extended real numbers
The extended real numbers are the real numbers together with +∞ (or simply ∞) and -∞. This set is usually denoted by ˉℝ or [-∞,∞], and the elements +∞ and -∞ are called plus and minus infinity, respectively. (N.B., “ˉℝ” may sometimes the algebraic closure of ℝ; see the special notations in algebra.)
The real numbers are in certain contexts called finite as contrast to ∞.
0.0.1 Order on ˉℝ
The order (http://planetmath.org/TotalOrder) relation on ℝ extends to ˉℝ by defining that for any x∈ℝ, we have
-∞ | < | x, | ||
x | < | ∞, |
and that -∞<∞. For a∈ℝ, let us also define intervals
(a,∞] | = | {x∈ℝ:x>a}, | ||
[-∞,a) | = | {x∈ℝ: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 ˉℝ is not an ordered ring although ℝ 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, products 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 value is defined as
|±∞|=+∞. |
0.0.5 Topology
The topology of ˉR is given by the usual base of ℝ together with with intervals of type [-∞,a), (a,∞]. This makes ˉℝ into a compact topological space. ˉℝ can also be seen to be homeomorphic to the interval [-1, 1], via the map x↦(2/π)arctanx. Consequently, every continuous function f:ˉℝ→ˉℝ has a minimum and maximum.
0.0.6 Examples
-
1.
By taking x=-1 in the , we obtain the relations
(-1)⋅(±∞) = ∓∞.
Title | extended real numbers |
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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 | ˉℝ |
Defines | infinite |
Defines | infinity |
Defines | finite |