The hierarchy of numbers: each ring contains the next
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.
Timeline: key transcendence proofs 1844–1934
From Liouville's artificial construction (1844) to the Gelfond-Schneider theorem (1934), transcendence theory grew from curiosity to a major branch of number theory.
Algebraic vs transcendental: what makes a number algebraic?
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 |
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