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 closureMathworldPlanetmath 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 x, we have

- < x,
x < ,

and that -<.  For  a, let us also define intervals

(a,] = {x:x>a},
[-,a) = {x:x<a}.

0.0.2 Addition

For any real number x, we define

x+(±) = (±)+x=±,

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 x is a positive real number, then

x(±) = (±)x=±.

Similarly, if x is a negative real number, then

x(±) = (±)x=.

Furthermore, for and -, we define

(+)(+) = (-)(-)=+,
(+)(-) = (-)(+)=-.

In many areas of mathematics, productsMathworldPlanetmathPlanetmathPlanetmath like 0 are left undefined.  However, a special case is measure theory, where it is convenient to define

0(±) = (±)0=0.

0.0.4 Absolute value

For and -, the absolute valueMathworldPlanetmathPlanetmath is defined as


0.0.5 Topology

The topologyMathworldPlanetmath of R¯ is given by the usual base of together with with intervals of type  [-,a),  (a,].  This makes ¯ into a compactPlanetmathPlanetmath topological space. ¯ can also be seen to be homeomorphic to the interval  [-1, 1], via the map x(2/π)arctanx. Consequently, every continuous functionMathworldPlanetmathPlanetmath f:¯¯ has a minimum and maximum.

0.0.6 Examples

  1. 1.

    By taking  x=-1  in the , we obtain the relationsMathworldPlanetmath

    (-1)(±) = .
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 infiniteMathworldPlanetmath
Defines infinityMathworldPlanetmath
Defines finite