Lindemann-Weierstrass theorem
The Lindemann-Weierstrass theorem is a theorem in mathematics that is very useful in establishing the transcendence of numbers. The theorem states:
If α1,...,αn are distinct algebraic numbers, and β1,...,βn are any nonzero algebraic numbers, then
The transcendence of e and π are direct corollaries of this theorem. To show the transcendence of e, note that if e were algebraic, there would exist rational_numbers β0,...,βn, not all zero, such that
Since every rational number is algebraic, this violates the Lindemann-Weierstrass theorem, and so e must be transcendental.
To show the transcendence of π, suppose that π was algebraic. Since the set of all algebraic numbers forms a field, this implies that πi and 2πi are also algebraic. Taking β1 = β2 = 1, α1 = πi, α2 = 2πi, the Lindemann-Weierstrass theorem gives us the equation (see Euler's formula)
and this contradiction establishes the transcendence of π.
The theorem is named for Carl Louis Ferdinand von Lindemann and Karl Weierstraß
Referenced By
Carl Louis Ferdinand von Lindemann | E (mathematical constant) | E - base of natural logarithm | Ferdinand Lindeman | Ferdinand von Lindemann | History of Pi | Karl Weierstrass | Karl Weierstraß | List of mathematical proofs | List of mathematical topics (J-L) | List of number theory topics | List of proofs | Ludolph transcendental number | Transcendental number | Trascendental number | Weierstrass
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