Рационалан број — разлика између измена
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{{Short description|Количник два цела броја}}{{rut}}
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[[File:U%2B211A.svg|right|thumb|120px|A symbol for the set of rational numbers]]
[[File:Number-systems.svg|thumb|250px|The rational numbers (<math>\mathbb{Q}</math>) are included in the [[real numbers]] (<math>\mathbb{R}</math>), while themselves including the [[integers]] (<math>\mathbb{Z}</math>), which in turn include the [[natural numbers]] (<math>\mathbb{N}</math>)]]
У [[mathematics|математици]], '''рационалан број''' (понекад у разговору употребљавамо '''разломак''') је [[number|број]] који се може записати као [[quotient|однос]] два цела броја ''a''/''b'', где ''b'' није [[0 (број)|нула]]. <ref name="Rosen">{{cite book |last = Rosen |first=Kenneth |title=Discrete Mathematics and its Applications |year=2007 |edition=6th |publisher=McGraw-Hill |location=New York, NY|isbn=978-0-07-288008-3 |pages=105, 158–160}}</ref> For example, {{math|{{sfrac|−3|7}}}} is a rational number, as is every integer (e.g. {{math|5 {{=}} {{sfrac|5|1}}}}). The [[set (mathematics)|set]] of all rational numbers, also referred to as "'''the rationals'''",<ref>{{cite book |title=Elements of Pure and Applied Mathematics |edition=illustrated |first1=Harry |last1=Lass |publisher=Courier Corporation |year=2009 |isbn=978-0-486-47186-0 |page=382 |url=https://books.google.com/books?id=WAY_AwAAQBAJ}} [https://books.google.com/books?id=WAY_AwAAQBAJ&pg=PA382 Extract of page 382]</ref> the '''field of rationals'''<ref>{{cite book |title=The Collected Works of Julia Robinson |first1=Julia |last1=Robinson |publisher=American Mathematical Soc |year=1996 |isbn=978-0-8218-0575-6 |page=104 |url=https://books.google.com/books?id=_33D84OENIAC}} [https://books.google.com/books?id=_33D84OENIAC&pg=PA104 Extract of page 104]</ref> or the '''field of rational numbers''' is usually denoted by a boldface {{math|'''Q'''}} (or [[blackboard bold]] <math>\mathbb{Q}</math>, Unicode {{unichar|1D410|MATHEMATICAL BOLD CAPITAL Q}} or {{unichar|211A|DOUBLE-STRUCK CAPITAL Q}});<ref>{{cite web|last1=Rouse|first1=Margaret|title=Mathematical Symbols|url=http://searchdatacenter.techtarget.com/definition/Mathematical-Symbols|access-date=1 April 2015}}</ref> it was thus denoted in 1895 by [[Giuseppe Peano]] after ''[[wikt:quoziente|quoziente]]'', Italian for "[[quotient]]", and first appeared in Bourbaki's ''Algèbre''.<ref name=":0" />
Сваки рационалан број може бити написан на бесконачан број начина, на пример <math>\frac{3}{6} = \frac{2}{4} = \frac{1}{2}</math>. Најједноставнији облик је када [[бројилац]] и [[именилац]] немају заједничког [[дељење|делитеља]] (узајамно су [[прост број|прости]]), а сваки рационалан број различит од нуле има тачно једну једноставну форму са позитивним имениоцем. Рационални бројеви имају децимални развој са периодичним понављањем група цифара. Овде се рачуна и случај када нема децимала или када се од неког места 0 понавља бесконачно. Ово је истинито за сваку целобројну основу већу од 1. Другим речима, ако је развој исписа неког броја у некој бројној основи периодичан, он је периодичан у свим основама, а број је рационалан. [[Реалан број]] који није рационалан се зове [[ирационалан број|ирационалан]]. [[Скуп]] свих рационалних бројева, који чине [[поље (математика)|поље]], означава се са <math>\mathbb{Q}</math>. Користећи скуповну нотацију <math>\mathbb{Q}</math> се дефинише као: <math>\mathbb{Q} = \left\{\frac{m}{n} : m \in \mathbb{Z}, n \in \mathbb{Z}, n \ne 0 \right\},</math> где је <math>\mathbb{Z}</math> скуп [[цео број|целих бројева]].
