# Proposition: Addition Of Rational Numbers

Any rational number $$x\in\mathbb Q$$ is by the corresponding proposition an equivalence class. $x:=\frac ab,$ where $$a$$ and $$b\neq 0$$ denote integers representing the equivalence class $$x$$. Given two rational numbers $$x:=\frac ab$$, $$y:=\frac cd$$, $$a,c\in \mathbb Z$$, $$b,d\in \mathbb Z\setminus\{0\}$$, the addition of rational numbers is defined using the multiplication of the integers $$a\cdot d$$, $$b\cdot d$$, and $$c\cdot b$$ and the addition of the integers $$a\cdot d+c\cdot b$$:

$\begin{array}{rcl} x+y&:=&\frac {a\cdot d + c\cdot b}{b\cdot d}, \end{array}$

where $$\frac {ad + cb}{bd}$$ is also a rational number, called the sum of the rational numbers $$x$$ and $$y$$. The sum exists and is well-defined, i.e. it does not depend on the specific representatives $$\frac ab$$ and $$\frac cd$$ of $$x$$ and $$y$$.

Proofs: 1

Definitions: 1
Proofs: 2 3 4 5 6 7 8 9 10 11 12 13 14
Propositions: 15 16 17 18 19 20 21 22 23

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### References

#### Bibliography

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