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extended real numbers∗
matte†
2013-03-21 16:14:28
The extended real numbers are the real numbers together with +∞ (or simply
∞) and −∞. This set is usually denoted by R or [−∞, ∞], and the elements
+∞ and −∞ are called plus and minus infinity, respectively. (N.B., “R” may
sometimes mean 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 R
The order relation on R extends to R by defining that for any x ∈ R, we have
−∞ <
x,
x
∞,
<
and that −∞ < ∞. For a ∈ R, let us also define intervals
(a, ∞]
[−∞, a)
0.0.2
=
{x ∈ R : x > a},
= {x ∈ R : x < a}.
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
R is not an ordered ring although R is.
∗ hExtendedRealNumbersi created: h2013-03-21i by: hmattei version: h34441i Privacy
setting: h1i hDefinitioni h28-00i h12D99i
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1
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 · (±∞)
0.0.4
(±∞) · 0 = 0.
=
Absolute value
For ∞ and −∞, the absolute value is defined as
| ± ∞| = +∞.
0.0.5
Topology
The topology of R is given by the usual base of R together with with intervals
of type [−∞, a), (a, ∞]. This makes R into a compact topological space.
R can also be seen to be homeomorphic to the interval [−1, 1], via the map
x 7→ (2/π) arctan x. Consequently, every continuous function f : R → R has a
minimum and maximum.
0.0.6
Examples
1. By taking x = −1 in the product rule, we obtain the relations
(−1) · (±∞)
2
=
∓∞.