extended real numbers (original) (raw)
The real numbers are in certain contexts called finite as contrast to ∞.
0.0.1 Order on ℝ¯
The order (http://planetmath.org/TotalOrder) 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,∞] | = | {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 ℝ¯ is not an ordered ringalthough 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
| (+∞)⋅(+∞) | = | (-∞)⋅(-∞)=+∞, |
|---|---|---|
| (+∞)⋅(-∞) | = | (-∞)⋅(+∞)=-∞. |
0.0.4 Absolute value
0.0.5 Topology
The topology
of R¯ is given by the usual base of ℝtogether 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↦(2/π)arctanx. Consequently, everycontinuous function
f:ℝ¯→ℝ¯ has a minimum and maximum.
0.0.6 Examples
- By taking x=-1 in the , we obtain the relations
(-1)⋅(±∞) = ∓∞.
- By taking x=-1 in the , we obtain the relations