Proof

(related to Proposition: Multiplication Of Rational Numbers)

Note that since the products of integers $$ac$$ and $$bd$$ are also integers, it follows from the definition of rational numbers that for the rational numbers $$x,y \in \mathbb Q$$ with $$x:=\frac ab$$, $$y:=\frac cd$$ for some integers $$a,b,c,d\in\mathbb Z$$ with $$b\neq 0$$ and $$d\neq 0$$, also the number $x\cdot y=\frac{ac}{bd}$ is a rational number. It remains to be shown that the definition of multiplication of rational numbers is well-defined, i.e. it does not depend on the specific representatives of $$x$$ and $$y$$. Assume we have different representatives $x=\frac{a_1}{b_1}=\frac{a_2}{b_2},~y=\frac{c_1}{d_1}=\frac{c_2}{d_2}.\quad\quad( * )$ It follows from the definition of rational numbers that $$a_1=\frac{a_2b_1}{b_2}$$ and $$c_1=\frac{c_2d_1}{d_2}.$$ Therefore, we have by virtue of commutativity of multiplying integers and because integer one is neutral with respect to the multiplication of integers that $\begin{array}{rcll} x\cdot y&=&\frac{a_1}{b_1}\cdot\frac{c_1}{d_1}&\text{by definition of rational numbers}\\ &=&\frac{\frac{a_2b_1}{b_2}}{b_1}\cdot\frac{\frac{c_2d_1}{d_2}}{d_1}&\text{according to }(*)\\ &=&\frac{\frac{a_2b_1}{b_2}}{1}\cdot \frac 1{b_1}\cdot\frac{\frac{c_2d_1}{d_2}}{1}\cdot \frac 1{d_2}&\text{by definition of multiplying rational numbers}\\ &=&\frac{a_2b_1\cdot 1}{1\cdot b_2\cdot b_1}\cdot\frac{c_2d_1\cdot 1}{1\cdot d_2\cdot d_1}&\text{by definition of multiplying rational numbers}\\ &=&\frac{a_2\cdot 1}{b_2}\cdot\frac{b_1}{b_1}\cdot\frac{c_2\cdot 1}{d_2}\cdot\frac{d_1}{d_1}&\text{by commutativity of multiplying integers}\\ &=&\frac{a_2}{b_2}\cdot\frac{b_1}{b_1}\cdot\frac{c_2}{d_2}\cdot\frac{d_1}{d_1}&\text{because }1\text{ is neutral with respect to multiplication of integers}\\ &=&\frac{a_2\cdot 1}{b_2\cdot 1}\cdot\frac{c_2\cdot 1}{d_2\cdot 1}&\text{by definition of rational numbers}\\ &=&\frac{a_2}{b_2}\cdot\frac{c_2}{d_2}&\text{because }1\text{ is neutral with respect to multiplication of integers}\\ \end{array}$

Thank you to the contributors under CC BY-SA 4.0!

Github:

References

Bibliography

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