If a number irrational, can it be rational in some rational-based number system?

Question

The number $\frac{22}{7}$ is irrational in our base-$10$ system, but in, say, base-$14$, it is rational (it comes out to $3.2$ in that system).

It's easy for fractions that are irrational as decimals, as you can just represent them in a base that's double the denominator of the fraction. However, what if I have a number like $\pi$, or $\log(2)$?

For those numbers, it could easily be represented as a rational number if it is in base-($\pi\cdot 2$) or base-($\log(2)\cdot 2$), but is it possible to represent them in any rational-based number system?

Answer 1

Always refer to definitions, a number $x$ is called irrational iff $\forall p,q \in \Bbb Z : x\neq\frac pq$, that is when you can't express it as ratio of two integers, not based on how it looks using a different number system. EDIT: This means that you can never find two integers to precisely equal $\pi$ for example, $\frac 31$, $\frac{22}{7}$, $\frac{333}{106}$, $\frac{355}{113}$, $\frac{103993}{33102}$ $\dots$ won't equal $\pi$, they're all finite decimals, the real irrational $\pi$, has unending decimals.

Answer 2

Whether a number is rational or not is independent of the base in which the number may be expressed.

On the other hand the fraction $\frac ab: a,b\in \mathbb Z, b\gt 0$ may terminate or eventually recur when expressed as a "decimal" (Hardy could find no better word - Hardy and Wright, Introduction to the Theory of Numbers). Choosing the base $b$ automatically ensures that the expression terminates. This doesn't work if $b=1$, but then you have an integer anyway.

Answer 3

The number $22/7$ is not irrational, regardless of the number system one uses to express it. On the other hand $\pi$ is irrational, regardless of the number system one uses to express it. They are different numbers: $22/7$ is a good rational approximation to $\pi$, but they are not equal.

It is unfortunate that students are taught to think of "nonrepetition" as the essential quality that distinguishes rational numbers from irrational numbers. It is true that if a number is rational, then its decimal representation will eventually either terminate or repeat, but it might take a very long time, and just looking at a string of digits is not enough evidence to conclude whether or not the string represents the beginning of a repeating decimal or a non-repeating decimal. The real distinction between rational and irrational numbers lies in whether it is possible to express the number as a ratio of integers. If it is possible, then the number is rational; if it is not possible, then the number is irrational. The fact that rational numbers correspond to decimal representations that terminate or repeat is an important and interesting consequence of the definition, but it is not really the essence of the distinction.

For more on this, see my answer and the discussion in the comments beneath it at https://math.stackexchange.com/a/2073186/124095.