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{{short description|Quotient of two integers}}
{{redirect|Rationals||Rational (disambiguation)}}
{{Refimprove|date=September 2013}}
[[File:Number-systems.svg|thumb|The rational numbers (ℚ) are included in the [[real numbers]] (ℝ), while themselves including the [[integers]] (ℤ), which in turn include the [[natural numbers]] (ℕ)]]

In [[mathematics]], a '''rational number''' is a [[number]] that can be expressed as the [[quotient]] or [[fraction (mathematics)|fraction]] {{math|''p''/''q''}} of two [[integer]]s, a [[numerator]] {{math|''p''}} and a non-zero [[denominator]] {{math|''q''}}.<ref name="Rosen">{{cite book |last = Rosen |first=Kenneth |year= |title=Discrete Mathematics and its Applications |edition=6th |publisher=McGraw-Hill |location=New York, NY|isbn=978-0-07-288008-3 |pages=105, 158–160}}</ref> Since {{math|''q''}} may be equal to&nbsp;1, every integer is a rational number. The [[set (mathematics)|set]] of all rational numbers, often referred to as "'''the rationals'''", the '''field of rationals''' or the '''field of rational numbers''' is usually denoted by a boldface {{math|'''Q'''}} (or [[blackboard bold]] <math>\mathbb{Q}</math>, Unicode ℚ);<ref>{{cite web|last1=Rouse|first1=Margaret|title=Mathematical Symbols|url=http://searchdatacenter.techtarget.com/definition/Mathematical-Symbols|accessdate=1 April 2015}}</ref> it was thus denoted in 1895 by [[Giuseppe Peano]] after ''[[wikt:quoziente|quoziente]]'', Italian for "[[quotient]]".

The [[decimal expansion]] of a rational number always either terminates after a finite number of [[numerical digit|digits]] or begins to [[repeating decimal|repeat]] the same finite [[sequence]] of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for [[decimal|base 10]], but also for any other integer [[radix|base]] (e.g. [[binary numeral system|binary]], [[hexadecimal]]).

A [[real number]] that is not rational is called [[irrational number|irrational]]. Irrational numbers include [[square root of 2|{{math|{{sqrt|2}}}}]], [[Pi|{{pi}}]], [[E (mathematical constant)|{{math|''e''}}]], and [[Golden ratio|{{math|''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'')}} such that {{math|''q'' ≠ 0}}, for the [[equivalence relation]] defined by {{math|(''p''<sub>1</sub>, ''q''<sub>1</sub>) ~ (''p''<sub>2</sub>, ''q''<sub>2</sub>)}} [[iff|if, and only if]] {{math|''p''<sub>1</sub>''q''<sub>2</sub> {{=}} ''p''<sub>2</sub>''q''<sub>1</sub>}}. With this formal definition, the fraction {{math|''p''/''q''}} becomes the standard notation for the equivalence class of {{math|(''p'', ''q'')}}.

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>

In [[mathematical analysis]], the rational numbers form a [[dense set|dense subset]] of the real numbers. The real numbers can be constructed from the rational numbers by [[completion (metric space)|completion]], using [[Cauchy sequence]]s, [[Dedekind cut]]s, or infinite [[decimal]]s (see [[Construction of the real numbers]]).

==Terminology==
The term ''rational'' in reference to the set '''Q''' refers to the fact that a rational number represents a ''[[ratio]]'' of two integers. In mathematics, "rational" is often used as a noun abbreviating "rational number". The adjective ''rational'' sometimes means that the [[coefficient]]s are rational numbers. For example, a [[rational point]] is a point with rational [[coordinates]] (that is a point whose coordinates are rational numbers); a ''rational matrix'' is a [[matrix (mathematics)|matrix]] of rational numbers; a ''rational polynomial'' may be a polynomial with rational coefficients, although the term "polynomial over the rationals" is generally preferred, for avoiding confusion with "[[rational fraction|rational expression]]" and "[[rational function]]" (a polynomial is a rational expression and defines a rational function, even if its coefficients are not rational numbers). However, a [[rational curve]] ''is not'' a curve defined over the rationals, but a curve which can be parameterized by rational functions.

