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
| ZAHL | MINIMALPOLYNOM |
|---|---|
| √2 = 1,41421... | x^2 - 2 = 0 |
| φ = 1,61803... | x^2 - x - 1 = 0 |
| ∛5 = 1,70997... | x^3 - 5 = 0 |
| i = √(-1) | x^2 + 1 = 0 |
| π = 3,14159... | kein Polynom existiert |
| e = 2,71828... | kein Polynom existiert |
| e^π = 23,1406... | kein Polynom existiert |