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In high school, in definition of rational number we used to say,

"A number, which can be written in the form $\frac{p}{q}$, where $p,q(\neq0)\in\Bbb{Z}$ is called rational number."

But now I am realising that this is not a definition but it is a characterisation of rational numbers. Because the above definition starts with "a number", which means we have already defined real numbers before giving this definition.

So how to really define rational numbers? One way we have constructed $\Bbb{Q}$ as quotient field of $\Bbb{Z}$. But this is a long construction. So please give little clarification that what should be an answer if one asks "define rational number."

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    $\begingroup$ Yeah, quotient field is the way we usually do this. Often, it is taught before the notion of a quotient field, and then the definition is explicitly in terms of equivalence classes of pairs $(p,q)$, which is really the same as a quotient field. $\endgroup$ Commented Jun 22, 2020 at 4:34
  • $\begingroup$ im not understanding you can just change the wording of your definition slightly to say "a quantity, which can be..." $\endgroup$ Commented Jun 22, 2020 at 4:34
  • $\begingroup$ I don't think that the "a number" in the definition you quote necessarily stands for "a real number". In any case it should not, because that would really be circular, as any construction of the real numbers I know relies on already knowing what a rational number is. $\endgroup$ Commented Jun 22, 2020 at 4:35
  • $\begingroup$ I doubt if there is any definition of $\mathbb Q$ simpler than the quotient of $\mathbb Z$. $\endgroup$ Commented Jun 22, 2020 at 4:38
  • $\begingroup$ I think you sum it up quite well : One can either define $\mathbb{Q}$ as an extension of $\mathbb{Z}$ formed by fractions of integers (in other words, the quotient field), or as subfield of $\mathbb{R}$. Both points of view are useful in different contexts. $\endgroup$ Commented Jun 22, 2020 at 4:39

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You have to decide where to start, in other words you have to decide what your starting axioms are.

Starting with the real numbers is perfectly acceptable, they have pretty straightforward axioms (in summary: the real numbers are a complete ordered field). And then you can really define rational numbers inside the real numbers: before that you have to first define the integers $\mathbb Z$ inside the real numbers, and once you've done that then the definition of $\mathbb Q$ that you wrote makes perfect sense.

Or you can start with Peano's axioms for the natural numbers, from there build up the integers, and from there build up the rational numbers.

If you really follow each of these through step by step then yes, they are long constructions. If you want a really long construction, start instead from axioms of set theory, next build up the natural numbers, and then continue as before.

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