==Arithmetic==
{{see also|Fraction (mathematics)#Arithmetic with fractions}}
===Irreducible fraction===
Every rational number may be expressed in a unique way as an [[irreducible fraction]] {{math|''a''/''b''}}, where {{mvar|a}} and {{mvar|b}} are [[coprime integers]], and {{math|''b'' > 0}}. This is often called the [[canonical form]].

Starting from a rational number {{math|''a''/''b''}}, its canonical form may be obtained by dividing {{mvar|a}} and {{mvar|b}} by their [[greatest common divisor]], and, if {{math|''b'' < 0}}, changing the sign of the resulting numerator and denominator.

===Embedding of integers===
Any integer {{math|''n''}} can be expressed as the rational number {{math|''n''/1}}, which is its canonical form as a rational number.

===Equality===
:<math>\frac{a}{b} = \frac{c}{d}</math> if and only if <math>ad = bc.</math>

If both fractions are in canonical form then 
:<math>\frac{a}{b} = \frac{c}{d}</math> if and only if <math>a = c</math> and <math>b = d</math>
<!--Examples: 
:<math>\frac{1}{3} = \frac{2}{6}</math>
:<math>\frac{-1}{2} = \frac{1}{-2}</math>
:<math>\frac{0}{1} = \frac{0}{2}</math>-->

===Ordering===
If both denominators are positive, and, in particular, if both fractions are in canonical form,
:<math>\frac{a}{b} < \frac{c}{d}</math> if and only if <math>ad < bc.</math>

If either denominator is negative, each fraction with a negative denominator must first be converted into an equivalent form with a positive denominator by changing the signs of both its numerator and denominator.

===Addition===
Two fractions are added as follows:
:<math>\frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}.</math>

If both fractions are in canonical form, the result is in canonical form if and only if {{mvar|b}} and {{mvar|d}} are [[coprime integers]].

===Subtraction===
:<math>\frac{a}{b} - \frac{c}{d} = \frac{ad-bc}{bd}.</math>

If both fractions are in canonical form, the result is in canonical form if and only if {{mvar|b}} and {{mvar|d}} are [[coprime integers]].

===Multiplication===
The rule for multiplication is:
:<math>\frac{a}{b} \cdot\frac{c}{d} = \frac{ac}{bd}.</math>

Even if both fractions are in canonical form, the result may be a [[reducible fraction]].

===Inverse===
Every rational number {{math|''a''/''b''}} has an [[additive inverse]], often called its ''opposite'',
:<math> - \left( \frac{a}{b} \right) = \frac{-a}{b}.</math>
If {{math|''a''/''b''}} is in canonical form, the same is true for its opposite.

A nonzero rational number {{math|''a''/''b''}} has a [[multiplicative inverse]], also called its ''reciprocal'',
:<math> \left(\frac{a}{b}\right)^{-1} = \frac{b}{a}. </math>
If {{math|''a''/''b''}} is in canonical form, then the canonical form of its reciprocal is either {{tmath|\frac{b}{a} }} or {{tmath|\frac{-b}{-a} }}, depending on the sign of {{mvar|a}}.

===Division===
If {{math|''b''}}, {{math|''c''}}, and {{math|''d''}} are nonzero, the division rule is 
:<math>\frac{\frac{a}{b}} {\frac{c}{d}} = \frac{ad}{bc}.</math>

Thus, dividing {{math|''a''/''b''}} by {{math|''c''/''d''}} is equivalent to multiplying {{math|''a''/''b''}} by the [[multiplicative inverse|reciprocal]] of {{math|''c''/''d''}}:
:<math>\frac{ad}{bc} = \frac{a}{b} \cdot \frac{d}{c}.</math>

===Exponentiation to integer power===
If {{math|''n''}} is a non-negative integer, then
:<math>\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}.</math>
The result is in canonical form if the same is true for {{math|''a''/''b''}}.
In particular, 
:<math>\left(\frac{a}{b}\right)^0 = 1.</math>

If {{math|''a'' ≠ 0}}, then
:<math>\left(\frac{a}{b}\right)^{-n} = \frac{b^n}{a^n}.</math>
If {{math|''a''/''b''}} is in canonical form, the canonical form of the result is <math>\frac{b^n}{a^n}</math> if either {{math|''a'' > 0}} or {{mvar|n}} is even. Otherwise, the canonical form of the result is <math>\frac{-b^n}{-a^n}.</math>

==Continued fraction representation==
{{Main|Continued fraction}}
A '''finite continued fraction''' is an expression such as
:<math>a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{ \ddots + \cfrac{1}{a_n} }}},</math>
where ''a<sub>n</sub>'' are integers. Every rational number ''a''/''b'' can be represented as a finite continued fraction, whose [[coefficient]]s ''a<sub>n</sub>'' can be determined by applying the [[Euclidean algorithm]] to (''a'',''b'').

