A transcendental numberIn 1874 Georg Cantor found that the algebraic numbers are countable whereas the real numbers are not. Hence there must exist non-algebraic - that is transcendental - numbers. Some people claim this as an example of a nonconstructive proof - the existence of something is shown without giving any idea of how to construct it. However I find that most proofs that are supposed to be nonconstructive are nothing of the kind. Certainly in this case it is possible to use Cantor's proof to construct a non-algebraic real number.
This construction is achieved by the applet below. The algebraic numbers, that is the roots of polynomials with integer coefficients, are generated in sequence. For the Nth number of this sequence the Nth decimal digit is found, and then a different digit is printed. Hence the number printed differs from all algebraic numbers, and so is transcendental. Of course the decimal expansion of this number goes on forever, so it is better to think of the algorithm as representing the number, rather than what is printed out.
How it worksFor a polynomial anxn+an-1xn-1+..a1x+a0 with integer coefficients define the height as