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| [[File:Converging Sequence example.svg|320px|thumb|The plot of a convergent sequence {''a<sub>n</sub>''} is shown in blue. Visually we can see that the sequence is converging to the limit 0 as ''n'' increases.]]
| | '''Closure''' describes the case when the results of a mathematical operation are always defined. For example, in ordinary arithmetic, addition has closure. Whenever one adds two numbers, the answer is a number. The same is true of multiplication. Division does '''not''' have closure, because division by 0 is not defined. In the [[natural numbers]], subtraction does not have closure, but in the [[integers]] subtraction does have closure. Subtraction of two numbers can produce a negative number, which is not a natural number, but it is an integer. |
| [[File:Cauchy sequence illustration.svg|350px|thumb| The plot of a [[Cauchy sequence]] (''x<sub>n</sub>''), shown in blue, as ''x<sub>n</sub>'' versus ''n''. Visually, we see that the sequence appears to be converging to a limit point as the terms in the sequence become closer together as ''n'' increases. In the [[real numbers]] every Cauchy sequence converges to some limit.]]
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| In [[mathematics]], a [[sequence]] is an ordered list of events.<ref>{{cite web|author= Weisstein, Eric W.|title=Sequence|publisher=MathWorld--A Wolfram Web Resource|url= http://mathworld.wolfram.com/Sequence.html}}</ref> Each event consists of a mathematical object, which is a [[number]] in many cases. In some cases, the sequence tends towards a '''limit'''. In this case, the sequence is said to be '''convergent''', otherwise it is '''divergent'''.<ref>Courant, Richard (1961). ''Differential and Integral Calculus'' Volume I. Glasgow: Blackie & Son, Ltd., p. 29.</ref>
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| In a convergent sequence, the [[Absolute value|absolute]] difference between the value of the current element and the limit, <math>|x_{n} - lim(X)| </math> will decrease as the sequence progresses.
| | One can sometimes make closure of a mathematical object by including new elements. The integers are a closure of the natural numbers by including negative numbers. The [[real numbers]] are a closure of the [[rational numbers]] by including square roots of positive numbers. The [[complex numbers]] are a closure of the real numbers by including square roots of negative numbers. |
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| ==Formal Definition== | | Sometimes, if one includes an element to make closure, it makes more changes. For example, if one includes [[infinity]], <math>\infty = 1 / 0 </math>, that is closure of division, the laws of addition and subtraction are changed. There is no inversion of addition for <math>\infty</math>. |
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| We call <math>x</math> the '''limit''' of the sequence <math>(x_n)</math> if the following condition holds:
| | Ordinary closure is called finite closure. There is also infinite closure. The definition of a [[topological space]] mentions infinite closure. Open spaces have (finite) closure of [[intersection]]. Open sets have infinite closure of [[union (set theory)|union]]. That is, in mathematical notation, if A<sub>0</sub>, and A<sub>1</sub> and ... A<sub>''n''</sub> ... are open sets, then B and C are open sets: |
| :*For each [[real number]] <math>\epsilon > 0</math>, there exists a [[natural number]] <math>N</math> such that, for every natural number <math>n > N</math>, we have <math>|x_n - x| < \epsilon</math>.
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| In other words, for every measure of closeness <math>\epsilon</math>, the sequence's terms are eventually that close to the limit. The sequence <math>(x_n)</math> is said to '''converge to''' or '''tend to''' the limit <math>x</math>, written <math>x_n \to x</math> or <math>\lim_{n \to \infty} x_n = x</math>.
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| If a sequence converges to some limit, then it is '''convergent'''; otherwise it is '''divergent'''.
| | * <math>B = A_0 \cap A_1 </math> |
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| ==History==
| | * <math>C = {\cup_n} A_n </math> |
| The Greek philosopher [[Zeno of Elea]] is famous for formulating [[Zeno's paradoxes|paradoxes that involve limiting processes]].<ref>Craig, Edward (1998) ''Routledge Encyclopedia of Philosophy: Genealogy to Iqbal''. London: Routledge, p. 773.</ref>
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| [[Leucippus]], [[Democritus]], [[Antiphon (person)|Antiphon]], [[Eudoxus of Cnidus|Eudoxus]] and [[Archimedes]] developed the [[method of exhaustion]], which uses an infinite sequence of approximations to determine an area or a volume. Archimedes succeeded in summing what is now called a [[geometric series]].
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| [[Isaac Newton|Newton]] dealt with series in his works on ''Analysis with infinite series'' (written in 1669, circulated in manuscript, published in 1711), ''Method of fluxions and infinite series'' (written in 1671, published in English translation in 1736, Latin original published much later) and ''Tractatus de Quadratura Curvarum'' (written in 1693, published in 1704 as an Appendix to his ''Optiks''). In the latter work, Newton considers the binomial expansion of (''x''+''o'')<sup>''n''</sup> which he then linearizes by ''taking limits'' (letting ''o''→0).
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| In the 18th century, [[mathematician]]s like [[Leonhard Euler|Euler]] succeeded in summing some ''divergent'' series by stopping at the right moment; they did not much care whether a limit existed, as long as it could be calculated. At the end of the century, [[Joseph Louis Lagrange|Lagrange]] in his ''Théorie des fonctions analytiques'' (1797) opined that the lack of rigour precluded further development in calculus. [[Carl Friedrich Gauss|Gauss]] in his etude of [[hypergeometric series]] (1813) for the first time rigorously investigated under which conditions a series converged to a limit.
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| The modern definition of a limit (for any ε there exists an index ''N'' so that ...) was given by [[Bernhard Bolzano]] (''Der binomische Lehrsatz'', Prague 1816, little noticed at the time) and by [[Karl Weierstraß]] in the 1870s.
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| ==References== | |
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| [[Category:Mathematics]] | | [[Category:Mathematics]] |
Closure describes the case when the results of a mathematical operation are always defined. For example, in ordinary arithmetic, addition has closure. Whenever one adds two numbers, the answer is a number. The same is true of multiplication. Division does not have closure, because division by 0 is not defined. In the natural numbers, subtraction does not have closure, but in the integers subtraction does have closure. Subtraction of two numbers can produce a negative number, which is not a natural number, but it is an integer.
One can sometimes make closure of a mathematical object by including new elements. The integers are a closure of the natural numbers by including negative numbers. The real numbers are a closure of the rational numbers by including square roots of positive numbers. The complex numbers are a closure of the real numbers by including square roots of negative numbers.
Sometimes, if one includes an element to make closure, it makes more changes. For example, if one includes infinity, [math]\displaystyle{ \infty = 1 / 0 }[/math], that is closure of division, the laws of addition and subtraction are changed. There is no inversion of addition for [math]\displaystyle{ \infty }[/math].
Ordinary closure is called finite closure. There is also infinite closure. The definition of a topological space mentions infinite closure. Open spaces have (finite) closure of intersection. Open sets have infinite closure of union. That is, in mathematical notation, if A0, and A1 and ... An ... are open sets, then B and C are open sets:
- [math]\displaystyle{ B = A_0 \cap A_1 }[/math]
- [math]\displaystyle{ C = {\cup_n} A_n }[/math]