# 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:

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%)

$\frac{5}{4}$ (-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 $\pi$:

*$g$

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.

*$e$

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!