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Mathematics-Online lexicon:

real numbers

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The set $ \mathbb{R}$ of all (finite and infinite) decimal numbers is referred to as real numbers. Real numbers can be identified with the points of the real line. A number $ x$ corresponds to the distance from the origin, while its algebraic sign indicates whether $ x$ belongs to the positive or negative half line.

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The rational numbers $ \mathbb{Q}$ are dense in $ \mathbb{R}$, that is every irrational number $ x\in \mathbb{R}\backslash \mathbb{Q}$ can be approximated by fractions to an arbitrary degree of accuracy. Unlike $ \mathbb{Q}$ however, $ \mathbb{R}$ is not countable.

Together with addition and multiplication, the real numbers form a field. Moreover, $ \mathbb{R}$ is complete, that is every convergent sequence of real numbers has a limit in $ \mathbb{R}$.

(Authors: Höllig/Kopf/Abele)

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  automatically generated 6/11/2007