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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 \(y\).

Table of Contents

Proofs: 1

1. Proposition: Multiplication of Rational Numbers Is Associative
2. Proposition: Multiplication Of Rational Numbers Is Commutative
3. Proposition: Multiplication Of Rational Numbers Is Cancellative
4. Proposition: Existence of Rational One (Neutral Element of Multiplication of Rational Numbers)
5. Proposition: Uniqueness Of Rational One
6. Proposition: Existence of Inverse Rational Numbers With Respect to Multiplication
7. Proposition: Uniqueness of Inverse Rational Numbers With Respect to Multiplication
8. 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

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