As another application of the abstract concept of vector spaces, there is an amazing theorem due to Weierstrass and Lindemannn.
Theorem 7.3.34 Suppose a1,
In other words, the
There is a proof of this in the appendix. It is long and hard but only depends on elementary considerations other than some algebra involving symmetric polynomials. See Theorem F.3.5.
A number is transcendental, as opposed to algebraic, if it is not a root of a polynomial which has integer (rational) coefficients. Most numbers are this way but it is hard to verify that specific numbers are transcendental. That π is transcendental follows from
By the above theorem, this could not happen if π were algebraic because then iπ would also be algebraic. Recall these algebraic numbers form a field and i is clearly algebraic, being a root of x2 + 1. This fact about π was first proved by Lindemannn in 1882 and then the general theorem above was proved by Weierstrass in 1885. This fact that π is transcendental solved an old problem called squaring the circle which was to construct a square with the same area as a circle using a straight edge and compass. It can be shown that the fact π is transcendental implies this problem is impossible.1