Extended real number line

From Academic Kids

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 : xa} 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 ab and if both a + c and b + c are defined, then a + cb + c.
  • if ab and c > 0 and both a × c and b × c are defined, then a × cb × 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.zh:扩展的实数轴

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