Are elements of $\mathbb{Q}$ unique by value?

Question

I usually visualize elements $\frac{a}{b} \in\mathbb{Q}$ as $(a,b) \in \mathbb{Z}\times(\mathbb{Z}\setminus \{0\})$. By this construction, I think it's pretty clear that elements of $\mathbb{Q}$ may have the same value but have a different representation (i.e. $\frac{1}{2},\frac{2}{4}$) and therefore I consider different ordered pairs unique when I think of elements, despite having the same value. However, I find that when I think of $\mathbb{Q}$ as a whole, I tend to think of congruence classes and $\frac{1}{2}=\frac{2}{4}$.

I still think the former is more accurate, but in a general case does it really matter (if we're not specifically looking for relationships between numerators and denominators and some other factors)?

Answer 1

The set of rational numbers is not the same as $\mathbb{Z}\times(\mathbb{Z}\setminus\{0\})$, but rather $$ \mathbb{Q}=(\mathbb{Z}\times(\mathbb{Z}\setminus\{0\}))/\sim$$ where we say that $(a,b)\sim(c,d)$ if $ad=bc$.

Therefore there are infinitely many pairs $(a,b)$ corresponding to a given rational number. However, we can choose a canonical representative of each equivalence class by requiring that $b>0$ and $\gcd(a,b)=1$.

Answer 2

There is no significant difference between the two versions. You can think of $\frac{a}{b}$ as a funny way of writing $(a,b)$, stacked vertically and with a vinculum as separator, instead of stacked horizontally with a comma as separator, and with two outside parentheses. Each denotes an ordered pair.

The "value" of $(a,b)$, or of its stacked version, is the equivalence class it determines.

Answer 3

The map is not the territory.

The way you present an element is not the same way as the element itself. The fact that $\frac12$ and $\frac24$ are equal does not mean that $\frac12$ and $\frac24$ are different. These are just two ways of presenting the same number. Just as $1+2$ and $4-1$ are the same way writing $3$.

The point is that we have a canonical way of writing every rational number: $q$ can be written uniquely as $\frac rs$ where $r\in\Bbb Z$ and $s\in\Bbb N\setminus\{0\}$ are coprime. Now observe that $\frac24$ does not have this presentation, where $\frac12$ does.

Answer 4

I take this to be a philosophical question. If we have two different formalisms for a thing, which one is "really" that thing? Or are different formalisms interchangeable, as long as the things they designate have the same behavior?

Set theory is full of problems like this. The natural number $3$ is not the same set as the integer $3$ which is not the same set as the real number $3$. There are "canonical injections" that allow mathematicians to ignore these set-theoretical anomalies. Yet in set theory it's axiomatic (literally!) that two sets are the same if and only if they have the same elements. It's standard in math to regard this issue as unimportant and never mention it. That is as it should be, because this is a philosophical problem and not a mathematical one.

The philosopher Paul Benecerraf wrote a famous essay called What Numbers Could Not Be that discusses this problem.

You can formalize the idea that "two things are the same thing if they have the same behaviors" through Category theory; and I have heard that this issue relates to the philosophical doctrine of mathematical structuralism.

Barry Mazur wrote an essay on this topic, titled When is one thing equal to some other thing? It's a far more worthwhile read than anything I could say on this topic.