Approximations

Irrational

ExactValueApproximation
$\sqrt{2}$$1.414$$\dfrac{17}{12}, \dfrac{99}{70}$
$\sqrt{3}$$1.732$$\dfrac{26}{15}, \dfrac{31}{18}$
$\pi$$3.14159$$\dfrac{22}{7}, \dfrac{355}{113}$
$\pi^2$$9.87$$\dfrac{49}{5}, \dfrac{355}{36}$
$\pi^3$$31.0063$$31$
$e$$2.71828$$\dfrac{19}{7}, \dfrac{87}{32}$
$e^2$$7.389$$\dfrac{22}{3}$
$e^3$$20.0855$$20$
$\phi$$\dfrac{1+\sqrt{5}}{2} \approx 1.618$$\dfrac{13}{8}, \dfrac{21}{13}, \cdots, \dfrac{Fib(n+1)}{Fib(n)}$

Logarithm

ExactValueApproximation
$\ln 2$$0.693$$0.69$
$\ln 10$$2.303$$2.30$
$\log e$$0.434$$0.43$
$\log 2$$0.301$$0.30$
$\log 3$$0.477$$0.48$
$\log 5$$0.699$$0.70$
$\log 7$$0.845$$0.85$
$\log 11$$1.041$$1.04$

Note for purists: $\log x$ has base-$10$ if not specified.

Small Angle Trigo

ExpressionApproximation
$\sin \theta$$\theta$
$\tan \theta$$\theta$
$\cos \theta$$1 - \dfrac{\theta^2}{2}$
$\csc \theta$$1 / \theta$
$\cot \theta$$1 / \theta$
$\sec \theta$$1 + \dfrac{\theta^2}{2}$

Natural Fractions

ExactApproximation
$1/3$$0.333$
$1/6$$0.167$
$1/7$$0.143$
$1/8$$0.125$
$1/9$$0.111$
$1/11$$0.091$
$1/13$$0.077$