## Constants, Identities, and Variations

(under Construction)

Calculus:

If we have the following conditions:

1. $f(x)$ is continuous on $[a,b]$,
2. $f(a)$ and $f(b)$ are of different signs,
Then there exists a point $\xi\in(a,b)$ such that $f(\xi)=0$.

Euler's Formula for Complex Numbers:

$z = cos\ \theta + i\ sin\ \theta = e^{i\theta}$ (standard form)
$z = r (\cos(\theta)+ i \sin(\theta))$ (polar form)
$z = r e^{i\theta}$ (exponential form)
When $\theta = \pi$, Euler's formula evaluates to: $e^{i\pi} + 1 = 0$,

which is known as Euler's Identity