What do rationals represent?

Question

While learning about the construction of number systems, I realized that I had many misunderstandings of crucial concepts which I was learning intuitively. I recently learned about the construction of the rationals as a set of equivalence classes of ordered pairs of integers (m,n), with n ≠ 0. Also I know how additon and multiplication are defined on that set. I know that the rationals exhibit good mathematical properties but what I wonder is what is their interpertation? What do they represent? For example what a half represents? Should I be content with the explanation that is given to 5 year olds about dividing pizza in half and taking one piece of them? Is there a defined real world meaning for rationals?

Answer 1

The rigorous mathematical definition of rational numbers is that they are equivalence classes of ordered pairs of integers $(m,n)$ with $n\neq 0$ for the equivalence relation $$(m,n)\equiv (o,p)\iff mp = no.$$

I don't really understand how you first wrote the rigorous mathematical definition of $\mathbb Q$, then asked is there a rigorous mathematical definition of rational numbers...

Another way you can look at the rational numbers is that they are, given the set of natural numbers and the operations of addition and multiplication, the smallest set containing $\mathbb N$ and satisfying all the axioms for a field (they are the "smallest field containing the natural numbers").

---

As for the real world meaning of rationals... In a strict sense of the word, no mathematical concept has a direct "meaning" in nature, but I think that generally, the natural use of fractions comes from the fact that nature itself contains them. Your example with a pie is a completely legitimate way of justifying why rationals are used in real life: it is because there exists such a thing as "half a pie."

Answer 2

I differ slightly from the other comments and answers in that I don't think the real-world meaning you gave is different from the formal meaning. Maybe we can break it into steps. What I hope is that each step can be deduced from the next.

1. Two-thirds of a pizza is what you get when you divide a pizza into three equal pieces, then take two of them.

2. Two-thirds of a pizza has the property that, given three of them, you have two pizzas altogether. This property characterizes "two-thirds of a pizza." That is, if you have some percentage of a pizza, such that when you take three, you can make exactly two whole pizzas, then you have two-thirds of a pizza.

3. "Two-thirds of a pizza" is the unique solution to the equation 3x = 2 when x is interpreted as "numbers of pizzas."

4. Whatever formal model (I am using the word "model" in its everyday sense, not its mathematical sense) you have in mind for the integers, you can extend it to a model in which all equations of the form ax = b with $a \neq 0$ have solutions by using the "equivalence class trick" that you mentioned. In that larger model, the equation $3x = 2$ has a unique solution which we call 2/3.

In summary, I wouldn't take (the equivalence class of the pair (2, 3)) to be very different from (the unique solution to $3x = 2$) which is directly linked to pizza-land when $x$ is interpreted as pizza.