The [[decimal expansion]] of a rational number either terminates after a finite number of [[numerical digit|digits]] (example: {{math|{{sfrac|3|4}} {{=}} 0.75}}), or eventually begins to [[repeating decimal|repeat]] the same finite [[sequence]] of digits over and over (example: {{math|{{sfrac|9|44}} {{=}} 0.20454545...}}).<ref>{{Cite web|title=Rational number|url=https://www.britannica.com/science/rational-number|access-date=2020-08-11|website=Encyclopedia Britannica|language=en}}</ref> Conversely, any repeating or terminating decimal represents a rational number. These statements are true in [[decimal|base 10]], and in every other integer [[radix|base]] (for example, [[binary numeral system|binary]] or [[hexadecimal]]).
A [[real number]] that is not rational is called [[irrational number|irrational]].<ref name=":0">{{Cite web|last=Weisstein|first=Eric W.|title=Rational Number|url=https://mathworld.wolfram.com/RationalNumber.html|access-date=2020-08-11|website=mathworld.wolfram.com|language=en}}</ref> Irrational numbers include {{math|[[square root of 2|{{sqrt|2}}]]}}, [[Pi|{{pi}}]], {{math|[[E (mathematical constant)|''e'']]}}, and {{math|[[Golden ratio|''φ'']]}}. The [[decimal expansion]] of an irrational number continues without repeating. Since the set of rational numbers is [[countable set|countable]], and the set of real numbers is [[uncountable set|uncountable]], [[almost all]] real numbers are irrational.<ref name="Rosen"/>
Rational numbers can be [[Formalism (mathematics)|formally]] defined as [[equivalence class]]es of pairs of integers {{math|(''p'', ''q'')}} with {{math|''q'' ≠ 0}}, using the [[equivalence relation]] defined as follows:
: <math>\left( p_1, q_1 \right) \sim \left( p_2, q_2 \right) \iff p_1 q_2 = p_2 q_1.</math>
The fraction {{math|{{sfrac|''p''|''q''}}}} then denotes the equivalence class of {{math|(''p'', ''q'')}}.<ref name=":1">{{Cite book|last=Biggs|first=Norman L.|title=Discrete Mathematics|publisher=Oxford University Press|year=2002|isbn=978-0-19-871369-2|location=India|pages=75–78}}</ref>
Rational numbers together with [[addition]] and [[multiplication]] form a [[field (mathematics)|field]] which contains the [[integer]]s, and is contained in any field containing the integers. In other words, the field of rational numbers is a [[prime field]], and a field has [[characteristic zero]] if and only if it contains the rational numbers as a subfield. Finite [[field extension|extensions]] of {{math|'''Q'''}} are called [[algebraic number field]]s, and the [[algebraic closure]] of {{math|'''Q'''}} is the field of [[algebraic number]]s.<ref name="Gilbert">{{cite book |last1=Gilbert |first1=Jimmie |last2=Linda |first2=Gilbert|author2-link=Linda Gilbert Saucier |year=2005 |title=Elements of Modern Algebra |edition=6th |publisher=Thomson Brooks/Cole |location=Belmont, CA |isbn=0-534-40264-X |pages=243–244}}</ref>
== Етимологија ==
Although nowadays ''rational numbers'' are defined in terms of ''ratios'', the term ''rational'' is not a [[morphological derivation|derivation]] of ''ratio''. On the opposite, it is ''ratio'' that is derived from ''rational'': the first use of ''ratio'' with its modern meaning was attested in English about 1660,<ref>{{cite book|title=Oxford English Dictionary|edition=2nd|date=1989|publisher=Oxford University Press}} Entry '''ratio''', ''n.'', sense 2.a.</ref> while the use of ''rational'' for qualifying numbers appeared almost a century earlier, in 1570.<ref>{{cite book|title=Oxford English Dictionary|edition=2nd|date=1989|publisher=Oxford University Press}} Entry '''rational''', ''a. (adv.)'' and ''n.''