# Proposition: Multiplication Of Rational Numbers

According the definition of rational numbers, we can identify rational numbers $$x,y \in \mathbb Q$$ with equivalence classes $$x:=\frac ab$$, $$y:=\frac cd$$ for some integers $$a,b,c,d\in\mathbb Z$$ with $$b\neq 0$$ and $$d\neq 0$$.

Based on the multiplication of integers, we define a new multiplication operation "$$\cdot$$" for all rational numbers by setting

$\begin{array}{ccl} x\cdot y=\frac ab \cdot \frac cd &:=& \frac{a\cdot c}{b\cdot d}. \end{array}$

where $$\frac{ac}{bd}$$ is also a rational number, called the product of the rational numbers $$x$$ and $$y$$. The product 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

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

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### References

#### Bibliography

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