(related to Proposition: Addition of Rational Numbers Is Cancellative)

Because the addition of rational numbers is commutative, it is without loss of generality sufficient to show the right cancellation property, i.e. \[x+z=y+z\Longleftrightarrow x=y,~~~~~~(x,y,z\in\mathbb Q).\]

By definition of rational numbers, 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:

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

Assume the equation \(x+z=y+z\) holds. We have to show that \(x=y\), and we will do it by virtue of the following mathematical definitions and concepts: * definition of adding rational numbers, * cancellation law for multiplying integers, * distributivity law for integers, * associativity law for multiplying integers, * commutativity law for multiplying integers, and * cancellation law for adding integers. \[\begin{array}{rcll} x+z=y+z&\Leftrightarrow&\frac ab+\frac ef=\frac cd+\frac ef&\text{by definition of rational numbers}\\ &\Leftrightarrow&\frac {af+eb}{bf}+\frac {cf+ed}{df}& \text{by definition of adding rational numbers}\\ &\Leftrightarrow&(af+eb)df=(cf+ed)bf&\text{by definition of rational numbers}\\ &\Leftrightarrow&(af+eb)d=(cf+ed)b&\text{by cancellation law for multiplying integers}\\ &\Leftrightarrow&(af)d+(eb)d=(cf)b+(ed)b&\text{by distributivity law for integers}\\ &\Leftrightarrow&afd+ebd=cfb+edb&\text{by associativity law for multiplying integers}\\ &\Leftrightarrow&adf+ebd=cbf+ebd&\text{by commutativity law for multiplying integers}\\ &\Leftrightarrow&adf=cbf&\text{by cancellation law for adding integers}\\ &\Leftrightarrow&ad=cb&\text{by cancellation law for multiplying integers}\\ &\Leftrightarrow&\frac ab=\frac cd&\text{by definition of rational numbers}\\ &\Leftrightarrow&x=y&\text{by definition of rational numbers}\\ \end{array}\]

Altogether, we have shown that the addition of integers is cancellative \[ x+z=y+z\Rightarrow x=y,~~~~~~(x,y,z\in\mathbb Q),\] and its conversion \[x=y\Rightarrow x+z=y+z,~~~~~~(x,y,z\in\mathbb Q).\]

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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.