# Definition: Order Relation of Real Numbers

According to the definition of real numbers, any real number $$x \in\mathbb R$$ can be represented as residue classes of rational Cauchy sequences: $$\begin{array}{ccl} x&=&(x_n)_{n\in\mathbb N} + I. \end{array}$$

Based on the order relation for rational numbers, we have tree cases of representing a real number, called: * positive real number $x > 0$, if and only if there is an $N\in\mathbb N$ such that $x_n > 0$ for all $n > N,$1 * zero $x=0$, if and only if for all rational $\epsilon > 0$ there is an $N\in\mathbb N$ such that $-\epsilon < x_n < \epsilon$ for all $n > N,$ * negative real number $x < 0$, if and only if there is an $N\in\mathbb N$ such that $x_n < 0$ for all $n > N.$

Using the definition of subtraction of real numbers, we can define the order relation for real numbers as follows: * $x > y$ if and only if $x-y > 0,$ * $x = y$ if and only if $x-y = 0,$ * $x < y$ if and only if $x-y < 0.$

### Notation

• The set of all positive real numbers is denoted by $\mathbb R^+.$
• The set of all negative real numbers is denoted by $\mathbb R^-.$

Corollaries: 1

Axioms: 1
Chapters: 2
Corollaries: 3 4 5 6
Definitions: 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 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
Examples: 49
Explanations: 50
Lemmas: 51 52
Parts: 53
Problems: 54
Proofs: 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99
Propositions: 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149
Sections: 150 151
Theorems: 152 153 154

Github: ### References

#### Bibliography

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

#### Footnotes

1. The first $0$ in all three cases means the real zero, the second $0$ is the rational zero).