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
'The strength and weakness of physicists is that we believe in what we can measure. And if we can't measure it, then we say it probably doesn't exist. And that closes us off to an enormous amount of phenomena that we may not be able to measure because they only happened once. For example, the Big Bang. ... That's one reason why they scoffed at higher dimensions for so many years. Now we realize that there's no alternative... '