(related to Proposition: Distributivity Law For Rational Numbers)

By the definition of rational numbers, the rational numbers \(x,y,z \in \mathbb Q\) are identified by pairs \(x:=\frac ab\), \(y:=\frac cd\) and \(z:=\frac ef\) of some integers \(a,b,c,d,e,f\in\mathbb Z\) with \(b\neq 0\), \(d\neq 0\), and \(f\neq 0\). Because multiplying rational numbers is commutative, it is sufficient to show the left-distributivity law \[x\cdot(y+z)=(x\cdot y)+(x\cdot z).\]

The left-distributivity law can be proven using the following mathematical definitions and concepts: * definition of adding rational numbers, * definition of multiplying rational numbers, * distributivity law for integers, * associativity law for multiplying integers, and * commutativity law for multiplying integers. The proof follows:

\[\begin{array}{ccll} x\cdot(y+z)&=&\frac ab\cdot\left(\frac cd + \frac ef\right)&\text{by definition of rational numbers}\\ &=&\frac ab\cdot\frac{cf+ed}{df}&\text{by definition of adding rational numbers}\\ &=&\frac{a(cf+ed)}{b(df)}&\text{by definition of multiplying rational numbers}\\ &=&\frac{a(cf)+a(ed)}{b(df)}&\text{by distributivity law of integers}\\ &=&\frac{(ac)f+(ae)d}{bdf}&\text{by associativity law for multiplying integers}\\ &=&\frac{(ac)f}{(bd)f}+\frac{(ae)d}{bdf}&\text{by definition of adding rational numbers}\\ &=&\frac{(ac)f}{(bd)f}+\frac{(ae)d}{(bf)d}&\text{by commutativity law for multiplying integers}\\ &=&\frac{ac}{bd}+\frac{ae}{bf}&\text{by definition of rational numbers}\\ &=&(x\cdot y)+(x\cdot z)&\text{by definition of rational numbers} \end{array} \]

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  1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013