- Types of numbers at the time of Liouville: Integers, Rational Numbers, Algebraic numbers (roots of polynomials with rational coefficients), transcendental numbers (numbers that are not roots of any polynomial with rational coefficients)
- The concept of transcendental numbers existed before any were known
- Liouville proved that you can't approximate any algebraic number really well with rational numbers
- Theorem: If you find a sequence $\frac{p_m}{q_m} \to \alpha$ such that $\mid \alpha - \frac{p_m}{q_m} \mid \leq \frac{1}{{q_m}^n}$ for $m$ sufficiently large then $\alpha$ is not the root of a polynomial of degree $n$

Proof of Liouville's theorem:

Suppose $f(x) = a_0 + a_1x + a_2x^2 +...+a_nx^n$ and $f(z) = 0$ and $z_m = \frac{p_m}{q_m} \to z$

$\\ \frac{f(z_m)-f(z)}{z_m-z} = a_1({z_m}-z) + a_2({z_m}^2 - z^2) + a_3({z_m}^3 +...+ a_n({z_m}^n - z^n)$

$\\$ if $f(z) = 0$ $\frac{f(z_m)}{z_m-z} = a_1 + a_2(z_m-z_) + a_3({z_m}^2- z^2) +..$

$\\$ for m large enough $\mid z_m - z_n \mid < 1$

$\\ \frac{f(z_m)}{z_m-z} | < |a_1| + 2|a_2|(z_m-z_n) + 3|a_3|{(|z|+1)}^2... n|a_n|{(|z|+1)}^{n-1}$

$\\$ for m large enough $|\frac{f(z_m)}{z_m-z}| = M$; M is some number.

$\\ z_m = \frac{p_m}{q_m} \to z$ ; $q_m$ will eventually be greater than M

$\\ (z-z_m) > \frac{|f(z_m)|}{M} > \frac{f(z_m)}{q_m}$

$\\ f(z_m) = f(\frac{p_m}{q_m}$ This is an integer represented by a finite sequence which is $> \frac{1}{{q_m}^n}$

$\\ z_m$ cannot be a root of a degree n polynomial