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| {{redirect|Positive infinity|the band|Positive Infinity}}
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| {{Unreferenced|date=January 2012}}
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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 −∞ (read as '''positive [[infinity]]''' and '''negative infinity''' respectively). These new elements are not real numbers. It is useful in describing various [[limit of a function|limiting behavior]]s in [[calculus]] and [[mathematical analysis]], especially in the theory of [[measure (mathematics)|measure]] and [[integral|integration]]. The affinely extended real number system is denoted '''{{Overline|R}}''' or [−∞, +∞].
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| When the meaning is clear from context, the symbol +∞ is often written simply as ∞.
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| ==Motivation==
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| ===Limits===
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| 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
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| :<math>f(x) = x^{-2}.</math>
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| The graph of this function has a horizontal [[asymptote]] at y = 0. Geometrically, as we move farther and farther to the right along the ''x''-axis, the value of 1/''x''<sup>2</sup> approaches 0. This limiting behavior is similar to the [[limit of a function]] at a [[real number]], except that there is no real number to which ''x'' approaches.
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| By adjoining the elements +∞ and −∞ to '''R''', we allow a formulation of a "limit at infinity" with [[topology|topological]] properties similar to those for '''R'''.
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| To make things completely formal, the [[Cauchy sequences]] definition of '''R''' allows us to define +∞ as the set of all sequences of rationals which, for any K>0, from some point on exceed K. We can define −∞ similarly.
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| ===Measure and integration===
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| In [[measure theory]], it is often useful to allow sets which have infinite measure and integrals whose value may be infinite.
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| Such measures arise naturally out of calculus. For example, in assigning a [[measure (mathematics)|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
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| :<math>\int_1^{\infty}\frac{dx}{x}</math>
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| the value "infinity" arises. Finally, it is often useful to consider the limit of a sequence of functions, such as
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| :<math>f_n(x) = \begin{cases} 2n(1-nx), & \mbox{if } 0 \le x \le \frac{1}{n} \\ 0, & \mbox{if } \frac{1}{n} < x \le 1\end{cases}</math>
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| 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.
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| ==Order and topological properties==
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| The affinely extended real number system turns into a [[totally ordered set]] by defining −∞ ≤ ''a'' ≤ +∞ for all ''a''. This order has the desirable property that every subset has a [[supremum]] and an [[infimum]]: it is a [[complete lattice]].
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| This induces the [[order topology]] on '''{{Overline|R}}'''. In this topology, a set ''U'' is a [[neighborhood (topology)|neighborhood]] of +∞ if and only if it contains a set {''x'' : ''x'' > ''a''} for some real number ''a'', and analogously for the neighborhoods of −∞. '''{{Overline|R}}''' is a [[Compact space|compact]] [[Hausdorff space]] [[homeomorphism|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'''.
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| With this topology the specially defined [[Limit of a function|limits]] for ''x'' tending to +∞ and −∞, and the specially defined concepts of limits equal to +∞ and −∞, reduce to the general topological definitions of limits.
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| ==Arithmetic operations==
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| The arithmetic operations of '''R''' can be partially extended to '''{{Overline|R}}''' as follows:
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| :<math> | |
| \begin{align}
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| a + \infty = +\infty + a & = +\infty, & a & \neq -\infty \\
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| a - \infty = -\infty + a & = -\infty, & a & \neq +\infty \\
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| a \cdot (\pm\infty) = \pm\infty \cdot a & = \pm\infty, & a & \in (0, +\infty] \\
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| a \cdot (\pm\infty) = \pm\infty \cdot a & = \mp\infty, & a & \in [-\infty, 0) \\
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| \frac{a}{\pm\infty} & = 0, & a & \in \mathbb{R} \\
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| \frac{\pm\infty}{a} & = \pm\infty, & a & \in (0, +\infty) \\
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| \frac{\pm\infty}{a} & = \mp\infty, & a & \in (-\infty, 0)
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| \end{align}
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| </math>
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| Here, "''a'' + ∞" means both "''a'' + (+∞)" and "''a'' − (−∞)", while "''a'' − ∞" means both "''a'' − (+∞)" and "''a'' + (−∞)".
