ចំនួនពិត៖ ភាពខុសគ្នារវាងកំណែនានា

{{more footnotes|date=April 2016}}

[[File:Latex real numbers.svg|right|thumb|120px|A symbol of the set of '''real numbers''' (ℝ)]]

 

 

The real numbers include all the [[rational number]]s, such as the [[integer]] −5 and the [[fraction (mathematics)|fraction]] 4/3, and all the [[irrational number]]s, such as {{sqrt|2}} (1.41421356…, the [[square root of 2]], an irrational [[algebraic number]]). Included within the irrationals are the [[transcendental number]]s, such as [[pi|{{pi}}]] (3.14159265…). Real numbers can be thought of as points on an infinitely long [[line (geometry)|line]] called the [[number line]] or [[real line]], where the points corresponding to [[integers]] are equally spaced. Any real number can be determined by a possibly infinite [[decimal representation]], such as that of 8.632, where each consecutive digit is measured in units one tenth the size of the previous one. The [[real line]] can be thought of as a part of the [[complex plane]], and [[complex number]]s include real numbers.

The reals are [[uncountable set|uncountable]]; that is: while both the set of all [[natural number]]s and the set of all real numbers are [[infinite set]]s, there can be no [[one-to-one function]] from the real numbers to the natural numbers: the [[cardinality]] of the set of all real numbers (denoted <math>\mathfrak c</math> and called [[cardinality of the continuum]]) is strictly greater than the cardinality of the set of all natural numbers (denoted <math>\aleph_0</math> [[aleph number#Aleph-naught|'aleph-naught']]). The statement that there is no subset of the reals with cardinality strictly greater than <math>\aleph_0</math> and strictly smaller than <math>\mathfrak c</math> is known as the [[continuum hypothesis]] (CH). It is known to be neither provable nor refutable using the axioms of [[Zermelo–Fraenkel set theory]] and the [[axiom of choice]] (ZFC), the standard foundation of modern mathematics, in the sense that some models of ZFC satisfy CH, while others violate it.

 

[[File:Number-systems.svg|thumb|Real numbers (R) include the rational (Q), which include the integers (Z), which include the natural numbers (N)]]

 

The development of [[calculus]] in the 18th century used the entire set of real numbers without having defined them cleanly. The first rigorous definition was given by [[Georg Cantor]] in 1871. In 1874, he showed that the set of all real numbers is [[uncountable|uncountably infinite]] but the set of all [[algebraic number]]s is [[countable|countably infinite]]. Contrary to widely held beliefs, his first method was not his famous [[Cantor's diagonal argument|diagonal argument]], which he published in 1891. See [[Cantor's first uncountability proof]].

 

{{Main|Construction of the real numbers}}

 

* [[Self-adjoint operator]]s on a [[Hilbert space]] (for example, self-adjoint square complex [[matrix (math)|matrices]]) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their [[eigenvector|eigenvalues]] are real and they form a real [[associative algebra]]. [[Positive-definite]] operators correspond to the positive reals and [[normal operator]]s correspond to the complex numbers.

 

{{portal|Mathematics|Algebra|Number theory|Analysis}}

* [[Continued fraction]]

* [[Real analysis]]

 

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{{Reflist|30em}}

 

* [[Georg Cantor]], 1874, "{{lang|de|Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen}}", ''{{lang|de|Journal für die Reine und Angewandte Mathematik}}'', volume 77, pages 258&ndash;262.

* [[Solomon Feferman]], 1989, ''The Number Systems: Foundations of Algebra and Analysis'', AMS Chelsea, ISBN 0-8218-2915-7.

* {{citation|first=Carol|last=Schumacher|title=ChapterZero / Fundamental Notions of Abstract Mathematics|year=1996|publisher=Addison-Wesley|isbn=0-201-82653-4}}.

 

* {{SpringerEOM|title=Real number|id=p/r080060}}

* [http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Real_numbers_1.html The real numbers: Pythagoras to Stevin]

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[[Category:Real algebraic geometry]]

[[Category:Elementary mathematics]]