What are the transcendentals, and are they related to the imaginaries?
The answers on previous page didn't perhaps tell very well what's the point of trancendentals, focusing more on the power series business...
Consider numbers in general. Children first learn that numbers are 1,2,3,... Then they find out about 0 and negative numbers, -1,-2,-3,... and then the rational numbers, and finally about real numbers like squareroot 2, pi and e and so on, even though they're not very aware what real number really means.
Now as square root 2 can easily be proven to be irrational and is the first number known to be irrational, question arises:
are all irrationals squareroots? Or more generally
are all irrationals n-roots? The answer to both these questions is no, and can most easily be seen when considering numbers like sqrt 2+ sqrt 3 (exercise: prove that sqrt n is rational if and only if n is a square of a natural number).
All of these counter examples are nonweird, because they are roots of polynomials, sqrt 2 +sqrt 3 for example is root of (x^2 -5)^2 -24. While they are perfectly good answers, they aren't entirely satisfying, because we wanted to ask if there's really weird real numbers. So it's natural to reformulate the question: are there any real numbers which aren't roots of polynomials (with rational coefficients)? Numbers which are such roots are called
algebraic, and the rest are called
trancendental. For all I know, no number was known to be trancendental before late 19th century when pi and
e were proven to be ones.
Now back to the numbers in general, the concept of number has been upgraded by adding something to the previous definition: First you had just the natural numbers, then you add 0 and the negative numbers and get integers, and then you add the fractions and get the rationals. Now tempting expansion would be to add the roots of polynomials, but the existence of trancendentals shows that it's insufficient.
Trancendetality of pi is also important, because as a corollary a circle can't be squared (with compass and ruler). The problem is pretty much as old as the western civilization.
You don't need the axiom of choice that often at all in "basic" mathematics. For example, you don't need it when "choosing" from a finite set.
I wouldn't be too sure about it, although it isn't used explicitly. Think this standard undergraduate exercise as an example: You have to prove that x in R^n is at the boundary of a nonempty set A if and only if there's a sequence of A's points which converges to x.
The proof of course is that you take x_k from the intersection of A and B(x;1/k), where k=1,2,3,..., and the sequence (x_k) converges to x.
But: Do you use the axiom of choice when picking that x_k? In all honesty, I don't know if it's used. However this example, I think, is enough to show that the matter isn't as easy and clear cut as it is often seen to be.