4 ms·

If a number system has a transcendental number as its base, would these numbers still be called transcendental in that number system?

by zkmon 10mo ago

moefh 10mo ago
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.
frutiger 10mo ago
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).
- kinkyusa 10mo ago
  [dead]
- tocs3 10mo ago
  Base pi: https://en.wikipedia.org/wiki/Non-integer_base_of_numeration#Base_%CF%80 https://en.wikipedia.org/wiki/Non-integer_base_of_numeration... Base e: https://en.wikipedia.org/wiki/Non-integer_base_of_numeration#Base_e https://en.wikipedia.org/wiki/Non-integer_base_of_numeration...
- JadeNB 10mo ago
  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.)
- jibal 10mo ago
  > I think the elements of the base need to be enumerable (proof needed but it feels natural) Proof of what? Needed for what? The elements of the number system are the base raised to non-negative integer powers, which of course is an enumerable set. > transcendental numbers are not enumerable Category mistake ... sets can be enumerable or not; numbers are not the sort of thing that can be enumerable or not. (The set of transcendental numbers is of course not enumerable [per Georg Cantor], but that doesn't seem to be what you're talking about.)
kinkyusa 10mo ago
[dead]
gizmo686 10mo ago
The notion of transcendental is not related to how we right numbers. However, in abstract algebra, we generalize the notion of algebraic/transental to arbitrary fields. In such a framework, a number is only transental relative to a particular field. For instance, the standard statement that pi us transcendental would become the pi is transcendental in Q (the rational numbers). However, pi is trivially not transcendental over Q(pi), which is the smallest field possible after adding pi to the rational numbers. A more interesting question is if e is transcendental over Q(pi); as far as I am aware that is still an open problem.

If a number system has a transcendental number as its base, would these numbers still be called transcendental in that number system?

by zkmon 10mo ago

If a number system has a transcendental number as its base, would these numbers still be called transcendental in that number system?

moefh 10mo ago

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.

frutiger 10mo ago

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).

kinkyusa 10mo ago
[dead]
tocs3 10mo ago
Base pi: https://en.wikipedia.org/wiki/Non-integer_base_of_numeration#Base_%CF%80 https://en.wikipedia.org/wiki/Non-integer_base_of_numeration... Base e: https://en.wikipedia.org/wiki/Non-integer_base_of_numeration#Base_e https://en.wikipedia.org/wiki/Non-integer_base_of_numeration...
JadeNB 10mo ago
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.)
jibal 10mo ago
> I think the elements of the base need to be enumerable (proof needed but it feels natural) Proof of what? Needed for what? The elements of the number system are the base raised to non-negative integer powers, which of course is an enumerable set. > transcendental numbers are not enumerable Category mistake ... sets can be enumerable or not; numbers are not the sort of thing that can be enumerable or not. (The set of transcendental numbers is of course not enumerable [per Georg Cantor], but that doesn't seem to be what you're talking about.)

kinkyusa 10mo ago

[dead]

gizmo686 10mo ago

The notion of transcendental is not related to how we right numbers. However, in abstract algebra, we generalize the notion of algebraic/transental to arbitrary fields. In such a framework, a number is only transental relative to a particular field. For instance, the standard statement that pi us transcendental would become the pi is transcendental in Q (the rational numbers). However, pi is trivially not transcendental over Q(pi), which is the smallest field possible after adding pi to the rational numbers. A more interesting question is if e is transcendental over Q(pi); as far as I am aware that is still an open problem.