Extended real number line
The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). It is useful in mathematical analysis, especially in integration theory. The extended real number line is denoted by R or [−∞,+∞].
The extended real number line turns into a totally ordered set by defining −∞ ≤ a ≤ +∞ for all a. This order has the nice property that every subset has a supremum and an infimum: it is a complete lattice. The total order induces a topology on R. In this topology, a set U is a neighborhood of +∞ if and only if it contains a set {x : x ≥ a} for some real number a, and analogously for the neighborhoods of −∞. R is a compact Hausdorff space homeomorphic to the unit interval [0,1].
The arithmetical operations of R can be partly extended to R as follows:
- a + ∞ = ∞ + a = ∞ if a ≠ −∞
- a − ∞ = −∞ + a = −∞ if a ≠ +∞
- a × +∞ = +∞ × a = +∞ if a > 0
- a × +∞ = +∞ × a = −∞ if a < 0
- a × −∞ = −∞ × a = −∞ if a > 0
- a × −∞ = −∞ × a = +∞ if a < 0
- a / ±∞ = 0 if −∞ < a < +∞
- ±∞ / a = ±∞ if 0 < a < +∞
- +∞ / a = −∞ if −∞ < a < 0
- −∞ / a = +∞ if −∞ < a < 0
The expressions ∞ − ∞, 0 × ±∞ and ±∞ / ±∞ are usually left undefined. Also, 1 / 0 is not defined as +∞ (because −∞ would be just as good a candidate). These rules are modeled on the laws for infinite limits.
Note that with these definitions, R is not a field and not even a ring. However, it still has several convenient properties:
- a + (b + c) and (a + b) + c are either equal or both undefined.
- a + b and b + a are either equal or both undefined.
- a × (b × c) and (a × b) × c are either equal or both undefined.
- a × b and b × a are either equal or both undefined
- a × (b + c) and (a × b) + (a × c) are equal if both are defined.
- if a ≤ b and if both a + c and b + c are defined, then a + c ≤ b + c.
- if a ≤ b and c > 0 and both a × c and b × c are defined, then a × c ≤ b × c.
In general, all laws of arithmetic are valid in R as long as all occurring expressions are defined.
By using the intuition of limits, several functions can be naturally extended to R. For instance, one defines exp(−∞) = 0, exp(+∞) = +∞, ln(0) = −∞, ln(+∞) = ∞ etc.