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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. It is also important not to mix up the set of all rational numbers \(\mathbb Q\) with the set of “all fractions”. This is because 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. \(\frac 12=\frac 24=\frac 5{10}=\ldots\)

| | | | | Contributors: bookofproofs | References: [696]

1.Proof: (related to "Definition of Rational Numbers")

2.Proposition: Addition Of Rational Numbers

3.Proposition: Multiplication Of Rational Numbers

4.Definition: Subtraction of Rational Numbers

5.Proposition: Distributivity Law For Rational Numbers


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Bibliography (further reading)

[696] Kramer Jürg, von Pippich, Anna-Maria: “Von den natürlichen Zahlen zu den Quaternionen”, Springer-Spektrum, 2013

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