Mo logo [home] [lexicon] [problems] [tests] [courses] [auxiliaries] [notes] [staff] german flag
Mathematics-Online lexicon:

total order of real numbers

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 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)

see also:

  automatically generated 6/11/2007