Please forgive the lack of technical vocabulary.
So if I have any limited understanding of transcendental numbers, it's that if one were to be given only algebraic numbers and were permitted to use any type and amount of algebraic operations on those algebraic numbers, then a number like π would be inaccessible from those operations.
It's as if π and the algebraic numbers are on different "tracks".
But say one were given π to begin with. Obviously, 2π, 1/π, π^2, etc. are all accessible from that starting point, but what about the vast majority of other transcendental numbers that have nothing to do with π.
Given two random transcendental numbers that are independently arrived at (i.e. they're not like π and 3π), are these two numbers always mutually accessible to and from each other with algebraic operations or not?
To put it another way, given any two transcendental numbers A and B, is there always an algebraic method to arrive at B from A, without using B itself, but only algebraic numbers and other A-derived transcendental numbers?(I say that last part to avoid this trivial solution -> (A/A)*B=B
If not, does that mean that the set of transcendental numbers is filled with an infinite amount of algebraically isolated "tracks", if that metaphor works?
(As I was writing this, I realized I may be mistaken in my limited understanding of transcendental numbers, because I did not know that numbers like 2^√2 were transcendental, which seems easily accessible from algebraic numbers. Nonetheless, I'll post and please correct me.)
I assume that by "accessible" you mean in the same sense that the algebraic numbers are reachable from rationals; so, using the basic operations of addition, multiplication, subtraction, and division, plus taking roots of polynomials with coefficients we've already "reached".
In that case, no, transcendental numbers aren't always reachable from one another. This is because of the difference between countable and uncountable sets - there are only countable many "accessing" steps we can take to get to a new number, but there are uncountably many transcendental numbers. So, in fact: if you took two transcendental numbers at random, it is almost certain that neither will be reachable from the other.