Can $\pi$ be rational in some base radix

I am from a physics background and my mathematics is not very good, so pardon my insolence with the question.

Editing based on the comments : We know that $\pi$ in decimal (i.e. base 10) is transcendental. Is it possible to have a radix base in which $\pi$ can be rational in that base ?

PS : My question is in general for any transcendental number.

Solution 1:

I think you're getting confused on the difference between a representation that does not terminate, a number being irrational, and a number being transcendental.

Whether the representation terminates depends on the base. For example, in base $10$, this does not terminate: $$\frac{1}{3} = 0.3333333333333\ldots$$ But in base $3$, we have $$\frac{1}{10} = 0.1.$$

There is some overlap, but this is is separate concept from irrational numbers. We say a number $x$ is rational if there are integers $a$ and $b \neq 0$ such that $$\frac{a}{b} = x.$$ If there are no such integers, then the number is irrational. It doesn't matter if the representation is terminating or non-terminating in a given base, as long as the integers $a$ and $b$ exist, the number is rational.

One-third is a rational number, because we can use $a = 1$ and $b = 3$ (to use the most obvious choice, there are others, of course), and this even though its representation in base $10$ does not terminate.

But $\pi$ is irrational, though it can be approximated well enough (in a purely practical sense) with rational numbers. Its representation will not terminate in any integer base. I would even go so far as to say that the only way for $\pi$ to have a terminating representation in base $x$ is for $x = n \pi$, where $n$ is a nonzero integer.

I could be wrong about that last bit, but the fact remains that $\pi$ has no truly precise representation as a ratio of two integers. In general, any irrational number represented in an integer base will have a non-terminating representation.

As for transcendental numbers, I will just say that all transcendental numbers are irrational and thus any transcendental number represented in an integer base will have a non-terminating representation.