Originally Posted by ThePieMan
Thanks for the response. That cleared up some misunderstanding I had about this topic. From what I understand, a transcendental number is a number that is not a solution of any algebraic solution. Or are their further definitions of transcendence?
Yep that's totally right, although obviously to be complete we have to specify that the coefficients are integers; otherwise e wouldn't be transcendental because it's a root of x  e = 0, for example.
In a sense it's an extension of irrational numbers. Irrational numbers cannot be written as a/b; this is equivalent to saying they are not the root of any first degree polynomial, specifically bx  a = 0. Transcendental numbers are the same but it's for any degree.
That's not so hard once you understand what countability is. Once you've got the basic idea, hopefully you can build on it with relative ease via reading.
Two sets A and B have the same size if you can make a function that maps each element of A to a unique element in B, in other words a one to one mapping (the technical word you will see for this type of function is 'bijection'). A simple analogy: counting ten sheep. You map sheep one to the number 1, sheep two to the number 2, etcetera, and you end up with a one to one mapping between A = {the sheep} and B = {1, 2, ... , 10}. Then we say the size of the set of sheep is 10.
It's the same with infinite sets. The most basic infinite set is the naturals, {1, 2, 3, ... }. We say a set is countable if it has the same size (by the above definition) as the naturals.
For instance, the integers, {..., 2, 1, 0, 1, 2, ...}. If we zig zag back and forth, we can make a function that sends
0 to 1,
1 to 2,
1 to 3,
2 to 4,
2 to 5,
3 to 6,
3 to 7,
and so on. The important thing is that this is onetoone; each integer now has an associated natural number, and each natural number an associated integer. Thus the size of the set of integers is the same as the size of the naturals, in other words, the integers are countable.
How about all natural coordinates, (n, m)? These are also countable. A neat way to do it is to send each pair to the natural number (2^n)*(3^m). The crucial thing is that it never maps two different pairs to the same natural number (because prime factorisations are unique). I brought up this example because it isn't actually onetoone, but it's still okay. It's not onetoone because some naturals don't get mapped to (for instance, 5) by any pair. However, heuristically, we know that the set of pairs of naturals is clearly infinite, so it must be at least as big as the naturals; and this mapping has managed to 'fit them' into the naturals, with extra room in fact, so it can't be any bigger than the naturals. Therefore it's the same size.
This illustrates that to demonstrate countability, you just have to map your set to the naturals in a way that doesn't send two elements to the same natural (the technical word is 'injection'); it doesn't actually have to be onetoone, as long as you fit your set inside.
The rationals are also countable. The reals however are not, which is quite an astonishing fact (that some infinities are bigger than others). I've explained why before, you can see here. Clearly the reals are just the combination of two sets: the transcendentals and the nontranscendentals. As the nontranscendentals can be counted (a good challenge is to try to prove this roughly yourself; start by considering if the information about the coefficients can be counted), the transcendentals cannot; if they could be, then the nontranscendentals and transcendentals together could be counted, i.e. all the real numbers could be counted, which is not true. Intuitively, this means that almost all real numbers, if you pick one at random, are transcendental; algebraic numbers are, in a sense, 'rare'.


Bookmarks