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Rational number

From Wikipedia, the free encyclopedia
Quotient of two integers
"Rationals" redirects here. For other uses, seeRational (disambiguation).
The rational numbers⁠Q{\displaystyle \mathbb {Q} }⁠ are included in thereal numbers⁠R{\displaystyle \mathbb {R} }⁠, which are included in thecomplex numbers⁠C{\displaystyle \mathbb {C} }⁠, while rationals include theintegers⁠Z{\displaystyle \mathbb {Z} }⁠, which in turn include thenatural numbers⁠N{\displaystyle \mathbb {N} }⁠.

Inmathematics, arational number is anumber that can be expressed as thequotient orfraction⁠pq{\displaystyle {\tfrac {p}{q}}}⁠ of twointegers, anumeratorp and a non-zerodenominatorq.[1] For example,⁠37{\displaystyle {\tfrac {3}{7}}}⁠ is a rational number, as is every integer (for example,−5=−51{\displaystyle -5={\tfrac {-5}{1}}}). Theset of all rational numbers is often referred to as "the rationals",[2] and isclosed underaddition,subtraction,multiplication, anddivision by a nonzero rational number. It is afield under these operations and therefore also calledthefield of rationals[3] or thefield of rational numbers. It is usually denoted by boldfaceQ, orblackboard bold⁠Q.{\displaystyle \mathbb {Q} .}⁠

A rational number is areal number. The real numbers that are rational are those whosedecimal expansion either terminates after a finite number ofdigits (example:3/4 = 0.75), or eventually begins torepeat the same finitesequence of digits over and over (example:9/44 = 0.20454545...).[4] This statement is true not only inbase 10, but also in every other integerbase, such as thebinary andhexadecimal ones (seeRepeating decimal § Extension to other bases).

Areal number that is not rational is calledirrational.[5] Irrational numbers include thesquare root of 2(⁠2{\displaystyle {\sqrt {2}}}⁠),π,e, and thegolden ratio (φ). Since the set of rational numbers iscountable, and the set of real numbers isuncountable,almost all real numbers are irrational.[1]

The field of rational numbers is the unique field that contains theintegers, and is contained in any field containing the integers. In other words, the field of rational numbers is aprime field. A field hascharacteristic zero if and only if it contains the rational numbers as a subfield. Finiteextensions of⁠Q{\displaystyle \mathbb {Q} }⁠ are calledalgebraic number fields, and thealgebraic closure of⁠Q{\displaystyle \mathbb {Q} }⁠ is the field ofalgebraic numbers.[6]

Inmathematical analysis, the rational numbers form adense subset of the real numbers. The real numbers can be constructed from the rational numbers bycompletion, usingCauchy sequences,Dedekind cuts, or infinitedecimals (seeConstruction of the real numbers).

Terminology

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In mathematics, "rational" is often used as a noun abbreviating "rational number". The adjectiverational sometimes means that thecoefficients are rational numbers. For example, arational point is a point with rationalcoordinates (i.e., a point whose coordinates are rational numbers); arational matrix is amatrix of rational numbers, though it sometimes also refers to a matrix whose entries are rational functions; arational polynomial may be a polynomial with rational coefficients, although the term "polynomial over the rationals" is generally preferred, to avoid confusion between "rational expression" and "rational function" (apolynomial is a rational expression and defines a rational function, even if its coefficients are not rational numbers). However, arational curveis not a curve defined over the rationals, but a curve which can be parameterized by rational functions.

Etymology

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Although nowadaysrational numbers are defined in terms ofratios, the termrational is not aderivation ofratio. On the contrary, it isratio that is derived fromrational: the first use ofratio with its modern meaning was attested in English about 1660,[7] while the use ofrational for qualifying numbers appeared almost a century earlier, in 1570.[8] This meaning ofrational came from the mathematical meaning ofirrational, which was first used in 1551, and it was used in "translations of Euclid (following his peculiar use ofἄλογος)".[9][10]

This unusual history originated in the fact thatancient Greeks "avoided heresy by forbidding themselves from thinking of those [irrational] lengths as numbers".[11] So such lengths wereirrational, in the sense ofillogical, that is "not to be spoken about" (ἄλογος in Greek).[12]

Arithmetic

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See also:Fraction (mathematics) § Arithmetic with fractions

Irreducible fraction

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Every rational number may be expressed in a unique way as anirreducible fraction⁠ab,{\displaystyle {\tfrac {a}{b}},}⁠ wherea andb arecoprime integers andb > 0. This is often called thecanonical form of the rational number.

