asterisked ones are the ones that come up most often:
(Note that “(=)” doesn’t mean “exactly equal”, just “less than 0.5% away”.)
| Number | Decimal (error) | Fraction (error) |
|---|---|---|
| *$\sqrt2$ | 1.4 (+1%) | $\frac{7}{5}$ (+1%) or $\frac{10}{7}$ (+1%) |
| *$\sqrt3$ | 1.7 (+2%) | $\frac{7}{4}$ (-1%) |
| $\sqrt5$ | 2.2 (+2%) | $\frac{9}{4}$ (-1%) |
| $\sqrt6$ | 2.4 (+2%) | $\frac{22}{9}$ (=) or $\frac{49}{20}$ (=) |
| $\sqrt7$ | 2.6 (+2%) | $\frac{8}{3}$ (-1%) or $\frac{53}{20}$ (=) |
| $\sqrt8$ | 2.8 (+1%) | $\frac{14}{5}$ (+1%) or $\frac{20}{7}$ (+1%) |
| *$\frac{\sqrt2}{2}$ | 0.71 (=) | $\frac{7}{10}$ (+1%) or $\frac{5}{7}$ (+1%) |
| *$\frac{\sqrt3}{2}$ | 0.87 (=) | $\frac{7}{8}$ (+1%) or $\frac{13}{15}$ (=) |
Intermediate numbers worth knowing: $\pi$ and $g$
You also probably want to be able to rattle off a few things to do with $g$ and $\pi$:
| Number | Decimal (error) | Fraction (error) |
|---|---|---|
| *$g$ | 10 (-2%) or 9.8 | |
| *$2\pi$ | 6.3 (=) | $\frac{44}{7}$ (=) |
| *$\pi$ | 3.1 (+1%) | $\frac{22}{7}$ (=) |
| $\frac{\pi}{2}$ | 1.6 (-2%) | $\frac{11}{7}$ (=) |
| $\frac{\pi}{3}$ | 1.05 (=) | $\frac{22}{21}$ (=) |
| $\frac{\pi}{4}$ | 0.79 (-1%) | $\frac{11}{14}$ (=) |
Advanced numbers worth knowing: $\exp$ and $\log$
Now we’re into advanced ninjary! The natural logs of 2 and 3 come up all the time, so it’s worth knowing them; everything else is just gravy.
| Number | Decimal (error) | Fraction (error) |
|---|---|---|
| *$e$ | 2.7 (+1%) | $\frac{19}{7}$ (=) |
| $e^2$ | 7.4 | $\frac{37}{5}$ (=) |
| $e^3$ | 20 (=) | |
| $e^4$ | 55 (-1%) | |
| $e^{1/2}$ | 1.65 (=) | $\frac{5}{3}$ (-1%) |
| $e^{1/3}$ | 1.4 (=) | $\frac{7}{5}$ (=) |
| *$\ln 2$ | 0.7 (-1%) | $\frac{7}{10}$ (-1%) |
| *$\ln 3$ | 1.1 (=) | $\frac{11}{10}$ (=) |
| $\ln 5$ | 1.6 (+1%) | |
| $\ln 7$ | 1.95 (=) | $\frac{39}{20}$ (=) |
In next week’s ninja secret, I’ll show you how to use the errors when you’re combining estimates. I bet you can’t wait!
* Edited 2024-10-04 to fix the tables.