<sup>1</sup>, sense 5.a.</ref> This meaning of ''rational'' came from the mathematical meaning of ''irrational'', which was first used in 1551, and it was used in "translations of Euclid (following his peculiar use of {{lang|grc|ἄλογος}})".<ref>{{cite book|title=Oxford English Dictionary|edition=2nd|date=1989|publisher=Oxford University Press}} Entry '''irrational''', ''a.'' and ''n.'', sense 3.</ref><ref>{{Cite web|date=2017-05-09|first=Peter|last= Shor|authorlink=Peter Shor|title=Does rational come from ratio or ratio come from rational|url=https://english.stackexchange.com/questions/217956/does-rational-come-from-ratio-or-ratio-come-from-rational/218079#218079|access-date=2021-03-19|website=Stack Exchange|language=en-US}}</ref>
This unusual history originated in the fact that [[Greek mathematics|ancient Greeks]] "avoided heresy by forbidding themselves from thinking of those [irrational] lengths as numbers".<ref>{{Cite web|last1=Coolman|first1=Robert|date=2016-01-29|title=How a Mathematical Superstition Stultified Algebra for Over a Thousand Years|url=https://nautil.us/blog/how-a-mathematical-superstition-stultified-algebra-for-over-a-thousand-years|access-date=2021-03-20|language=en-US}}</ref> So such lengths were ''irrational'', in the sense of ''illogical'', that is "not to be spoken about" ({{lang|grc|ἄλογος}} in Greek).<ref>{{cite book|last1=Kramer|first1=Edna|title=The Nature and Growth of Modern Mathematics|date=1983|publisher=Princeton University Press|page=28}}</ref>
== Аритметика ==
[[Датотека:Fracciones.gif|мини||
Два рационална броја (разломка) <math>\frac{a}{b}</math> и <math>\frac{c}{d}</math> су једнаки ако и само ако важи <math>ad = bc\,</math>.
Линија 35 ⟶ 46:
За сваки позитивни рационални број постоји бесконачно много начина да се број овако представи и то се зову египатски разломци. Код старих Египћана је овакав начин представљања био основа за све математичке радње.
== Формална конструкција ==
== Спољашње везе ==▼
[[File:Rational Representation.svg|thumb|right|250px|Дијаграм који приказује репрезентацију еквивалентних класа парова целих бројева]]
{{Commonscat|Rational numbers}}▼
The rational numbers may be built as [[equivalence class]]es of [[ordered pair]]s of [[integer]]s.<ref name=":1" /><ref name=":2">{{Cite web|title=Fraction - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Fraction|access-date=2021-08-17|website=encyclopediaofmath.org}}</ref>
More precisely, let {{math|('''Z''' × ('''Z''' \ {0}))}} be the set of the pairs {{math|(''m'', ''n'')}} of integers such {{math|''n'' ≠ 0}}. An [[equivalence relation]] is defined on this set by
: <math>\left(m_1, n_1 \right) \sim \left(m_2, n_2 \right) \iff m_1 n_2 = m_2 n_1.</math><ref name=":1" /><ref name=":2" />
Addition and multiplication can be defined by the following rules:
:<math>\left(m_1, n_1\right) + \left(m_2, n_2\right) \equiv \left(m_1n_2 + n_1m_2, n_1n_2\right),</math>
:<math>\left(m_1, n_1\right) \times \left(m_2, n_2\right) \equiv \left(m_1m_2, n_1n_2\right).</math><ref name=":1" />
This equivalence relation is a [[congruence relation]], which means that it is compatible with the addition and multiplication defined above; the set of rational numbers {{math|'''Q'''}} is the defined as the [[quotient set]] by this equivalence relation, {{math|1=('''Z''' × ('''Z''' \ {0})) / ~}}, equipped with the addition and the multiplication induced by the above operations. (This construction can be carried out with any [[integral domain]] and produces its [[field of fractions]].)<ref name=":1" />
The equivalence class of a pair {{math|(''m'', ''n'')}} is denoted {{math|{{sfrac|''m''|''n''}}}}.
