 Transcendental number theory

Transcendental number theory studies numbers that satisfy no nonzero polynomial equation with integer coefficients, and stronger independence properties among numbers and special-function values. The distinction is algebraic: $\sqrt{2}$ is irrational but algebraic because it satisfies $x^2-2=0$.

A useful example is the Liouville constant $L={\sum }_{n\ge 1}10^{-n!}$. Truncating after $N$ terms gives a rational $p_N/q_N$ with $q_N=10^{N!}$ andFor any fixed algebraic degree, these approximations eventually contradict the rational-approximation lower bound in Liouville approximation theorem. Thus exceptionally good approximation can certify transcendence.

Other results, such as the six exponentials theorem, constrain collections of exponential values using linear independence of their inputs. Proving each number irrational is weaker than proving it transcendental, and proving separate transcendence is weaker than proving algebraic independence of a collection.