Extended real number line
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In mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ (pronounced "positive infinity" and "negative infinity"). These new elements are not real numbers. It is useful in describing various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration. The affinely extended real number system is denoted R or [−∞, +∞]. The affinely extended real number system should be distinguished from the projectively extended real numbers by having two infinities, rather than one. When the meaning is clear from context, the symbol +∞ is often written simply as ∞.
MotivationLimitsWe often wish to describe the behavior of a function f(x), as either the argument x or the function value f(x) gets "very big" in some sense. For example, consider the function
By adjoining the elements +∞ and -∞ to R, we allow ourselves to formulate a definition of such a "limit at infinity" with similar topological properties as real-valued limits. Measure and integrationIn measure theory, it is often useful to allow sets which have infinite measure and integrals whose value may be infinite. Such measures arise naturally out of calculus. For example, if we are to assign a measure to R that agrees with the usual length of intervals, this measure must be larger than any finite real number. Also, when considering infinite integrals, such as
Order and topological propertiesThe affinely extended real number system 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. This induces the order 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]. Thus the topology is metrizable corresponding (for a given homeomorphism) to the ordinary metric on this interval. There is no metric which is an extension of the ordinary metric on R. With this topology the specially defined limits for x tending to +∞ and -∞, and the specially defined concept of a limit being +∞ and -∞, reduce to the general topological definition of limit. Arithmetic operationsThe arithmetic operations of R can be partially extended to R as follows:
Here, "a + ∞" means both "a + (+∞)" and "a − (−∞)", and "a − ∞" means both "a − (+∞)" and "a + (−∞)". The expressions ∞ − ∞, 0 × (±∞) and ±∞ / ±∞ (called indeterminate forms) are usually left undefined. These rules are modeled on the laws for infinite limits. However, in the context of probability or measure theory, 0 × (±∞) is often defined as 0. Note that 1 / 0 is not defined as either +∞ or −∞, because although it is true that whenever f(x) → 0 for a continuous function f(x), we must have that 1/f(x) is eventually in every neighborhood of the set {−∞, +∞}, it is not true that 1/f(x) must tend to one of these points. An example is f(x) = 1/(sin(1/x)). (Its modulus 1/|f(x)|, nevertheless, does approach +∞.) Algebraic propertiesNote that with these definitions, R is not a field and not even a ring. However, it still has several convenient properties:
In general, all laws of arithmetic are valid in R as long as all occurring expressions are defined. MiscellaneousSeveral functions can be continuously extended to R by taking limits. For instance, one defines exp(−∞) = 0, exp(+∞) = +∞, ln(0) = −∞, ln(+∞) = +∞ etc. Also, some discontinuities can be removed. For example, the function Failed to parse (Missing texvc executable; please see math/README to configure.): x^{-2}
can be made continuous (under some definitions of continuity) by setting the value to +∞ for x = 0, and 0 for x = +∞ and x = -∞. The function Failed to parse (Missing texvc executable; please see math/README to configure.): x^{-1}
can not be made continuous (because the function approaches -∞ as x approaches 0 from below, and +∞ as x approaches 0 from above).
Compare the real projective line, which does not distinguish between +∞ and −∞. As a result, on one hand a function may have limit ∞ on the real projective line, while in the affinely extended real number system only the absolute value of the function has a limit, e.g. in the case of the function Failed to parse (Missing texvc executable; please see math/README to configure.): x^{-1}
at x = 0. On the other hand Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{x \to -\infty}{f(x)}
and Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{x \to +\infty}{f(x)}
correspond on the real projective line to only a limit from the right and one from the left, respectively, with the full limit only existing when the two are equal. Thus ex and arctan (x) cannot be made continuous at x= ∞ on the real projective line.
See alsoReferences
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