A transcendental number

In 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.

You do not appear to have Java enabled on this browser. (Note that Java is no longer supplied by default on Windows systems). The latest Java environment can be downloaded and installed from the Java website

How it works

For a polynomial anxn+an-1xn-1+..a1x+a0 with integer coefficients define the height as
H= n+
Then for each H>0 there are a finite number of polynomials with this height. The program generates each of these in turn. Then there is the difficult bit of finding the positive real roots to the required accuracy. Fortunately Java has a BigDecimal type, which allows arithmetic to arbitrary precision. Finding roots of polynomials consists of two parts. (1) Localising each root (2) Closing in on the root. For (1) we use the continued fractions method of Akritas. For (2) we use the Newton-Rapherson method. Note that this only works for squarefree polynomials, that is those without a repeated root. However, we can ignore polynomials which are not squarefree as, their roots will also be roots of squarefree polynomials. Note that this applet is computationally very intensive, so it might be an idea to check the Pause box if you want to give your CPU a rest.

Source Code