== Other representations ==

* [[common fraction]]: <math>\frac{8}{3}</math>
* [[mixed numeral]]: <math>2\tfrac{2}{3}</math>
* [[repeating decimal]] using a [[vinculum (symbol)|vinculum]]: <math>2.\overline{6}</math>
* repeating decimal using [[parentheses]]: <math>2.(6)</math>
* [[continued fraction]] using traditional typography: <math>2 + \cfrac{1}{1 + \cfrac{1}{2} }</math>
* continued fraction in abbreviated notation: [2; 1, 2]
* [[egyptian fraction]]: <math> 2 + \frac{1}{2} + \frac{1}{6} </math>
* [[Fundamental theorem of arithmetic#Canonical representation of a positive integer|prime power decomposition]]: <math>2^3\times3^{-1}</math>
* [[quote notation]]: '''3!6'''

are different ways to represent the same rational value.

==Formal construction==
[[File:Rational Representation.svg|thumb|right|300px|A diagram showing a representation of the equivalent classes of pairs of integers]]
The rational numbers may be built as [[equivalence class]]es of [[ordered pair]]s of [[integer]]s. 

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|(''m''<sub>1</sub>, ''n''<sub>1</sub>) ~ (''m''<sub>2</sub>, ''n''<sub>2</sub>)}} if and only if {{math|1=''m''<sub>1</sub>''n''<sub>2</sub> = ''m''<sub>2</sub>''n''<sub>1</sub>}}.

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>

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]].) 

The equivalence class of a pair {{math|(''m'', ''n'')}} is denoted <math>\frac mn.</math>  
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_1n_2=m_2n_1.</math> this means that <math>\frac{m_1}{n_1}=\frac{m_2}{n_2}</math> if and only <math>m_1n_2=m_2n_1.</math>

Every equivalence class <math>\frac mn</math> 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 ''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>\frac n1.</math>

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>(m_1n_2 \le n_1m_2 \quad\text{if}\quad n_1n_2>0)\quad \text{or}\quad (m_1n_2 \ge n_1m_2 \quad\text{if} \quad n_1n_2 < 0).</math>

==Properties==
[[File:Diagonal argument.svg|thumb|right|200px|A diagram illustrating the countability of the positive rationals]]
The set '''Q''' of all rational numbers, together with the addition and multiplication operations shown above, forms a [[field (mathematics)|field]].

'''Q''' has no [[field automorphism]] other than the identity.

With the order defined above, '''Q''' is an [[ordered field]] that has no subfield other than itself, and is the smallest ordered field, in the sense that every ordered field contains a unique subfield [[isomorphism|isomorpic]] to '''Q'''.

'''Q''' is a [[prime field]], which is a field that has no subfield other than itself.<ref>{{cite book |url=https://books.google.co.il/books?id=WHjO9K6xEm4C&pg=PA578 |title=Encyclopedic Dictionary of Mathematics, Volume 1 |page=578 |location=London, England |publisher=MIT Press |isbn=0-2625-9020-4 |first=Nihon |last=Sūgakkai |year=1993}}</ref> The rationals are the smallest field with [[characteristic (algebra)|characteristic]] zero. Every field of characteristic zero contains a unique subfield isomorpic to '''Q'''.

'''Q''' is the [[field of fractions]] of the [[integer]]s '''Z'''.<ref>{{cite book |last1=Bourbaki |first1=N. |title=Algebra II: Chapters 4 - 7 |date=2003 |publisher=Springer Science & Business Media |page=A.VII.5}}</ref> The [[algebraic closure]] of '''Q''', i.e. the field of roots of rational polynomials, is the field of [[algebraic number]]s.

