Mathematical Constants: e, i, √2, and π

Euler's number, e

$e \approx 2.71828\ldots$ is an irrational number that shows up constantly in growth and decay, including continuously compounded interest. One way to define it is as an infinite sum:

e = \sum_{n=0}^{\infty} \dfrac{1}{n!} = \dfrac{1}{0!} + \dfrac{1}{1!} + \dfrac{1}{2!} + \cdots

The imaginary unit, i

$i$ is defined by $i = \sqrt{-1}$ , so $i^2 = -1$ . Higher powers of $i$ repeat with a period of 4:

i^0=1, \quad i^1=i, \quad i^2=-1, \quad i^3=-i, \quad i^4=1, \quad \ldots

To find $i^n$ for a large $n$ , divide by 4 and use the remainder.

Worked example: find $i^{45}$ .

45 = 4 \times 11 + 1 \quad\Rightarrow\quad i^{45} = i^1 = i

Negative powers cycle the same way, just starting from $i^{-1}=\tfrac{1}{i}=-i$ :

i^{-1}=-i, \quad i^{-2}=-1, \quad i^{-3}=i, \quad i^{-4}=1

Pythagoras' constant, √2

$\sqrt{2} \approx 1.41421\ldots$ is the diagonal length of a unit square, and was the first number ever proven irrational.

Archimedes' constant, π

$\pi$ is the ratio of a circle's circumference to its diameter, for every circle regardless of size:

\pi = \dfrac{C}{d} = \dfrac{C}{2r} \approx 3.14159\ldots