The Lindemann-Weierstrass Theorem states that if a finite set of algebraic numbers is linearly independent over the rationals, then the exponentials of those numbers are algebraically independent. Proved by Ferdinand von Lindemann and extended by Karl Weierstrass, it implies that both e and pi are transcendental numbers and, in particular, settled the ancient problem of squaring the circle by straightedge and compass in the negative.
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StatementThe Lindemann-Weierstrass theorem states that if a set of algebraic numbers is linearly independent over the rational numbers, then their exponentials are algebraically independent over the algebraic numbers; it is the theorem that proves e and pi are both transcendental. 1 Proof YearLindemann proved the restricted single-exponential case in 1882 (enough to show pi transcendental); Weierstrass generalized it to the full theorem in 1885, giving the theorem its usual joint name. Classification
Statement FormCharacterization Theorem 1 Lindemann-Weierstrass Theorem
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Source Lindemann-Weierstrass Theorem (Wikipedia)
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1. Lindemann-Weierstrass Theorem (Wikipedia)
Wikimedia FoundationLead section, opening sentence
In transcendental number theory, the Lindemann-Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers.
Lead section, naming paragraph
Lindemann proved in 1882 that eα is transcendental for every non-zero algebraic number α, thereby establishing that π is transcendental (see below). Weierstrass proved the above more general statement in 1885.
Principal Result Of: Ferdinand von Lindemann, Lead sentence of the article on the result
The theorem is named for Ferdinand von Lindemann and Karl Weierstrass.
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