>the mistaken belief that an irrational number is one whose decimal representation doesn't repeat
...which is true for any base-n representation where n is a natural (even rational) number. And that's kind of implied most of the time, so it seems like a useful definition. Where would this lead to problems?
I think the problem is that the definition, while valid, makes you hyperfocus on irrational numbers this way.
Seldom do we prove that a number is irrational by inspecting its decimal expansion. This would be in most cases a very unnatural proof. Since irrationality is a negative property (meaning, one arising out of a negation: the number is not a ratio), most of the time you prove it by contradiction. But people who just know the "digits don't repeat" definition expect us to somehow be able to list all of the digits of an irrational number and show that this infinite list doesn't repeat, which is, of course, an impossible task.
It's _true_, but it's not good as a definition, because it's hard to reason about. It drags in all sorts of contingent facts about base-10 representations that are not usually of interest.
The equivalent property, that a number is irrational if it's not equal to m÷n for any integers m and n, is much simpler. So we use that as the definition, and from that simple and intrinsic definition, we prove the _theorem_ that the decimal representation of an irrational number never repeats.
My phrasing was bad. I should have said "the mistaken belief that an irrational number is * defined to be * one whose decimal representation doesn't repeat”.
Usually we define it like this: an irrational number is one that isn't a quotient of two integers. Starting from that definition, we then prove the _theorem_ that the decimal representation a number repeats if and only if the number is rational.
It's much easier to start from the intrinsic properties, and use those to prove things about the representation, than the other way around. But if you don't distinguish the representation from the thing itself, you can't tell which way you are going.
The proof that the usual definition is equivalent to the representation is fairly straightforward and easy, no matter which side you picked as the definition. And once the equivalence is established, all other proofs proceed naturally. It therefore matters a lot that we pick one as a definition and know which one we picked, but not so much which one we picked.
Now in fact the quotient definition is by far more interesting mathematically. There is also a clear foundational reason to prefer it, namely that you can easily construct and prove things about the rational numbers long before you construct the real numbers. However it is unlikely that anyone who is confused about the definition of a rational number has a clear understanding of how the reals are constructed, so that is not a particularly important consideration for them.
Furthermore the fact that foundational considerations argue for one construction over another has little bearing on what is pedagogically preferable. As a famous example, the easiest way to rigorously define logarithms is through the integral of 1/x. However explaining logarithms that way to someone who doesn't know them is a pedagogical disaster.
Thank you! This thread is full of people insisting on something wrong because they were taught incorrectly; an irrational number is defined in terms of integer ratios for a reason.
It's not like those people haven't worked with an irrational base before, either! Radians have an irrational base. When we talk about 2π radians, or 1/4π radians, that's exactly what we're doing.
It was unclear; I understand rational numbers to be ratios, and irrationals to be inexpressible as fractions, and I see an almost direct connection between that and digital representation in rational bases, so it seemed deeply confusing to see a consequence of the definition of rationals being refuted. The only way out was an irrational base.
...which is true for any base-n representation where n is a natural (even rational) number. And that's kind of implied most of the time, so it seems like a useful definition. Where would this lead to problems?