Comment by zkmon
6 hours ago
If a number system has a transcendental number as its base, would these numbers still be called transcendental in that number system?
6 hours ago
If a number system has a transcendental number as its base, would these numbers still be called transcendental in that number system?
Yes. A number is transcendental if it's not the root of a polynomial with integer coefficients; that's completely independent of how you represent it.
I think the elements of the base need to be enumerable (proof needed but it feels natural), and transcendental numbers are not enumerable (proof also needed).
Base pi: https://en.wikipedia.org/wiki/Non-integer_base_of_numeration...
Base e: https://en.wikipedia.org/wiki/Non-integer_base_of_numeration...
I think your parent comment was speaking of a "base-$\alpha$ representation", where $\alpha$ is a single transcendental number—no concerns about countability, though one must be quite careful about the "digits" in this base.
(I'm not sure what "the elements of the base need to be enumerable" means—usually, as above, one speaks of a single base; while mixed-radix systems exist, the usual definition still has only one base per position, and only countably many positions. But the proof of countability of transcendental numbers is easy, since each is a root of a polynomial over $\mathbb Q$, there are only countably many such polynomials, and every polynomial has only finitely many roots.)
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