Def/Rational number

 =Construction=

Consider the Cartesian product $\ZZ^2$, consisting of ordered pairs $(x,y) \in \ZZ$. Define a subset $X \subset \ZZ^2$ by: $$X = \{ (x,y) \mbox{ such that } y \neq 0 \}.$$ Define a relation on $X$ by: $(x,y) \sim (u,v)$ if $xv = yu$. We prove that this is an equivalence relation:
 * Reflexive: If $(x,y) \in X$, then $(x,y) \sim (x,y)$ since $xy = yx$ by the commutativity of multiplication.
 * Symmetric: If $(x,y) \in X$, and $(u,v) \in X$, and $(x,y) \sim (u,v)$, then $xv = yu$.  It follows that $(u,v) \sim (x,y)$ since $uy = vx$ by the commutativity of multiplication of integers.
 * Transitive: Suppose that $(x,y) \in X$, $(u,v) \in X$, and $(s,t) \in X$.  Suppose that $(x,y) \sim (u,v)$ and $(u,v) \sim (s,t)$.  Then $xv = yu$ and $ut = vs$.  It follows that $xvt = yut = yvs$.  Since $v \neq 0$, it follows that $xt = ys$.  Therefore $(x,y) \sim (s,t)$.

Hence the relation on $X$ is an equivalence relation. Define the set of rational numbers, denoted $\QQ$, to be the resulting set of equivalence classes.

If $(x,y) \in X$, one writes $\frac{x}{y}$ for the equivalence class containing $(x,y)$. Thus, fractions, in this construction, are defined as equivalence classes of pairs of integers. Familiar identities arise, for example: $$\frac{2}{4} = \frac{1}{2}, \mbox{ since } (2,4) \sim (1,2), \mbox{ since } 2 \cdot 2 = 4 \cdot 1.$$

=Addition and subtraction=

Addition of rational numbers can be defined as follows: given rational numbers $\frac{x}{y}$ and $\frac{u}{v}$, one defines the sum by: $$\frac{x}{y} + \frac{u}{v} = \frac{xv + yu}{yv}.$$ We must show that this definition is well-defined.

=Multiplication=

=Division=

=Reduction=