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In mathematics, a sequence of nested intervals can be intuitively understood as an ordered collection of intervals on the real number line with natural numbers as an index. In order for a sequence of intervals to be considered nested intervals, two conditions have to be met: 1. * Every interval in the sequence is contained in the previous one ( is always a subset of ). 2. * The length of the intervals get arbitrarily small (meaning the length falls below every possible threshold after a certain index ).

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rdfs:label	Nested intervals (en)
rdfs:comment	In mathematics, a sequence of nested intervals can be intuitively understood as an ordered collection of intervals on the real number line with natural numbers as an index. In order for a sequence of intervals to be considered nested intervals, two conditions have to be met: 1. * Every interval in the sequence is contained in the previous one ( is always a subset of ). 2. * The length of the intervals get arbitrarily small (meaning the length falls below every possible threshold after a certain index ). (en)
sameAs	Nested intervals Nested intervals
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Further dbt:Pi dbt:Reflist
Subject	Theorems in real analysis Sets of real numbers
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Nested_intervals?oldid=1074589682&ns=0
Wikipage page ID	7532405 (xsd:integer)
page length (characters) of wiki page	22837 (xsd:nonNegativeInteger)
Wikipage revision ID	1074589682 (xsd:integer)
Link from a Wikipage to another Wikipage	Root-finding algorithms Uncountable set Algorithm Approximation Archimedean property Noli turbare circulos meos! Babylonia Bisection Bisection method Bolzano–Weierstrass theorem Sequence Calculus Cantor's intersection theorem Cauchy sequence Differentiable function Differential calculus Differential equation Disjoint sets Disk (mathematics) Methods of computing square roots Physics Pi Without loss of generality Circle De Morgan's laws Square number Complete metric space Computer science Connectedness Continuous function Hermann Weyl Hexagon Newton's method Ordered field Empty set Engineering Gottfried Wilhelm Leibniz Field (mathematics) Infimum and supremum Absolutely integrable function Intersection (set theory) Interval (mathematics) Ludolph van Ceulen Isaac Newton Limit of a sequence Limit point Numerical analysis Mathematics Inverse second History of numbers Rational number Real line Real number Recursion Singleton Theorems in real analysis Sets of real numbers
has abstract	In mathematics, a sequence of nested intervals can be intuitively understood as an ordered collection of intervals on the real number line with natural numbers as an index. In order for a sequence of intervals to be considered nested intervals, two conditions have to be met: 1. * Every interval in the sequence is contained in the previous one ( is always a subset of ). 2. * The length of the intervals get arbitrarily small (meaning the length falls below every possible threshold after a certain index ). In other words, the left bound of the interval can only increase, and the right bound can only decrease. Historically - long before anyone defined nested intervals in a textbook - people implicitly constructed such nestings for concrete calculation purposes. For example, the ancient Babylonians discovered a method for computing square roots of numbers. In contrast, the famed Archimedes constructed sequences of polygons, that inscribed and surcumscribed a unit circle, in order to get a lower and upper bound for the circles circumference - which is the circle number Pi. The central question to be posed is the nature of the intersection over all the natural numbers, or, put differently, the set of numbers, that are found in every Interval (thus, for all ). In modern mathematics, nested intervals are used as a construction method for the real numbers (in order to complete the field of rational numbers). (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Nested_intervals
is Wikipage redirect of	Nested Intervals Theorem Nested interval property Nested intervals theorem Nested sequence of closed intervals Nested sequence of intervals Nested sequences of intervals
is Link from a Wikipage to another Wikipage of	0.999... Bisection method Bolzano–Weierstrass theorem Cantor's first set theory article Cardinal number Georg Cantor Computable number Finite intersection property Infimum and supremum Nested set model Nested Intervals Theorem Nested interval property Nested intervals theorem Nested sequence of closed intervals Nested sequence of intervals Nested sequences of intervals
is foaf:primaryTopic of	http://en.wikipedia.org/wiki/Nested_intervals

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