(related to Proposition: Existence of Rational Zero (Neutral Element of Addition of Rational Numbers))

By definition of rational numbers \(x\in\mathbb Q\), the rational number \(x\in\mathbb Q\) can be represented by a pair of integers \(a,b\in\mathbb Z\): \[x:=\frac ab,\quad b\in\mathbb Z\setminus\{0\}.\] Since \(0\in\mathbb Z\), i.e. the integer zero exists, the (rational) zero \(0\in\mathbb Q\) also exists, because it can be represented e.g. by the a pair of two integers: the (integer) zero and an arbitrary integer \(d\neq 0\): \[0=0_{\in\mathbb Q}:=\frac {0_{\in\mathbb Z}}{d},\quad d\in\mathbb Z\setminus\{0\}.\]

We will show that the (rational) zero is neutral with respect to the addition of rational numbers by virtue of the following mathematical definitions and concepts: * definition of addition of rational numbers "\( + \)", * \(\mathbb Z\) is an integral domain, in particular \(0_{\in\mathbb Z}\) is its only zero divisor, and * integer \(0_{\in\mathbb Z}\) is neutral with respect to addition of integers:

\[\begin{array}{rcll} x + 0&=&\frac ab+\frac 0d&\text{by definition of rational numbers}\\ &=&\frac{ad+0b}{bd}&\text{by definition of addition of rational numbers}\\ &=&\frac{ad+0}{bd}&\text{because 0 is the only zero divisor of integers}\\ &=&\frac{ad}{bd}&\text{because 0 is neutral with respect to the addition of integers}\\ &=&\frac{a}{b}&\text{because }\mathbb Z\text{ is an integral domain}\\ &=&x&\text{by definition of rational numbers} \end{array} \]

It remains to be shown that also the equation \(0+x=x\) holds for all \(x\in\mathbb Q\). It follows immediately from the commutativity of adding rational numbers.

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