The rational numbers: fractions and their operations

Fractions and their equivalence

In the lesson on number systems, the rational numbers were presented as the fractions of integers with nonzero denominator, and it was pointed out that a fraction cannot be defined as a division, because division is precisely what the integers do not allow. This lesson carries out the construction announced there. Its starting point is the integers of the lesson on the integers, with their sum, their product, their order and a property that will be used at every turn: the product of two nonzero integers is nonzero. It follows that a nonzero factor can be cancelled, because if $ab = ac$ with $a \neq 0$, then $a(b - c) = 0$, and therefore $b - c = 0$.

Definition (fraction). A fraction is an ordered pair $(p, q)$ of integers with $q \neq 0$, written $\frac{p}{q}$; the integer $p$ is its numerator and $q$ its denominator. The fractions $\frac{p}{q}$ and $\frac{r}{s}$ are said to be equivalent when their cross products coincide, that is, when $ps = qr$.

Thus $\frac{2}{3}$ and $\frac{4}{6}$ are equivalent, because $2 \cdot 6 = 3 \cdot 4$, whereas $\frac{2}{3}$ and $\frac{3}{4}$ are not, because $2 \cdot 4 \neq 3 \cdot 3$. The definition is not arbitrary. If fractions are to be quotients, the equality $\frac{p}{q} = \frac{r}{s}$ must be equivalent, after multiplying both sides by $qs$, to the equality $ps = qr$; the definition takes as its starting point this consequence, which contains only products of integers and therefore makes sense before division exists. Now, for equivalence to be able to play the role of an equality, it must behave like one: it must be an equivalence relation, in the sense of the lesson on sets, relations and functions.

If this is so, the general results of the lesson on sets, relations and functions hold for the equivalence of fractions, and there is no need to repeat them: by the theorem on equivalences and partitions, the classes of equivalent fractions form a partition of the set of fractions, so that each fraction belongs to exactly one class and two different classes have no fraction in common. Each class is a number.

Definition (rational number). A rational number is the equivalence class of a fraction, that is, the set of all fractions equivalent to it, and any one of them represents it; in the notation of the lesson on sets, relations and functions, the rational number represented by $\frac{p}{q}$ is the class $C\left( \frac{p}{q} \right)$, and here it is written simply $\frac{p}{q}$. The set of rational numbers, which is the quotient set of the fractions by their equivalence, is denoted by $\mathbb{Q}$. One writes $\frac{p}{q} = \frac{r}{s}$ when both fractions represent the same rational number, that is, when they are equivalent, as the lemma on equivalence classes proved there states.

Each integer $p$ is identified with the rational number represented by $\frac{p}{1}$. The identification does not confuse different integers, because $\frac{p}{1} = \frac{r}{1}$ is equivalent to $p \cdot 1 = 1 \cdot r$, that is, to $p = r$; in this sense, $\mathbb{Z} \subseteq \mathbb{Q}$. It will be checked later that the operations on rational numbers, applied to the integers, give the usual ones. The practical rule most often used to change the fraction without changing the number is the following.