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Mathematics-Online lexicon:

total order of real numbers

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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 $b$.}

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


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^+ =

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)


  automatically generated 6/11/2007