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extended real numbers
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(Definition)
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The extended real numbers are the real numbers together with $+\infty$ (or simply $\infty$ ) and $-\infty$ . This set is usually denoted by $\overline{\sR}$ or $[-\infty,\,\infty]$ , and the elements $+\infty$ and $-\infty$ are called plus and minus infinity, respectively. (N.B., ``$\overline{\sR}$ '' may sometimes mean the algebraic closure of $\mathbb{R}$ ; see the special notations in algebra.)
The real numbers are in certain contexts called finite as contrast to $\infty$ .
The order relation on $\sR$ extends to $\overline{\sR}$ by defining that for any $x\in \sR$ , we have \begin{eqnarray*} -\infty&<& x, \\ x &<& \infty, \end{eqnarray*}and that $-\infty < \infty$ . For $a\in\sR$ , let us also define intervals \begin{eqnarray*} (a,\,\infty{]} &=& \{x\in \sR: x>a \}, \\ {[}{-\infty},\,a) &=& \{x\in \sR: x<a \}. \end{eqnarray*}
For any real number $x$ , we define \begin{eqnarray*} x + (\pm\infty) &=& (\pm\infty) + x = \pm\infty, \end{eqnarray*}and for $+\infty$ and $-\infty$ , we define \begin{eqnarray*} (\pm \infty) + (\pm \infty) &=& \pm \infty. \end{eqnarray*}It should be pointed out that sums like $(+\infty) + (-\infty)$ are left undefined. Thus $\overline{\sR}$ is not an ordered ring although $\sR$ is.
If $x$ is a positive real number, then \begin{eqnarray*} x \cdot (\pm \infty) &=& (\pm\infty)\cdot x = \pm\infty. \end{eqnarray*}Similarly, if $x$ is a negative real number, then \begin{eqnarray*} x \cdot (\pm \infty) &=& (\pm \infty)\cdot x = \mp\infty. \end{eqnarray*}Furthermore, for $\infty$ and $-\infty$ , we define \begin{eqnarray*} (+\infty) \cdot(+\infty) &=& (-\infty)\cdot (-\infty) = +\infty, \\ (+\infty) \cdot (- \infty) &=& (-\infty)\cdot (+\infty) = -\infty. \end{eqnarray*} In many areas of mathematics, products like $0\cdot \infty$ are left undefined. However, a special case is measure theory, where it is convenient to define \begin{eqnarray*} 0\cdot (\pm \infty) &=& (\pm \infty) \cdot 0 = 0. \end{eqnarray*}
For $\infty$ and $-\infty$ , the absolute value is defined as $$ |\pm \infty| = +\infty. $$
The topology of $\overline{R}$ is given by the usual base of $\sR$ together with with intervals of type $[-\infty,\,a)$ , $(a,\,\infty]$ . This makes $\overline{\sR}$ into a compact topological space. $\overline{\sR}$ can also be seen to be homeomorphic to the interval $[-1,\,1]$ , via the map $x \mapsto (2/\pi) \arctan x$ . Consequently, every continuous function $f\colon \overline{\sR}\to \overline{\sR}$ has a minimum and maximum.
- By taking $x = -1$ in the product rule, we obtain the relations \begin{eqnarray*} (-1)\cdot (\pm\infty) &=& \mp \infty. \end{eqnarray*}
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"extended real numbers" is owned by matte. [ full author list (6) | owner history (2) ]
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Cross-references: continuous function, map, homeomorphic, topological space, compact, type, base, topology, absolute value, theory, measure, products, negative, positive, ordered ring, sums, intervals, relation, special notations in algebra, algebraic closure, plus, elements, real numbers
There are 97 references to this entry.
This is version 18 of extended real numbers, born on 2003-07-12, modified 2007-04-27.
Object id is 4441, canonical name is ExtendedRealNumbers.
Accessed 23391 times total.
Classification:
| AMS MSC: | 28-00 (Measure and integration :: General reference works ) | | | 12D99 (Field theory and polynomials :: Real and complex fields :: Miscellaneous) |
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Pending Errata and Addenda
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