Definition 4

The field of fractions of the ring $\mathbf{Z}$ of rational integers is called the field of rational numbers and is denoted by $\mathbf{Q}$. The elements of $\mathbf{Q}$ are called rational numbers.

Every rational number is thus of the form $a/b$ where $a$ and $b$ are rational integers with $b \neq 0$ (and we may even take $b > 0$ as the relation

proves). $\mathbf{Q}_+$ is used to denote the set of rational numbers of the form $a/b$ with $a \in \mathbf{N}$ and $b \in \mathbf{N}^*$.

We have the relations:

(1)

(2) (3) (4) (5)

The first two follow from the formulae $a/b + a'/b' = (ab' + a'b)/bb'$, $(a/b)(a'/b') = aa'/bb'$, $0 \in \mathbf{Q}_+$, $1 \in \mathbf{Q}_+$ and the fact that $\mathbf{N}$ is stable under addition and multiplication and $\mathbf{N}^*$ under multiplication. Obviously $0 \in \mathbf{Q}_+$, whence $0 \in (-\mathbf{Q}_+)$; let $x$ be in $\mathbf{Q}_+ \cap (-\mathbf{Q}_+)$; then there exist positive integers $a, b, a', b'$ with $b \neq 0,\ b' \neq 0$ and $x = a/b = -a'/b'$; then $ab' + ba' = 0$, whence $ab' = 0$ (for $ab' \geq 0$ and $ba' \geq 0$) and therefore $a = 0$ since $b' \neq 0$; in other words, $x = 0$. Finally, obviously $\mathbf{N} \subset \mathbf{Z}$ and $\mathbf{N} \subset \mathbf{Q}_+$. Conversely, if $x$ belongs to $\mathbf{Z} \cap \mathbf{Q}_+$, it is a rational integer; there exist two rational integers $a$ and $b$ with $a \geq 0,\ b > 0$ and $x = a/b$, whence $a = bx$; if $x \notin \mathbf{N}$, then $-x > 0$, whence $-a = b(-x) > 0$ and consequently $a < 0$ contrary to the hypothesis.

Given two rational numbers $x$ and $y$, we write $x \leq y$ if $y - x \in \mathbf{Q}_+$. It is easily deduced from formulae (1), (3) and (4) that $x \leq y$ is a total ordering on

Q, from (5) that this relation induces the usual order relation on Z. Finally, from (1) it follows that the relations $x \leq y$ and $x' \leq y'$ imply $x + x' \leq y + y'$ and from (2) that the relation $x \leq y$ implies $xz \leq yz$ for all $z \geq 0$ and $xz \geq yz$ for $z \leq 0$ (cf. VI, § 2, no. 1).

Let $x$ be a rational number. $x$ is said to be positive if $x \geq 0$, strictly positive if $x > 0$, negative if $x \leq 0$ and strictly negative if $x < 0$.† The set of positive rational numbers is $\mathbf{Q}_+$ and that of negative numbers is $-\mathbf{Q}_+$. If $\mathbf{Q}^*$ denotes the set of non-zero rational numbers, the set $\mathbf{Q}_+^*$ of strictly positive numbers is equal to $\mathbf{Q}^* \cap \mathbf{Q}_+$ and $-\mathbf{Q}_+^*$ is the set of strictly negative numbers.

The sets $\mathbf{Q}_+^*$ and $\{1, -1\}$ are subgroups of the multiplicative group $\mathbf{Q}^*$. Every rational number $x \neq 0$ can be expressed in one and only one way in the form $1.y, (-1).y$, where $y > 0$; hence the multiplicative group $\mathbf{Q}^*$ is the product of the subgroups $\mathbf{Q}_+^*$ and $\{1, -1\}$, the component of $x$ in $\mathbf{Q}_+^*$ is called the absolute value of $x$ and is denoted by $|x|$; the component of $x$ in $\{-1, 1\}$ (equal to 1 if $x > 0$, to $-1$ if $x < 0$) is called the sign of $x$ and denoted by $\operatorname{sgn}\, x$.

Usually these two functions are extended to the whole of $\mathbf{Q}$ by setting $|0| = 0$ and $\operatorname{sgn}\, 0 = 0$.