Mathematics-Online lexicon: total order of real numbers

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	Mathematics-Online lexicon:
	total order of real numbers

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overview

Real numbers can be compared (on the real line) with the order relation. For $a,b,c \in {\mathbb{R}}$ we define

$\displaystyle \begin{array}{ll} a < b: & \mbox{$a$\ is situated to the left of $b$,} \\ c > b: & \mbox{$c$\ is situated to the right of $b$.} \end{array}$

If such a comparison allows equality, the symbols $\le$ and $\ge$ are used.

$\includegraphics[width=.5\linewidth]{a_ordnung_reeller_zahlen}$

Positive real numbers are denoted by

$\displaystyle \mathbb{R}^+ = \{x\in\mathbb{R}:\ x>0\}$

and, analogously, negative real numbers by $\mathbb{R}^-$ . Moreover, $\mathbb{R}_0^+ = \mathbb{R}^+\cup\{0\}$ .

Real numbers are complete with respect to the order relation. This means that for every bounded set of real numbers there exists a least upper bound (supremum) and a greatest lower bound (infimum) in $\mathbb{R}$ .

(Authors: Höllig/Kimmerle/Abele)

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automatically generated 6/11/2007