(related to Proposition: Multiplication of Rational Numbers Is Associative)

Let \(x,y,z\in\mathbb Q\) be rational numbers, which by definition means that each rational number is an equivalence class of ordered pairs of integers represented by some integers \(a,b,c,d,e,f\in\mathbb Z\), with \(b\neq 0,d\neq 0,f\neq 0\)1:

\[\begin{array}{ccc}x:=\frac ab,&y:=\frac cd,&z:=\frac ef.\end{array}\]

In order to show the law \[(x\cdot y)\cdot z=x\cdot (y\cdot z)\] we use the following mathematical definitions and concepts: * definition of rational numbers. * definition of multiplying rational numbers, and * associativity law for multiplying integers. \[\begin{array}{rcll} (x\cdot y)\cdot z&=&\left(\frac ab\cdot \frac cd\right)\cdot \frac ef&\text{by definition of rational numbers}\\ &=&\frac {ac}{bd}\cdot \frac ef& \text{by definition of multiplying rational numbers}\\ &=&\frac {(ac)e}{(bd)f}& \text{by definition of multiplying rational numbers}\\ &=&\frac {a(ce)}{b(df)}& \text{by associativity of multiplying integers}\\ &=&\frac ab\cdot \frac{ce}{df}& \text{by definition of multiplying rational numbers}\\ &=&\frac ab\cdot \left(\frac cd\cdot \frac ef\right)& \text{by definition of multiplying rational numbers}\\ &=&x\cdot(y\cdot z)&\text{by definition of rational numbers} \end{array}\]

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


  1. Note that the symbol "\(0\)" denotes the zero defined for integers, and not the zero defined for rational numbers.