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Proposition: Definition of Real Numbers

Let \((x_n)_{n\in\mathbb N}\), \((y_n)_{n\in\mathbb N}\) be rational Cauchy sequences. We call them equivalent, if their difference is convergent to \(0\), formally

\[(x_n)_{n\in\mathbb N}\sim(y_n)_{n\in\mathbb N}\quad\Longleftrightarrow\quad \lim_{n\to\infty } y_n-x_n =0\]

The relation “\(\sim\)” defined above is an equivalence relation, i.e. for a given rational Cauchy sequence \((x_n)_{n\in\mathbb N}\) we can consider a whole set of rational Cauchy sequences \((y_n)_{n\in\mathbb N}\) equivalent to \((x_n)_{n\in\mathbb N}\):

\[x:=\{(y_n)_{n\in\mathbb N}\text{ is a rational Cauchy sequence},~ (y_n)_{n\in\mathbb N}\sim (x_n)_{n\in\mathbb N}\}\quad\quad ( * )\]

The set1 \(x\) is called a real number, and the rational Cauchy sequence \((x_n)_{n\in\mathbb N}\) is called a representation2 of the real number. The set of all real numbers is denoted by \(\mathbb R\).

For practical purposes, \( ( * ) \) it equivalent with the notation

\[x:=(x_n)_{n\in\mathbb N} + I, \]
where \(I\) is the set of all rational sequences convergent to \(0\).

1 Please note that real numbers are in fact sets.

2 This has very important practical consequences, in particular it means that the same real number can be represented in many ways, especially in any numeral system (e.g. decimal or binary).

| | | | | Contributors: bookofproofs | References: [696]

1.Proof: (related to "Definition of Real Numbers")

2.Proposition: Addition of Real Numbers

3.Definition: Subtraction of Real Numbers

4.Proposition: Multiplication of Real Numbers

5.Definition: Division of Real Numbers

6.Explanation: Why is it impossible to divide by \(0\)?

7.Proposition: Distributivity Law For Real Numbers

8.Proposition: Unique Solvability of \(a+x=b\)

9.Proposition: Unique Solvability of \(ax=b\)

10.Proposition: \(-(x+y)=-x-y\)

11.Proposition: \((xy)^{-1}=x^{-1}y^{-1}\)

12.Lemma: Unit Ring of All Rational Cauchy Sequences

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Bibliography (further reading)

[696] Kramer Jürg, von Pippich, Anna-Maria: “Von den natürlichen Zahlen zu den Quaternionen”, Springer-Spektrum, 2013

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