The set of all rational numbers is [[countable]], while the set of all real numbers (as well as the set of irrational numbers) is uncountable. Being countable, the set of rational numbers is a [[null set]], that is, [[almost all]] real numbers are irrational, in the sense of [[Lebesgue measure]].

The rationals are a [[densely ordered]] set: between any two rationals, there sits another one, and, therefore, infinitely many other ones. For example, for any two fractions such that 
:<math>\frac{a}{b} < \frac{c}{d}</math>
(where <math>b,d</math> are positive), we have
:<math>\frac{a}{b} < \frac{a + c}{b + d} < \frac{c}{d}.</math>

Any [[totally ordered]] set which is countable, dense (in the above sense), and has no least or greatest element is [[order isomorphism|order isomorphic]] to the rational numbers.

==Real numbers and topological properties==
The rationals are a [[dense set|dense subset]] of the real numbers: every real number has rational numbers arbitrarily close to it. A related property is that rational numbers are the only numbers with [[finite set|finite]] expansions as [[continued fraction|regular continued fractions]].

By virtue of their order, the rationals carry an [[order topology]]. The rational numbers, as a subspace of the real numbers, also carry a [[subspace topology]]. The rational numbers form a [[metric space]] by using the [[absolute difference]] metric {{nowrap|''d''(''x'',''y'') {{=}} {{!}}''x'' − ''y''{{!}},}} and this yields a third topology on '''Q'''. All three topologies coincide and turn the rationals into a [[topological field]]. The rational numbers are an important example of a space which is not [[locally compact]]. The rationals are characterized topologically as the unique [[countable]] [[Topological property|metrizable space]] without [[isolated point]]s. The space is also [[totally disconnected space|totally disconnected]]. The rational numbers do not form a [[completeness (topology)|complete metric space]]; the [[real numbers]] are the completion of '''Q''' under the metric {{nowrap|''d''(''x'',''y'') {{=}} {{!}}''x'' − ''y''{{!}},}} above.

==''p''-adic numbers==
{{Main article|p-adic number}}
In addition to the absolute value metric mentioned above, there are other metrics which turn '''Q''' into a topological field:

Let ''p'' be a [[prime number]] and for any non-zero integer ''a'', let {{nowrap|{{!}}''a''{{!}}<sub>''p''</sub> {{=}} ''p''<sup>−''n''</sup>}}, where ''p<sup>n</sup>'' is the highest power of ''p'' [[divisor|dividing]] ''a''.

In addition set {{nowrap|{{!}}0{{!}}<sub>''p''</sub> {{=}} 0.}} For any rational number ''a''/''b'', we set {{nowrap|{{!}}''a''/''b''{{!}}<sub>''p''</sub> {{=}} {{!}}''a''{{!}}<sub>''p''</sub> / {{!}}''b''{{!}}<sub>''p''</sub>.}}

Then {{nowrap|''d<sub>p</sub>''(''x'',''y'') {{=}} {{!}}''x'' − ''y''{{!}}<sub>''p''</sub>}} defines a [[metric space|metric]] on '''Q'''.

The metric space ('''Q''',''d<sub>p</sub>'') is not complete, and its completion is the [[p-adic number|''p''-adic number field]] '''Q'''<sub>''p''</sub>. [[Ostrowski's theorem]] states that any non-trivial [[absolute value (algebra)|absolute value]] on the rational numbers '''Q''' is equivalent to either the usual real absolute value or a [[p-adic number|''p''-adic]] absolute value.

==See also==
*[[Floating point]]
*[[Ford circle]]s
*[[Niven's theorem]]
*[[Rational data type]]
*[[Glossary_of_arithmetic_and_diophantine_geometry#N|height of a rational number in lowest term = naive height]]
*[[Gaussian rational]]

==References==
<references/>

==External links==
{{Commonscat|Rational numbers}}
*{{springer|title=Rational number|id=p/r077620}}
*[http://mathworld.wolfram.com/RationalNumber.html "Rational Number" From MathWorld – A Wolfram Web Resource]

{{Algebraic numbers}}
{{Number systems}}
{{Rational numbers}}

[[Category:Elementary mathematics]]
[[Category:Field theory]]
[[Category:Fractions (mathematics)]]
[[Category:Rational numbers| ]]