Definition: Subtraction of Rational Numbers

According to the definition of rational numbers, we can represent rational numbers \(x,y\) by two integers each, formally \(x=\frac ab\) and \(y=\frac cd\) for some integers \(a,b,c,d\in\mathbb Z\) with \(b\neq 0\) and \(d\neq 0\). The subtraction of rational numbers, written \(x-y\), is defined as the addition of the first rational number \(x\) with the inverse of the second rational number with respect to addition \((-y)\), formally \[x-y=\frac ab-\frac cd:=\frac ab+\frac {-c}d=\frac{ad-cb}{bd}=x+(-y).\]

Definitions: 1

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