extended real numbers
The extended real numbers are the real numbers together with (or simply ) and . This set is usually denoted by or , and the elements and are called plus and minus infinity, respectively. (N.B., “” may sometimes the algebraic closure of ; see the special notations in algebra.)
The real numbers are in certain contexts called finite as contrast to .
0.0.1 Order on
The order (http://planetmath.org/TotalOrder) relation on extends to by defining that for any , we have
and that . For , let us also define intervals
0.0.2 Addition
For any real number , we define
and for and , we define
It should be pointed out that sums like are left undefined. Thus is not an ordered ring although is.
0.0.3 Multiplication
If is a positive real number, then
Similarly, if is a negative real number, then
Furthermore, for and , we define
In many areas of mathematics, products like are left undefined. However, a special case is measure theory, where it is convenient to define
0.0.4 Absolute value
For and , the absolute value is defined as
0.0.5 Topology
The topology of is given by the usual base of together with with intervals of type , . This makes into a compact topological space. can also be seen to be homeomorphic to the interval , via the map . Consequently, every continuous function has a minimum and maximum.
0.0.6 Examples
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1.
By taking in the , we obtain the relations
Title | extended real numbers |
---|---|
Canonical name | ExtendedRealNumbers |
Date of creation | 2013-03-22 13:44:44 |
Last modified on | 2013-03-22 13:44:44 |
Owner | matte (1858) |
Last modified by | matte (1858) |
Numerical id | 21 |
Author | matte (1858) |
Entry type | Definition |
Classification | msc 28-00 |
Classification | msc 12D99 |
Related topic | ImproperLimits |
Related topic | IntermediateValueTheoremForExtendedRealNumbers |
Related topic | ExampleOfNonCompleteLatticeHomomorphism |
Defines | plus infinity |
Defines | minus infinity |
Defines | |
Defines | infinite |
Defines | infinity |
Defines | finite |