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!


Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.


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