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Extended real number line
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<h2 id="motivation">Motivation</h2> <h3>Limits</h3> <p>The extended number line is often useful to describe the behavior of a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> f {\displaystyle f} when either the <a href="/facts/Argument_of_a_function/QkuPc28I">argument</a> x {\displaystyle x} or the function value f {\displaystyle f} gets "infinitely large" in some sense. For example, consider the function f {\displaystyle f} defined by </p> f ( x ) = 1 x 2 {\displaystyle f(x)={\frac {1}{x^{2}}}} . <p>The <a href="/facts/Graph_of_a_function/htYX0BGX">graph</a> of this function has a horizontal <a href="/facts/Asymptote/nvLPuxKa">asymptote</a> at y = 0 {\displaystyle y=0} . Geometrically, when moving increasingly farther to the right along the x {\displaystyle x} -axis, the value of 1 / x 2 {\textstyle {1}/{x^{2}}} <a href="/facts/Limit_of_a_function/HFbQ69e5">approaches</a> 0. This limiting behavior is similar to the <a href="/facts/Limit_of_a_function/HFbQ69e5">limit of a function</a> lim x → x 0 f ( x ) {\textstyle \lim _{x\to x_{0}}f(x)} in which the <a href="/facts/Real_number/R02gw5Pb">real number</a> x {\displaystyle x} approaches x 0 , {\displaystyle x_{0},} except that there is no real number that x {\displaystyle x} approaches when x {\displaystyle x} increases infinitely. Adjoining the elements + ∞ {\displaystyle +\infty } and − ∞ {\displaystyle -\infty } to R {\displaystyle \mathbb {R} } enables a definition of "limits at infinity" which is very similar to the usual defininion of limits, except that | x − x 0 | < ε {\displaystyle |x-x_{0}|<\varepsilon } is replaced by x > N {\displaystyle x>N} (for + ∞ {\displaystyle +\infty } ) or x < − N {\displaystyle x<-N} (for − ∞ {\displaystyle -\infty } ). This allows proving and writing </p> lim x → + ∞ 1 x 2 = 0 , lim x → − ∞ 1 x 2 = 0 , lim x → 0 1 x 2 = + ∞ . {\displaystyle {\begin{aligned}\lim _{x\to +\infty }{\frac {1}{x^{2}}}&=0,\\\lim _{x\to -\infty }{\frac {1}{x^{2}}}&=0,\\\lim _{x\to 0}{\frac {1}{x^{2}}}&=+\infty .\end{aligned}}} <h3>Measure and integration</h3> <p>In <a href="/facts/Measure_theory/yCq7zlI4">measure theory</a>, it is often useful to allow sets that have infinite <a href="/facts/Measure_(mathematics)/yCq7zlI4">measure</a> and integrals whose value may be infinite. </p><p>Such measures arise naturally out of calculus. For example, in assigning a measure to R {\displaystyle \mathbb {R} } that agrees with the usual length of <a href="/facts/Interval_(mathematics)/e9se3vTe">intervals</a>, this measure must be larger than any finite real number. Also, when considering <a href="/facts/Improper_integral/1YLuane0">improper integrals</a>, such as </p> ∫ 1 ∞ d x x {\displaystyle \int _{1}^{\infty }{\frac {dx}{x}}} <p>the value "infinity" arises. Finally, it is often useful to consider the limit of a sequence of functions, such as </p> f n ( x ) = { 2 n ( 1 − n x ) , if 0 ≤ x ≤ 1 n 0 , if 1 n < x ≤ 1 {\displaystyle f_{n}(x)={\begin{cases}2n(1-nx),&{\mbox{if }}0\leq x\leq {\frac {1}{n}}\\0,&{\mbox{if }}{\frac {1}{n}}<x\leq 1\end{cases}}} . <p>Without allowing functions to take on infinite values, such essential results as the <a href="/facts/Monotone_convergence_theorem/mYFH9yGY">monotone convergence theorem</a> and the <a href="/facts/Dominated_convergence_theorem/EDCoCBJY">dominated convergence theorem</a> would not make sense. </p> <h2 id="order-and-topological-properties">Order and topological properties</h2> <p>The extended real number system R ¯ {\displaystyle {\overline {\mathbb {R} }}} , defined as [ − ∞ , + ∞ ] {\displaystyle [-\infty ,+\infty ]} or R ∪ { − ∞ , + ∞ } {\displaystyle \mathbb {R} \cup \left\{-\infty ,+\infty \right\}} , can be turned into a <a href="/facts/Totally_ordered_set/xnTtmuYJ">totally ordered set</a> by defining − ∞ ≤ a ≤ + ∞ {\displaystyle -\infty \leq a\leq +\infty } for all a ∈ R ¯ {\displaystyle a\in {\overline {\mathbb {R} }}} . With this <a href="/facts/Order_topology/EHISytz6">order topology</a>, R ¯ {\displaystyle {\overline {\mathbb {R} }}} has the desirable property of <a href="/facts/Compact_space/d0cgXJH7">compactness</a>: Every <a href="/facts/Subset/SJGERmbA">subset</a> of R ¯ {\displaystyle {\overline {\mathbb {R} }}} has a <a href="/facts/Supremum/oXCKVopz">supremum</a> and an <a href="/facts/Infimum/oXCKVopz">infimum</a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> (the infimum of the <a href="/facts/Empty_set/ffEk3eA6">empty set</a> is + ∞ {\displaystyle +\infty } , and its supremum is − ∞ {\displaystyle -\infty } ). Moreover, with this <a href="/facts/Topological_space/v2ut7CbK">topology</a>, R ¯ {\displaystyle {\overline {\mathbb {R} }}} is <a href="/facts/Homeomorphic/jMfWg3Mn">homeomorphic</a> to the <a href="/facts/Unit_interval/W4dP8IJo">unit interval</a> [ 0 , 1 ] {\displaystyle [0,1]} . Thus the topology is <a href="/facts/Metrizable/BCLdo5Jf">metrizable</a>, corresponding (for a given homeomorphism) to the ordinary <a href="/facts/Metric_(mathematics)/RkUptSo8">metric</a> on this interval. There is no metric, however, that is an extension of the ordinary metric on R {\displaystyle \mathbb {R} } . </p><p>In this topology, a set U {\displaystyle U} is a <a href="/facts/Neighborhood_(mathematics)/XHKgrpkp">neighborhood</a> of + ∞ {\displaystyle +\infty } if and only if it contains a set { x : x > a } {\displaystyle \{x:x>a\}} for some real number a {\displaystyle a} . The notion of the neighborhood of − ∞ {\displaystyle -\infty } can be defined similarly. Using this characterization of extended-real neighborhoods, <a href="/facts/Limit_of_a_function/HFbQ69e5">limits</a> with x {\displaystyle x} tending to + ∞ {\displaystyle +\infty } or − ∞ {\displaystyle -\infty } , and limits "equal" to + ∞ {\displaystyle +\infty } and − ∞ {\displaystyle -\infty } , reduce to the general topological definition of limits—instead of having a special definition in the real number system. </p> <h2 id="arithmetic-operations">Arithmetic operations</h2> <p>The arithmetic operations of R {\displaystyle \mathbb {R} } can be partially extended to R ¯ {\displaystyle {\overline {\mathbb {R} }}} as follows:<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p> a ± ∞ = ± ∞ + a = ± ∞ , a ≠ ∓ ∞ a ⋅ ( ± ∞ ) = ± ∞ ⋅ a = ± ∞ , a ∈ ( 0 , + ∞ ] a ⋅ ( ± ∞ ) = ± ∞ ⋅ a = ∓ ∞ , a ∈ [ − ∞ , 0 ) a ± ∞ = 0 , a ∈ R ± ∞ a = ± ∞ , a ∈ ( 0 , + ∞ ) ± ∞ a = ∓ ∞ , a ∈ ( − ∞ , 0 ) {\displaystyle {\begin{aligned}a\pm \infty =\pm \infty +a&=\pm \infty ,&a&\neq \mp \infty \\a\cdot (\pm \infty )=\pm \infty \cdot a&=\pm \infty ,&a&\in (0,+\infty ]\\a\cdot (\pm \infty )=\pm \infty \cdot a&=\mp \infty ,&a&\in [-\infty ,0)\\{\frac {a}{\pm \infty }}&=0,&a&\in \mathbb {R} \\{\frac {\pm \infty }{a}}&=\pm \infty ,&a&\in (0,+\infty )\\{\frac {\pm \infty }{a}}&=\mp \infty ,&a&\in (-\infty ,0)\end{aligned}}} <p>For exponentiation, see <a href="/facts/Exponentiation/pac6QY2g">Exponentiation § Limits of powers</a>. Here, a + ∞ {\displaystyle a+\infty } means both a + ( + ∞ ) {\displaystyle a+(+\infty )} and a − ( − ∞ ) {\displaystyle a-(-\infty )} , while a − ∞ {\displaystyle a-\infty } means both a − ( + ∞ ) {\displaystyle a-(+\infty )} and a + ( − ∞ ) {\displaystyle a+(-\infty )} . </p><p>The expressions ∞ − ∞ {\displaystyle \infty -\infty } , 0 × ( ± ∞ ) {\displaystyle 0\times (\pm \infty )} , and ± ∞ / ± ∞ {\displaystyle \pm \infty /\pm \infty } (called <a href="/facts/Indeterminate_form/5Y3JWKjS">indeterminate forms</a>) are usually left <a href="/facts/Defined_and_undefined/Bc1Q3px4">undefined</a>. These rules are modeled on the laws for <a href="/facts/Limit_of_a_function/HFbQ69e5">infinite limits</a>. However, in the context of <a href="/facts/Probability_theory/mFBn51rE">probability</a> or measure theory, 0 × ± ∞ {\displaystyle 0\times \pm \infty } is often defined as 0.