Definition of Rational Numbers

Proposition: Definition of Rational Numbers

Let \((a,b),(c,d)\) be ordered pairs of integers, such that \(b\) and \(d\) do not equal the integer number zero \(0\). We consider the pairs \((a,b),(c,d)\) equivalent, if the integer products \(a\cdot d\) and \(b\cdot c\) are equal, formally

\[(a,b)\sim (c,d)\quad\Longleftrightarrow\quad a\cdot d = b\cdot c.\]

The relation “\(\sim\)” defined above is an equivalence relation, i.e. for a given ordered pair \((a,b)\in\mathbb Z\times\mathbb Z\setminus\{0\}\), we can consider a whole set of ordered pairs \((c,d)\in\mathbb Z\times\mathbb Z\setminus\{0\}\) equivalent to \((a,b)\):

\[x:=\{(c,d)\in\mathbb Z\times\mathbb Z\setminus\{0\}:\quad( c, d )\sim ( a, b )\}\quad\quad ( * )\]

The set¹ \(x\) is called a rational number. We say that the ordered pair \((a,b)\in\mathbb Z\times\mathbb Z\setminus\{0\}\) is representing the rational number \(x\). It is called a fraction or a ratio. Usually, we denote the fraction \((a,b)\) by \(\frac ab\). The integer \(a\) is called the nominator and the integer \(b\) is called the denominator of the fraction \(\frac ab\). The set of all rational numbers is denoted by \(\mathbb Q\).

A more common notation for the ordered pairs in \( ( * ) \) using ratios is

\[x:=\left\{\frac cd: \quad \frac cd\sim \frac ab,~b,d\neq 0\right\}.\]

¹ Please note that rational numbers are in fact sets. The set \(\mathbb Q\) is the set of “all fractions”: the same rational number can be represented by different fractions, especially if we multiply the nominators and denominators by a non-zero integer, e.g. \(\frac 12=\frac 24=\frac 5{10}=\ldots\). Since all these different fractions are equal, they represent the same rational number.