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| The expressions ∞ − ∞, 0 × (±∞) and ±∞ / ±∞ (called [[indeterminate form]]s) are usually left [[Defined and undefined|undefined]]. These rules are modeled on the laws for [[Limit_of_a_function#Limit_of_a_function_at_infinity|infinite limits]]. However, in the context of probability or measure theory, 0 × (±∞) is often defined as 0.
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| The expression 1/0 is not defined either as +∞ or −∞, because although it is true that whenever ''f''(''x'') → 0 for a [[continuous function]] ''f''(''x'') it must be the case that 1/''f''(''x'') is eventually contained in every [[neighborhood (topology)|neighborhood]] of the set {−∞, +∞}, it is ''not'' true that 1/''f''(''x'') must tend to one of these points. An example is ''f''(''x'') = (sin ''x'')/''x'' (as ''x'' goes to infinity). (The [[absolute value|modulus]] | 1/''f''(''x'') |, nevertheless, does approach +∞.)
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| ==Algebraic properties==
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| With these definitions '''{{Overline|R}}''' is '''not''' a [[field (mathematics)|field]], nor a [[ring (mathematics)|ring]], and not even a [[group (mathematics)|group]] or [[semigroup]]. However, it still has several convenient properties:
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| * ''a'' + (''b'' + ''c'') and (''a'' + ''b'') + ''c'' are either equal or both undefined.
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| * ''a'' + ''b'' and ''b'' + ''a'' are either equal or both undefined.
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| * ''a'' × (''b'' × ''c'') and (''a'' × ''b'') × ''c'' are either equal or both undefined.
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| * ''a'' × ''b'' and ''b'' × ''a'' are either equal or both undefined
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| * ''a'' × (''b'' + ''c'') and (''a'' × ''b'') + (''a'' × ''c'') are equal if both are defined.
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| * if ''a'' ≤ ''b'' and if both ''a'' + ''c'' and ''b'' + ''c'' are defined, then ''a'' + ''c'' ≤ ''b'' + ''c''.
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| * if ''a'' ≤ ''b'' and ''c'' > 0 and both ''a'' × ''c'' and ''b'' × ''c'' are defined, then ''a'' × ''c'' ≤ ''b'' × ''c''.
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| In general, all laws of arithmetic are valid in '''{{Overline|R}}''' as long as all occurring expressions are defined.
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| ==Miscellaneous==
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| Several [[function (mathematics)|functions]] can be [[continuity (topology)|continuously]] extended to '''{{Overline|R}}''' by taking limits. For instance, one defines [[exponential function|exp]](−∞) = 0, exp(+∞) = +∞, [[natural logarithm|ln]](0) = −∞, ln(+∞) = +∞ etc.
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| Some discontinuities may additionally be removed. For example, the function 1/''x''<sup>2</sup> 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 1/''x'' can ''not'' be made continuous because the function approaches −∞ as ''x'' approaches 0 from below, and +∞ as ''x'' approaches 0 from above.
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| 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 1/''x'' at ''x'' = 0. On the other hand
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| :<math>\lim_{x \to -\infty}{f(x)}</math> and <math>\lim_{x \to +\infty}{f(x)}</math>
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| 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<sup>''x''</sup> and arctan(''x'') cannot be made continuous at ''x'' = ∞ on the real projective line.
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| ==See also==
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| * [[Real projective line]], which adds positive infinity without negative infinity to the real number line.
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| * [[Division by zero]]
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| * [[Extended complex plane]]
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| * [[Improper integral]]
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| * [[Series (mathematics)]]
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| ==References==
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| * {{MathWorld|author= David W. Cantrell|title=Affinely Extended Real Numbers|urlname=AffinelyExtendedRealNumbers}}
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| {{Number Systems}}
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| [[Category:Infinity]]
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| [[Category:Real numbers]]
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