Real Numbers:Rational - Revision history

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[[File:Number-systems.svg|thumb|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>)]]

In [[mathematics]], a '''rational number''' is a [[number]] that can be expressed as the [[quotient]] or [[fraction (mathematics)|fraction]] {{math|{{sfrac|''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 |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> 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]]",{{Citation needed|date=August 2021}} and first appeared in Bourbaki's ''Algèbre''.<ref name=":0" />

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]]).{{Citation needed|date=August 2021}}

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>

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 (for more, see [[Construction of the real numbers]]).{{Citation needed|date=August 2021}}

==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|{{sfrac|''a''|''b''}}}}, where {{mvar|a}} and {{mvar|b}} are [[coprime integers]] and {{math|''b'' > 0}}. This is often called the [[canonical form]] of the rational number.

Starting from a rational number {{math|{{sfrac|''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.{{Citation needed|date=August 2021}}

===Embedding of integers===
Any integer {{math|''n''}} can be expressed as the rational number {{math|{{sfrac|''n''|1}}}}, which is its canonical form as a rational number.{{Citation needed|date=August 2021}}

===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><ref name=":1" />


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

On the other hand, if either denominator is negative, then 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.<ref name=":1" />

===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]].<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>

===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]].<ref name=":2" />{{Verify source|date=August 2021}}

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

where the result may be a [[reducible fraction]]—even if both original fractions are in canonical form.<ref name=":1" /><ref name=":2" />

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

A nonzero rational number {{math|{{sfrac|''a''|''b''}}}} has a [[multiplicative inverse]], also called its ''reciprocal'',
:<math> \left(\frac{a}{b}\right)^{-1} = \frac{b}{a}. </math>
If {{math|{{sfrac|''a''|''b''}}}} is in canonical form, then the canonical form of its reciprocal is either {{math|{{sfrac|''b''|''a''}}}} or {{math|{{sfrac|−''b''|−''a''}}}}, depending on the sign of {{mvar|a}}.{{Citation needed|date=August 2021}}

===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|{{sfrac|''a''|''b''}}}} by {{math|{{sfrac|''c''|''d''}}}} is equivalent to multiplying {{math|{{sfrac|''a''|''b''}}}} by the [[multiplicative inverse|reciprocal]] of {{math|{{sfrac|''c''|''d''}}}}:
:<math>\frac{ad}{bc} = \frac{a}{b} \cdot \frac{d}{c}.</math><ref name=":2" />{{Verify source|date=August 2021}}

===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|{{sfrac|''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|{{sfrac|''a''|''b''}}}} is in canonical form, the canonical form of the result is {{math|{{sfrac|''bn''|''an''}}}} if {{math|''a'' > 0}} or {{mvar|n}} is even. Otherwise, the canonical form of the result is {{math|{{sfrac|−''bn''|−''an''}}}}.{{Citation needed|date=August 2021}}

==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.<ref name=":1" /><ref name=":2" />

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''1, ''n''1)}} and {{math|(''m''2, ''n''2)}} belong to the same equivalence class (that is are equivalent) if and only if {{math|''m''1''n''2 {{=}} ''m''2''n''1}}. This means that {{math|{{sfrac|''m''1|''n''1}} {{=}} {{sfrac|''m''2|''n''2}}}} if and only {{math|''m''1''n''2 {{=}} ''m''2''n''1}}.<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>

← Older revision		Revision as of 21:52, 21 October 2021
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	# Giese, Martin; Schönegge, Arno (December 1995). Any two countable densely ordered sets without endpoints are isomorphic - a formal proof with KIV (PDF) (Technical report). Retrieved 17 August 2021.		# Giese, Martin; Schönegge, Arno (December 1995). Any two countable densely ordered sets without endpoints are isomorphic - a formal proof with KIV (PDF) (Technical report). Retrieved 17 August 2021.
	# Weisstein, Eric W. "p-adic Number". mathworld.wolfram.com. Retrieved 2021-08-17.		# Weisstein, Eric W. "p-adic Number". mathworld.wolfram.com. Retrieved 2021-08-17.
		+
		+	== Licensing ==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikipedia.org/wiki/Rational_number Rational number, Wikipedia] under a CC BY-SA license