A number is transcendental if it is not the root of any polynomial equation with integer coefficients. pi satisfies no equation like x^2 - 3x + 1 = 0. e satisfies no such equation. They exist beyond the reach of algebra. Despite being rare to name, transcendental numbers are the rule rather than the exception: almost every real number is transcendental.
Every rational number is algebraic. Every algebraic number is real. But the transcendentals, the numbers outside the algebraic ring, are vastly more numerous than all algebraic numbers combined.
From Liouville's artificial construction (1844) to the Gelfond-Schneider theorem (1934), transcendence theory grew from curiosity to a major branch of number theory.
Table showing algebraic numbers with their minimal polynomials versus transcendental numbers with no such polynomial
| NUMBER | MINIMAL POLYNOMIAL |
|---|---|
| sqrt(2) = 1.41421... | x^2 - 2 = 0 |
| phi = 1.61803... | x^2 - x - 1 = 0 |
| cbrt(5) = 1.70997... | x^3 - 5 = 0 |
| i = sqrt(-1) | x^2 + 1 = 0 |
| pi = 3.14159... | no polynomial exists |
| e = 2.71828... | no polynomial exists |
| e^pi = 23.1406... | no polynomial exists |
A number is transcendental if it satisfies no polynomial equation with integer coefficients. Liouville gave the first explicit example in 1844. Hermite proved e is transcendental in 1873. Lindemann proved pi is transcendental in 1882, finally settling the ancient squaring-the-circle problem as impossible. The Gelfond-Schneider theorem (1934) shows that a^b is transcendental whenever a is algebraic and not 0 or 1, and b is algebraic and irrational. Despite being the rule rather than the exception, proving any specific number transcendental remains extremely difficult.