The

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

If α_{1},...,α_{n}are distinct algebraic numbers, and β_{1},...,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 β_{0},...,β_{n}, not all zero, such that

*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 π*i* and 2π*i* are also algebraic. Taking β_{1} = β_{2} = 1, α_{1} = π*i*, α_{2} = 2π*i*, the Lindemann-Weierstrass theorem gives us the equation (see Euler's formula)

The theorem is named for Carl Louis Ferdinand von Lindemann and Karl Weierstraß