How do you prove a number, like e or pi, to be transcendental?

Asked by:
Nathan Douglass

Answer

There is no general theorem that identifies transcendental numbers to my knowledge. However, there are theorems that do so for special cases. Of these theorems, Gelfond's theorem is probably the most useful. It states: If a number 'r' is not the root of any polynomial equation with integer coefficients of any degree, then 'r' is transcendental.

Cantor proved the existence of transcendental numbers. Hermite (1873) proved that the number 'e' is transcendental. Lindemann (1882)proved that 'pi' is transcendental.
Answered by:
Vince Calder, Ph.D., Physical Chemist, retired

'There is no inductive method which could lead to the fundamental concepts of physics. Failure to understand this fact constituted the basic philosophical error of so many investigators of the nineteenth century.'