## Constants, Identities, and Variations

(under Construction)

#### Calculus:

If we have the following conditions:

- $f(x)$ is continuous on $[a,b]$,
- $f(a)$ and $f(b)$ are of different signs,

#### 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**

#### Rational, Irrational, Algebraic, and Transcendental

(as defined in Mathematics - defined elsewhere in other ways)

Rational numbers are expressed as the ratio of two integers or whole numbers: $r=\frac{p}{q}$.

Algebraic numbers are the roots of finite polynomials with integer coefficients: $a_nx^n+\dotsb+a_2x^2+a_1x+a_0=0$.

Every rational number is algebraic when it is a root of the equation $qx-p=0$. There are algebraic numbers which are not rational. The most famous one is $\sqrt2$, which is a root of $x^2-2=0$.

The irrational and transcendental numbers are defined by what they are not: members of $\mathbb R$ in some way:

- Irrational, not Rational, $\mathbb R\setminus\mathbb Q$
- Transcendental Numbers are not Algebraic (Algebraic Number), $\mathbb R\setminus\mathbb A$

Every transcendental number is irrational.

That transcendental numbers exist and that not all real numbers are algebraic was first proved by Joseph Liouville in 1844. The first number to be demonstrably transcendental is now called the Liouville constant:

$\quad\displaystyle\sum_{k=1}^{\infty}10^{-k!}=0.110\,001\,000\,000\,000\,000\,000\,001\,000\dotsc$

The constants $e$ and $\pi$ have been proven to be transcendental, but Transcendental Number Theory has not yet determined whether $e+\pi$ is transcendental.