Definition of a positive rational number

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In baby rudin page 3 it says:

Definition An ordered set is a set $ S $ in which an order is defined.
For example, $ \mathbb{Q} $ is an ordered set if $ r < s $ is defined to mean that $ s - r $ is a positive rational number.

My question is how a positive rational number is defined.
I looked for an answer in google and all I could find is that a number is positive if it is greater than zero.

Thanks

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3 Answers

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Axiom Approach

One approach is to assume as an axiom that the rationals have a subset $P$ called the positive rationals which is closed under the operations $+,\times$ and which has the properties: $$0\not\in P\\ \forall q\in\mathbb Q(q\in P\lor -q\in P\lor q=0)\\ \forall q\in\mathbb Q(q\notin P\lor -q\notin P) $$

This basically means, for every rational $q$, exactly one of the statements: $q=0,q\in P,-q\in P$ is true.

Definition Approach

Most foundation mathematics starts with the natural numbers and the Peano postulates. This particular approach uses the natural number starting at $1$. (You can start at $0$, but the approach becomes slightly more complicated.)

Then define the positive rationals as the set of all ordered pairs of natural numbers ( written $m/n$ ) modulo the equivalence relation:

$$m_1/n_1\equiv_r m_2/n_2\iff m_1n_2=m_2n_1$$

Then the rationals are defined as ordered pairs of positive rationals - written $p-q$ - modulo the equivalence:

$$p_1-q_1\equiv_n p_2-q_2\iff p_1+q_2=q_1+p_2$$

There is some slipperiness of language here. The positive rationals are not a subset of the rationals, but we can "embed" them in a natural way, so, somewhat loosely, we start re-using the term "positive rationals" to refer to rationals that correspond to actual "positive rationals."

Theoretically, we should give them different names, but in practice, there is no harm, past the definition phase.

(In this approach, we have actually defined rational number before $0$ or "integer." We can define "integers" as rational numbers of the form $n/1-m/1$ where $n,m$ are natural numbers.)

At each step, you need to show the equivalence relationships are actually equivalence relationships, then define the operations $+,\times$ in the new set, and prove the operations are well-defined, and that they agree with the values from the previous step when we do the "embedding." It's all very tedious.

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Assume that you know what is a positive integer.
Then define a rational number $r=p/q$ (where $(p,q)=1,q>0,p,q\in\mathbb Z$) is positive if $p$ is positive.

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You must first assume the existence of the natural numbers. You can either assume them to be given by God (as Kronecker quipped), or let them be given by the Zermelo-Fraenkel axioms for set theory: $0$ is identified with $\emptyset$ through the axiom which introduces the empty set ("$\exists \emptyset (x \in \emptyset \iff x \notin \emptyset)")$, and the rest of the natural numbers are given by the axiom of the infinite set ($\exists N \big(\emptyset \in N \wedge (x \in N \implies \{x\} \in N) \big)$). To clarify, identify $0$ with $\emptyset$, $1$ with $\{ \emptyset \}$, $2$ with $\big\{ \{ \emptyset \} \big\}$ and so on.

Next, introduce the integers as a quotient set. If $\Bbb N$ is the set given above, on $\Bbb N \times \Bbb N$ introduce the equivalence relation $(x,y) \sim (a,b) \iff x-y = a-b$. Each element of $\Bbb N \times \Bbb N / \sim$ (denoted by a hat above the pair) will be an integer number, for instance the integer $-2$ will be $\widehat {(3,5)} = \widehat {(148, 150)}$. In simpler words, $\widehat {(a,b)}$ will be $a-b$ but the precise way of saying it is the complicated one above because subtraction is not defined on $\Bbb N$ (what is $3-5$? not a natural number). Introduce an addition $\widehat {(a,b)} + \widehat {(x,y)} = \widehat {(a+x, b+y)}$ and a multiplication $\widehat {(a,b)} \cdot \widehat {(x,y)} = \widehat {(ax + by, ay + bx)}$. Check that with these operations $\Bbb N \times \Bbb N / \sim$ becomes a ring, denoted $\Bbb Z$. Note that $n \in \Bbb N$ can be identified with $\widehat {(n,0)} \in \Bbb Z$.

Finally, if $\Bbb Z ^* = \Bbb Z \setminus \{0\}$, define an equivalence on $\Bbb Z \times \Bbb Z ^*$ by $(a,b) \approx (x,y) \iff ay = bx$ and denote by $\Bbb Q$ the quotient set $\Bbb Z \times \Bbb Z ^* / \approx$. Informally, if we now denote these new equivalence classes by overlines, then $\overline {(a,b)}$ is what we usually write as $\frac a b$. Define addition as $\overline {(a,b)} + \overline {(x,y)} = \overline {(ay + bx, by)}$ and multiplication as $\overline {(a,b)} \cdot \overline {(x,y)} = \overline {(ax, by)}$. Check that with these operations $\Bbb Q$ becomes a field. Note that $z \in \Bbb Z$ can be identified with $\overline {(z,1)} \in \Bbb Q$ and that the multiplication and addition defined above extend the ones on $\Bbb Z$. Algebraically, $\Bbb Q$ is the fraction field of $\Bbb Z$.

To summarize, $\Bbb Q$ is a quotient set of a quotient set of pairs of natural numbers.

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