EXTENDED REAL NUMBER LINE

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 ∞.

Contents

Motivation

Limits

Measure and integration

Order and topological properties

Arithmetic operations

Algebraic properties

Miscellaneous

References

Motivation

Limits

We 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
:
The graph of this function has a horizontal asymptote of ''y'' = 0. Geometrically, as we move farther and farther to the right down the ''x''-axis, the value of '''' gets closer and closer to 0. This limiting behavior is similar to the limit of a function at a real number, except that there is no real number which ''x'' is approaching.
By adjoining the element +∞ to 'R', we allow ourselves to formulate a definition of such a "limit at infinity" which is topologically identical to the usual definition at a real number.

Measure and integration

In 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
:
the value "infinity" arises. Finally, we often wish to consider the limit of a sequence of functions, such as
:
Without allowing functions to take on infinite values, such essential results as the monotone convergence theorem and the dominated convergence theorem would not make sense.

Order and topological properties

The 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 operations

The arithmetic operations of 'R' can be partially extended to
'R' as follows:

★ ''a'' + ∞ = +∞ + ''a'' = +∞    if ''a'' ≠ −∞

★ ''a'' − ∞ = −∞ + ''a'' = −∞    if ''a'' ≠ +∞

★ ''a'' × ±∞ = ±∞ × ''a'' = ±∞    if ''a'' > 0

★ ''a'' × ±∞ = ±∞ × ''a'' = ∓∞    if ''a'' < 0

★ ''a'' / ±∞ = 0    if −∞ < ''a'' < +∞

★ ±∞ / ''a'' = ±∞    if 0 < ''a'' < +∞

★ ±∞ / ''a'' = ∓∞    if −∞ < ''a'' < 0
Here, "''a'' + ∞" means both "''a'' + (+∞)" and "''a'' - (−∞)", and "''a'' − ∞" means both "''a'' − (+∞)" and "''a'' + (−∞)".
The expressions ∞ − ∞, 0 × ±∞ and ±∞ / ±∞ are usually left undefined. These rules are modeled on the laws for . However, in the context of probability or measure theory, 0 × ±∞ is usually 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 converge to one of these points. An example is ''f''(''x'') = 1/(sin(1/''x'')).

Algebraic properties

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.

Miscellaneous

Several 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 $x^\{-2\}$ can be made continuous by setting the value to +∞ for ''x'' = 0, and 0 for ''x'' = +∞ and ''x'' = -∞. The function $x^\{-1\}$ 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 $x^\{-1\}$ at ''x'' = 0. On the other hand $lim\_\{x\; o\; -infty\}\{f(x)\}$ and $lim\_\{x\; o\; +infty\}\{f(x)\}$ 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 e^''x'' and arctan (''x'') cannot be made continuous at ''x''= ∞ on the real projective line.

References

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