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p><p>When dealing with both positive and negative extended real numbers, the expression 1 / 0 {\displaystyle 1/0} is usually left undefined, because, although it is true that for every real nonzero sequence f {\displaystyle f} that <a href="/facts/Limit_of_a_sequence/Ktge2RwL">converges</a> to 0, the <a href="/facts/Multiplicative_inverse/yczdYyCw">reciprocal</a> sequence 1 / f {\displaystyle 1/f} is eventually contained in every neighborhood of { ∞ , − ∞ } {\displaystyle \{\infty ,-\infty \}} , it is <i>not</i> true that the sequence 1 / f {\displaystyle 1/f} must itself converge to either − ∞ {\displaystyle -\infty } or ∞ . {\displaystyle \infty .} Said another way, if a <a href="/facts/Continuous_function/sKbl02pB">continuous function</a> f {\displaystyle f} achieves a zero at a certain value x 0 , {\displaystyle x_{0},} then it need not be the case that 1 / f {\displaystyle 1/f} tends to either − ∞ {\displaystyle -\infty } or ∞ {\displaystyle \infty } in the limit as x {\displaystyle x} tends to x 0 {\displaystyle x_{0}} . This is the case for the limits of the <a href="/facts/Identity_function/9AtsP0LC">identity function</a> f ( x ) = x {\displaystyle f(x)=x} when x {\displaystyle x} tends to 0, and of f ( x ) = x 2 sin ( 1 / x ) {\displaystyle f(x)=x^{2}\sin \left(1/x\right)} (for the latter function, neither − ∞ {\displaystyle -\infty } nor ∞ {\displaystyle \infty } is a limit of 1 / f ( x ) {\displaystyle 1/f(x)} , even if only positive values of x {\displaystyle x} are considered). </p><p>However, in contexts where only non-negative values are considered, it is often convenient to define 1 / 0 = + ∞ {\displaystyle 1/0=+\infty } . For example, when working with <a href="/facts/Power_series/vYh6sTps">power series</a>, the <a href="/facts/Radius_of_convergence/4Un0ANfP">radius of convergence</a> of a power series with <a href="/facts/Coefficient/5Hwq2Wuq">coefficients</a> a n {\displaystyle a_{n}} is often defined as the reciprocal of the <a href="/facts/Limit_inferior_and_limit_superior/QeAhlygs">limit-supremum</a> of the sequence ( | a n | 1 / n ) {\displaystyle \left(|a_{n}|^{1/n}\right)} . Thus, if one allows 1 / 0 {\displaystyle 1/0} to take the value + ∞ {\displaystyle +\infty } , then one can use this formula regardless of whether the limit-supremum is 0 or not. </p> <h2 id="algebraic-properties">Algebraic properties</h2> <p>With the arithmetic operations defined above, R ¯ {\displaystyle {\overline {\mathbb {R} }}} is not even a <a href="/facts/Semigroup/rgSdk2kt">semigroup</a>, let alone a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a>, a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> or a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> as in the case of R {\displaystyle \mathbb {R} } . However, it has several convenient properties: </p> <ul><li> a + ( b + c ) {\displaystyle a+(b+c)} and ( a + b ) + c {\displaystyle (a+b)+c} are either equal or both undefined.</li> <li> a + b {\displaystyle a+b} and b + a {\displaystyle b+a} are either equal or both undefined.</li> <li> a ⋅ ( b ⋅ c ) {\displaystyle a\cdot (b\cdot c)} and ( a ⋅ b ) ⋅ c {\displaystyle (a\cdot b)\cdot c} are either equal or both undefined.