(related to Proposition: Addition 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 \[(x+y)+z=x+(y+z)\] we replace the symbols \(x,y,z\) by their representatives \(\frac ab,\frac cd,\frac ef\), and use the following mathematical definitions and concepts: * definition of adding rational numbers, * distributivity law for integers, * associativity law for multiplying integers, and * commutativity law for multiplying integers. \[\begin{array}{rcll} (x+y)+z&=&\left(\frac ab+\frac cd\right)+\frac ef&\text{by definition of rational numbers}\\ &=&\frac {ad + cb}{bd}+\frac ef& \text{by definition of adding rational numbers}\\ &=&\frac{(ad + cb)f + e(bd)}{(bd)f}&\text{by definition of adding rational numbers}\\ &=&\frac{adf + cbf + e(bd)}{(bd)f}&\text{by distributivity law for integers}\\ &=&\frac{adf + cbf + ebd}{bdf}&\text{by associativity of multiplying integers}\\ &=&\frac{adf + cfb + edb}{bdf}&\text{by commutativity of multiplying integers}\\ &=&\frac{adf + (cf + ed)b}{bdf}&\text{by distributivity law for integers}\\ &=&\frac{a(df) + (cf + ed)b}{b(df)}&\text{by associativity of multiplying integers}\\ &=&\frac ab+\frac{cf + ed}{df}&\text{by definition of adding rational numbers}\\ &=&\frac ab+\left(\frac cd + \frac ef\right)&\text{by definition of adding rational numbers}\\ &=&x+(y+z)&\text{by definition of rational numbers} \end{array}\]

Thank you to the contributors under CC BY-SA 4.0!




  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.