Two pairs {{math|(''m''<sub>1</sub>, ''n''<sub>1</sub>)}} and {{math|(''m''<sub>2</sub>, ''n''<sub>2</sub>)}} belong to the same equivalence class (that is are equivalent) if and only if {{math|''m''<sub>1</sub>''n''<sub>2</sub> {{=}} ''m''<sub>2</sub>''n''<sub>1</sub>}}. This means that {{math|{{sfrac|''m''<sub>1</sub>|''n''<sub>1</sub>}} {{=}} {{sfrac|''m''<sub>2</sub>|''n''<sub>2</sub>}}}} if and only {{math|''m''<sub>1</sub>''n''<sub>2</sub> {{=}} ''m''<sub>2</sub>''n''<sub>1</sub>}}.<ref name=":1" /><ref name=":2" />
Every equivalence class {{math|{{sfrac|''m''|''n''}}}} may be represented by infinitely many pairs, since
:<math>\cdots = \frac{-2m}{-2n} = \frac{-m}{-n} = \frac{m}{n} = \frac{2m}{2n} = \cdots.</math>
Each equivalence class contains a unique ''[[representative (mathematics)|canonical representative element]]''. The canonical representative is the unique pair {{math|(''m'', ''n'')}} in the equivalence class such that {{mvar|m}} and {{mvar|n}} are [[coprime]], and {{math|''n'' > 0}}. It is called the [[irreducible fraction|representation in lowest terms]] of the rational number.
The integers may be considered to be rational numbers identifying the integer {{mvar|n}} with the rational number {{math|{{sfrac|''n''|1}}}}.
A [[total order]] may be defined on the rational numbers, that extends the natural order of the integers. One has
:<math>\frac{m_1}{n_1} \le \frac{m_2}{n_2}</math>
if
:<math>(n_1n_2 > 0 \quad \text{and} \quad m_1n_2 \le n_1m_2)\qquad \text{or}\qquad (n_1n_2 < 0 \quad \text{and} \quad m_1n_2 \ge n_1m_2).</math>
== Референце ==
{{Reflist}}
== Литература ==
{{Refbegin|30em}}
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*{{citation|title=Theory of Algebraic Functions of One Variable|volume=39|series=History of mathematics|first1=Richard|last1=Dedekind|author1-link=Richard Dedekind|first2=Heinrich|last2=Weber|author2-link=Heinrich Martin Weber|publisher=American Mathematical Society|year=2012|isbn=978-0-8218-8330-3}}. — Translation into English by [[John Stillwell]] of ''Theorie der algebraischen Functionen einer Veränderlichen'' (1882).
*{{Citation|last=Gouvêa|first=F. Q.|author-link=F. Q. Gouvêa|date=March 1994|title=A Marvelous Proof|journal=[[American Mathematical Monthly]]|pages=203–222|volume=101|issue=3 |jstor=2975598|doi=10.2307/2975598}}
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*{{Citation|last=Hehner|first=Eric C. R.|author-link=Eric C. R. Hehner|last2=Horspool|first2=R. Nigel|author2-link=R. Nigel Horspool|year=1979|title=A new representation of the rational numbers for fast easy arithmetic|journal=[[SIAM Journal on Computing]]|pages=124–134 |volume=8 |issue=2 |doi=10.1137/0208011 |url=https://www.researchgate.net/publication/220617770|citeseerx=10.1.1.64.7714}}
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{{Refend}}
▲== Спољашње везе ==
* {{springer|title=Rational number|id=p/r077620}}
* [http://mathworld.wolfram.com/RationalNumber.html "Rational Number" From MathWorld – A Wolfram Web Resource]
* [http://math.stanford.edu/~conrad/248APage/handouts/algclosurecomp.pdf Completion of Algebraic Closure] – on-line lecture notes by Brian Conrad
* [https://web.archive.org/web/20161213093839/http://www.maths.gla.ac.uk/~ajb/dvi-ps/padicnotes.pdf An Introduction to ''p''-adic Numbers and ''p''-adic Analysis] - on-line lecture notes by Andrew Baker, 2007
* [http://homes.esat.kuleuven.be/~fvercaut/talks/pAdic.pdf Efficient p-adic arithmetic] (slides)
* [http://www.madore.org/~david/math/padics.pdf Introduction to p-adic numbers]
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