# Proposition: Definition of Rational Numbers

Let $$(a,b),(c,d)$$ be ordered pairs of integers, such that $$b$$ and $$d$$ do not equal the integer number zero $$0$$. We consider the pairs $$(a,b),(c,d)$$ equivalent, if the integer products $$a\cdot d$$ and $$b\cdot c$$ are equal, formally

$(a,b)\sim (c,d)\quad\Longleftrightarrow\quad a\cdot d = b\cdot c.$

The relation "$$\sim$$" defined above is an equivalence relation, i.e. for a given ordered pair $$(a,b)\in\mathbb Z\times\mathbb Z\setminus\{0\}$$, we can consider a whole set of ordered pairs $$(c,d)\in\mathbb Z\times\mathbb Z\setminus\{0\}$$ equivalent to $$(a,b)$$:

$x:=\{(c,d)\in\mathbb Z\times\mathbb Z\setminus\{0\}:\quad( c, d )\sim ( a, b )\}\quad\quad ( * )$

The set1 $$x$$ is called a rational number. We say that the ordered pair $$(a,b)\in\mathbb Z\times\mathbb Z\setminus\{0\}$$ is representing the rational number $$x$$. It is called a fraction or a ratio. Usually, we denote the fraction $$(a,b)$$ by $$\frac ab$$. The integer $$a$$ is called the nominator and the integer $$b$$ is called the denominator of the fraction $$\frac ab$$. The set of all rational numbers is denoted by $$\mathbb Q$$.

A more common notation for the ordered pairs in $$( * )$$ using ratios is

$x:=\left\{\frac cd: \quad \frac cd\sim \frac ab,~b,d\neq 0\right\}.$

Proofs: 1

Chapters: 1
Corollaries: 2 3
Definitions: 4 5 6 7 8 9 10 11 12 13
Examples: 14 15 16 17 18
Explanations: 19
Lemmas: 20
Motivations: 21
Parts: 22 23
Proofs: 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51
Propositions: 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Solutions: 66

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### References

#### Bibliography

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

#### Footnotes

1. Please note that rational numbers are in fact sets. The set $$\mathbb Q$$ is the set of "all fractions": the same rational number can be represented by different fractions, especially if we multiply the nominators and denominators by a non-zero integer, e.g. $$\frac 12=\frac 24=\frac 5{10}=\ldots$$. Since all these different fractions are equal, they represent the same rational number.