# Proof

(related to Proposition: Addition Of Rational Numbers)

Let $$x$$ and $$y$$ be rational numbers. By definition, it means that they are represented by some integers $$x=\frac ab$$, $$y=\frac cd$$, $$a,c\in \mathbb Z$$, $$b,d\in \mathbb Z\setminus\{0\}$$.

Note that $$ad$$, $$cb$$ and $$bd$$ are all integers, as they are the products of the respective integers $$a,d$$, $$c,d$$ and $$b,d$$. Also note that $$ad + cb$$ is an integer, as it is the sum of the integers $$ad$$ and $$cb$$. Moreover, we have $$bd\neq 0$$, since both $$b\neq 0$$ and $$d\neq 0$$ and their product cannot equal $$0$$, since integers form an integral domain. Therefore, the sum $\begin{array}{rcl} x+y=\frac {ad + cb}{bd}. \end{array}$ exists, because it denotes some new rational number, as it is represented by the integers $$ad + cb$$ and $$bd\neq 0$$.

It remains to be shown that the addition of rational numbers does not depend on the specific representatives of the numbers $$x$$ and $$y$$. Suppose, we have different representatives $\begin{array}{rcl} x=\frac{a_1}{b_1}=\frac{a_2}{b_2},~y=\frac{c_1}{d_1}=\frac{c_2}{d_2}.&&(*) \end{array}$ It follows from the definition of rational numbers that $$a_1=\frac{a_2b_1}{b_2}$$ and $$c_1=\frac{c_2d_1}{d_2}$$. We have to show that $x+y=\frac {a_1d_1 + c_1b_1}{b_1d_1}=\frac {a_2d_2 + c_2b_2}{b_2d_2}.$ In the following, we will use the following mathematical definitions and concepts: * definition of rational numbers, * definition of adding rational numbers (hypothesis), * commutativity law of multiplying integers , * integer one is neutral with respect to the multiplication of integers, * distributivity law for integers, and * multiplication of rational numbers:

$\begin{array}{rcll} x+y&=&\frac{a_1}{b_1}+\frac{c_1}{d_1}&\text{by definition of rational numbers}\\ &=&\frac{a_1d_1+c_1b_1}{b_1d_1}&\text{by hypothesis}\\ &=&\frac{a_1d_1\cdot 1+c_1b_1\cdot 1}{1\cdot b_1d_1}&\text{because }1\text{ is neutral with respect to multiplication of integers}\\ &=&\frac{(a_1d_1+c_1b_1)\cdot 1}{1\cdot b_1d_1}&\text{by distributivity law for integers}\\ &=&\frac{a_1d_1+c_1b_1}{1}\cdot \frac 1{b_1d_1}&\text{by definition of multiplying rational numbers}\\ &=&(a_1d_1+c_1b_1)\cdot \frac 1{b_1d_1}&\text{by definition of rational numbers}\\ &=&\left(\frac{a_2b_1}{b_2}d_1+\frac{c_2d_1}{d_2}b_1\right)\cdot \frac 1{b_1d_1}&\text{according to }(*)\\ &=&\left(\frac{a_2b_1\cdot d_1}{b_2\cdot 1}+\frac{c_2d_1\cdot b_1}{d_2\cdot 1}\right)\cdot \frac 1{b_1d_1}&\text{by definition rational numbers and of multiplying them}\\ &=&\left(\frac{a_2b_1d_1}{b_2}+\frac{c_2d_1b_1}{d_2}\right)\cdot \frac 1{b_1d_1}&\text{because }1\text{ is neutral with respect to multiplication of integers}\\ &=&\left(\frac{a_2b_1d_1d_2+c_2d_1b_1b_2}{b_2d_2}\right)\cdot \frac 1{b_1d_1}&\text{by hypothesis}\\ &=&\left(\frac{a_2d_2b_1d_1+c_2b_2b_1d_1}{b_2d_2}\right)\cdot \frac 1{b_1d_1}&\text{by commutativity law for multiplying integers}\\ &=&\left(\frac{(a_2d_2+c_2b_2)b_1d_1}{b_2d_2}\right)\cdot \frac 1{b_1d_1}&\text{by distributivity law for integers}\\ &=&\frac{a_2d_2+c_2b_2}{b_2d_2}\cdot \frac {b_1d_1\cdot 1}{b_1d_1}&\text{by definition of multiplying rational numbers}\\ &=&\frac{a_2d_2+c_2b_2}{b_2d_2}\cdot \frac {b_1d_1}{b_1d_1}&\text{because }1\text{ is neutral with respect to multiplication of integers}\\ &=&\frac{a_2d_2+c_2b_2}{b_2d_2}\cdot \frac {1}{1}&\text{by definition of rational numbers}\\ &=&\frac{(a_2d_2+c_2b_2)\cdot 1}{b_2d_2\cdot 1}&\text{by definition of multiplying rational numbers}\\ &=&\frac{a_2d_2+c_2b_2}{b_2d_2}&\text{because }1\text{ is neutral with respect to multiplication of integers}\\ &=&\frac{a_2}{b_2}+\frac{c_2}{d_2}&\text{by hypothesis}\\ \end{array}$

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### References

#### Bibliography

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