</li> <li> a ⋅ b {\displaystyle a\cdot b} and b ⋅ a {\displaystyle b\cdot a} are either equal or both undefined</li> <li> a ⋅ ( b + c ) {\displaystyle a\cdot (b+c)} and ( a ⋅ b ) + ( a ⋅ c ) {\displaystyle (a\cdot b)+(a\cdot c)} are equal if both are defined.</li> <li>If a ≤ b {\displaystyle a\leq b} and if both a + c {\displaystyle a+c} and b + c {\displaystyle b+c} are defined, then a + c ≤ b + c {\displaystyle a+c\leq b+c} .</li> <li>If a ≤ b {\displaystyle a\leq b} and c > 0 {\displaystyle c>0} and if both a ⋅ c {\displaystyle a\cdot c} and b ⋅ c {\displaystyle b\cdot c} are defined, then a ⋅ c ≤ b ⋅ c {\displaystyle a\cdot c\leq b\cdot c} .</li></ul> <p>In general, all laws of arithmetic are valid in R ¯ {\displaystyle {\overline {\mathbb {R} }}} as long as all occurring expressions are defined. </p> <h2 id="miscellaneous">Miscellaneous</h2> <p>Several functions can be <a href="/facts/Continuity_(topology)/sKbl02pB">continuously</a> <a href="/facts/Restriction_(mathematics)/JV59DhKy">extended</a> to R ¯ {\displaystyle {\overline {\mathbb {R} }}} by taking limits. For instance, one may define the extremal points of the following functions as: </p> exp ( − ∞ ) = 0 {\displaystyle \exp(-\infty )=0} , ln ( 0 ) = − ∞ {\displaystyle \ln(0)=-\infty } , tanh ( ± ∞ ) = ± 1 {\displaystyle \tanh(\pm \infty )=\pm 1} , arctan ( ± ∞ ) = ± π 2 {\displaystyle \arctan(\pm \infty )=\pm {\frac {\pi }{2}}} . <p>Some <a href="/facts/Singularity_(mathematics)/Y5sMQY4I">singularities</a> may additionally be removed. For example, the function 1 / x 2 {\displaystyle 1/x^{2}} can be continuously extended to R ¯ {\displaystyle {\overline {\mathbb {R} }}} (under <i>some</i> definitions of continuity), by setting the value to + ∞ {\displaystyle +\infty } for x = 0 {\displaystyle x=0} , and 0 for x = + ∞ {\displaystyle x=+\infty } and x = − ∞ {\displaystyle x=-\infty } . On the other hand, the function 1 / x {\displaystyle 1/x} can<i>not</i> be continuously extended, because the function approaches − ∞ {\displaystyle -\infty } as x {\displaystyle x} approaches 0 <a href="/facts/One-sided_limit/zLfr8gc5">from below</a>, and + ∞ {\displaystyle +\infty } as x {\displaystyle x} approaches 0 from above, i.e., the function not converging to the same value as its independent variable approaching to the same domain element from both the positive and negative value sides. </p><p>A similar but different real-line system, the <a href="/facts/Projectively_extended_real_line/9epfpLw2">projectively extended real line</a>, does not distinguish between + ∞ {\displaystyle +\infty } and − ∞ {\displaystyle -\infty } (i.e. infinity is unsigned).<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> As a result, a function may have limit ∞ {\displaystyle \infty } on the projectively extended real line, while in the extended real number system only the <a href="/facts/Absolute_value/dwJBpEs5">absolute value</a> of the function has a limit, e.g. in the case of the function 1 / x {\displaystyle 1/x} at x = 0 {\displaystyle x=0} . On the other hand, on the projectively extended real line, lim x → − ∞ f ( x ) {\displaystyle \lim _{x\to -\infty }{f(x)}} and lim x → + ∞ f ( x ) {\displaystyle \lim _{x\to +\infty }{f(x)}} correspond 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, the functions e x {\displaystyle e^{x}} and arctan ( x ) {\displaystyle \arctan(x)} cannot be made continuous at x = ∞ {\displaystyle x=\infty } on the projectively extended real line. