Is it possible that $\pi$ is finite in other numerical bases?

Question

In base $\pi$, the number $\pi$ is $1\cdot \pi^1 + 0\cdot \pi ^ 0 $, which is equal to $10$.

So, is $\pi$ an irrational number in all bases or not?

Answer 1

Generally when people talk about bases, they mean integer bases. For any integer base, $\pi$ has an infinite expansion. This is because if you could write $\pi$ as a finite expansion in base $b$ (an integer), then it has the form $$\sum_{n\in\mathbb{Z}}a_nb^n$$ where only finitely many $a_n$'s are nonzero (every $a_n$ is an integer less than $b$). But this is then just a finite sum of rational numbers, hence a rational number.

Edit: As Hagen said, the notion of rationality is independent of a "base". However, you can prove that a number is irrational if and only if its expansion in every integer base is infinite. This is because for any rational number $a/b$, you can always write the integer $a$ as a finite expansion $a_nb^n + a_{n-1}b^{n-1} + \cdots + a_1b + a_0$, and hence $a/b$ has the expansion $a_nb^{n-1} + a_{n-1}b^{n-2} + \cdots + a_1 + a_0b^{-1}$.

Answer 2

The concept of rationality/irrationality (or in the case of $\pi$ also: transcendence vs algebraicity). is defined without reference to bases: A number is rational iff there exists a nonzero integer such that multiplying the given number with this integer results in an integer.

Answer 3

The usual definitions of rational and irrational have nothing to do with the decimal representation of a number. Rather, a number is rational if it is the ratio of two positive integers, and irrational otherwise.

Using this definition, it turns out that a number is rational if and only if its decimal (base $10$) expansion is eventually repeating. This fact is true for any positive integer base greater than $1$, not just base $10$.

What about base $\boldsymbol{\pi}$?

Normally when we talk about bases, we mean a positive integer base greater than $1$. But of course if you allow an irrational base, then an irrational number no longer necessarily has a nonrepeating representation. Of course, you then have to define what exactly you mean by the digit representation of a number. What digits are allowed? And how do you choose which expansion to use if multiple expansions are possible?