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

'One thing I have learned in a long life: that all our science, measured against reality, is primitive and childlike - and yet it is the most precious thing we have.'