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Division_by_zero/wXK7t2Ge">Division by zero</a></li> <li><a href="/facts/Extended_complex_plane/4XgBzplC">Extended complex plane</a></li> <li><a href="/facts/Extended_natural_numbers/NUEQWBj1">Extended natural numbers</a></li> <li><a href="/facts/Improper_integral/1YLuane0">Improper integral</a></li> <li><a href="/facts/Infinity/cF506uD7">Infinity</a></li> <li><a href="/facts/Log_semiring/cSDL8zg6">Log semiring</a></li> <li><a href="/facts/Series_(mathematics)/yDnlsgIe">Series (mathematics)</a></li> <li><a href="/facts/Projectively_extended_real_line/9epfpLw2">Projectively extended real line</a></li> <li>Computer representations of extended real numbers, see <a href="/facts/Floating-point_arithmetic/eIckahxe">Floating-point arithmetic § Infinities</a> and <a href="/facts/IEEE_floating_point/dD3e7zAl">IEEE floating point</a></li></ul> <h2 id="notes">Notes</h2> <h2 id="further-reading">Further reading</h2> <ul><li>Aliprantis, Charalambos D.; Burkinshaw, Owen (1998), <a href="https://books.google.com/books?id=m40ivUwAonUC&pg=PA29"><i>Principles of Real Analysis</i></a> (3rd ed.), San Diego, CA: Academic Press, Inc., p. 29, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-12-050257-7, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1669668">1669668</a></li> <li>David W. Cantrell. <a href="https://mathworld.wolfram.com/AffinelyExtendedRealNumbers.html">"Affinely Extended Real Numbers"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Some authors use Affinely extended real number system and Affinely extended real number line, although the extended real numbers do not form an affine line. <a href="/wiki/Affine_line" target="_blank">/wiki/Affine_line</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Read as "positive infinity" and "negative infinity" respectively. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Wilkins, David (2007). "Section 6: The Extended Real Number System" (PDF). maths.tcd.ie. Retrieved 2019-12-03. <a href="https://www.maths.tcd.ie/~dwilkins/Courses/221/Extended.pdf" target="_blank">https://www.maths.tcd.ie/~dwilkins/Courses/221/Extended.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Oden, J. Tinsley; Demkowicz, Leszek (16 January 2018). Applied Functional Analysis (3 ed.). Chapman and Hall/CRC. p. 74. ISBN 9781498761147. Retrieved 8 December 2019. <a href="9781498761147" target="_blank">9781498761147</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Oden, J. Tinsley; Demkowicz, Leszek (16 January 2018). Applied Functional Analysis (3 ed.). Chapman and Hall/CRC. p. 74. ISBN 9781498761147. Retrieved 8 December 2019. <a href="9781498761147" target="_blank">9781498761147</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Oden, J. Tinsley; Demkowicz, Leszek (16 January 2018). Applied Functional Analysis (3 ed.). Chapman and Hall/CRC. p. 74. ISBN 9781498761147. Retrieved 8 December 2019. <a href="9781498761147" target="_blank">9781498761147</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Weisstein, Eric W. "Affinely Extended Real Numbers". mathworld.wolfram.com. Retrieved 2019-12-03. <a href="http://mathworld.wolfram.com/AffinelyExtendedRealNumbers.html" target="_blank">http://mathworld.wolfram.com/AffinelyExtendedRealNumbers.html</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Weisstein, Eric W. "Projectively Extended Real Numbers". mathworld.wolfram.com. Retrieved 2019-12-03. <a href="http://mathworld.wolfram.com/ProjectivelyExtendedRealNumbers.html" target="_blank">http://mathworld.wolfram.com/ProjectivelyExtendedRealNumbers.html</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Weisstein, Eric W. "Projectively Extended Real Numbers". mathworld.wolfram.com. Retrieved 2019-12-03. <a href="http://mathworld.wolfram.com/ProjectivelyExtendedRealNumbers.html" target="_blank">http://mathworld.wolfram.com/ProjectivelyExtendedRealNumbers.html</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> </ol>