How do you prove a number, like e or pi, to be transcendental?
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.
Vince Calder, Ph.D., Physical Chemist, retired
'One cannot help but be in awe when he contemplates the mysteries of eternity, of life, of the marvelous structure of reality. It is enough if one tries merely to comprehend a little of this mystery every day.'