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 welldefined, i.e. it does not depend on the specific representatives \(\frac ab\) and \(\frac cd\) of \(x\) and \(y\).
Table of Contents
Proofs: 1
 Proposition: Multiplication of Rational Numbers Is Associative
 Proposition: Multiplication Of Rational Numbers Is Commutative
 Proposition: Multiplication Of Rational Numbers Is Cancellative
 Proposition: Existence of Rational One (Neutral Element of Multiplication of Rational Numbers)
 Proposition: Uniqueness Of Rational One
 Proposition: Existence of Inverse Rational Numbers With Respect to Multiplication
 Proposition: Uniqueness of Inverse Rational Numbers With Respect to Multiplication
 Proposition: Multiplying Negative and Positive Rational Numbers
Mentioned in:
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
 Kramer Jürg, von Pippich, AnnaMaria: "Von den natürlichen Zahlen zu den Quaternionen", SpringerSpektrum, 2013