Lindemann-Weierstrass theorem
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The Lindemann-Weierstrass theorem is a theorem in mathematics that is very useful in establishing the transcendence of numbers. The theorem states:

If α₁,...,α_n are distinct algebraic numbers, and β₁,...,β_n are any nonzero algebraic numbers, then

The transcendence of e and &pi are direct corollaries of this theorem. To show the transcendence of e, note that if e were algebraic, there would exist rational_numbers β₀,...,β_n, 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

The original article can be found at: http://en.wikipedia.org/wiki/Lindemann-Weierstrass_theorem.
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