Starting from a rational number⁠ab,{\displaystyle {\tfrac {a}{b}},}⁠ its canonical form may be obtained by dividing botha andb by theirgreatest common divisor, and, ifb < 0, changing the sign of the resulting numerator and denominator.

Embedding of integers

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Any integern can be expressed as the rational number⁠n1,{\displaystyle {\tfrac {n}{1}},}⁠ which is its canonical form as a rational number.

Equality

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ab=cd{\displaystyle {\frac {a}{b}}={\frac {c}{d}}} if and only ifad=bc{\displaystyle ad=bc}

If both fractions are in canonical form, then:

ab=cd{\displaystyle {\frac {a}{b}}={\frac {c}{d}}} if and only ifa=c{\displaystyle a=c} andb=d{\displaystyle b=d}[13]

Ordering

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If both denominators are positive (particularly if both fractions are in canonical form):

ab<cd{\displaystyle {\frac {a}{b}}<{\frac {c}{d}}} if and only ifad<bc.{\displaystyle ad<bc.}

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.[13]

Addition

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Two fractions are added as follows:

ab+cd=ad+bcbd.{\displaystyle {\frac {a}{b}}+{\frac {c}{d}}={\frac {ad+bc}{bd}}.}

If both fractions are in canonical form, the result is in canonical form if and only ifb, d arecoprime integers.[13][14]

Subtraction

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ab−cd=ad−bcbd.{\displaystyle {\frac {a}{b}}-{\frac {c}{d}}={\frac {ad-bc}{bd}}.}

If both fractions are in canonical form, the result is in canonical form if and only ifb, d arecoprime integers.[14]

Multiplication

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The rule for multiplication is:

ab⋅cd=acbd.{\displaystyle {\frac {a}{b}}\cdot {\frac {c}{d}}={\frac {ac}{bd}}.}

where the result may be areducible fraction—even if both original fractions are in canonical form.[13][14]

Inverse

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Every rational number⁠ab{\displaystyle {\tfrac {a}{b}}}⁠ has anadditive inverse, often called itsopposite,

−(ab)=−ab.{\displaystyle -\left({\frac {a}{b}}\right)={\frac {-a}{b}}.}

If⁠ab{\displaystyle {\tfrac {a}{b}}}⁠ is in canonical form, the same is true for its opposite.

A nonzero rational number⁠ab{\displaystyle {\tfrac {a}{b}}}⁠ has amultiplicative inverse, also called itsreciprocal,

(ab)−1=ba.{\displaystyle \left({\frac {a}{b}}\right)^{-1}={\frac {b}{a}}.}

If⁠ab{\displaystyle {\tfrac {a}{b}}}⁠ is in canonical form, then the canonical form of its reciprocal is either⁠ba{\displaystyle {\tfrac {b}{a}}}⁠ or⁠−b−a,{\displaystyle {\tfrac {-b}{-a}},}⁠ depending on the sign ofa.

Division

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Ifb, c, d are nonzero, the division rule is

abcd=adbc.{\displaystyle {\frac {\,{\dfrac {a}{b}}\,}{\dfrac {c}{d}}}={\frac {ad}{bc}}.}

Thus, dividing⁠ab{\displaystyle {\tfrac {a}{b}}}⁠ by⁠cd{\displaystyle {\tfrac {c}{d}}}⁠ is equivalent to multiplying⁠ab{\displaystyle {\tfrac {a}{b}}}⁠ by thereciprocal of⁠cd:{\displaystyle {\tfrac {c}{d}}:}⁠[14]

adbc=ab⋅dc.{\displaystyle {\frac {ad}{bc}}={\frac {a}{b}}\cdot {\frac {d}{c}}.}

Exponentiation to integer power

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Ifn is a non-negative integer, then

(ab)n=anbn.{\displaystyle \left({\frac {a}{b}}\right)^{n}={\frac {a^{n}}{b^{n}}}.}

The result is in canonical form if the same is true for⁠ab.{\displaystyle {\tfrac {a}{b}}.}⁠ In particular,

(ab)0=1.{\displaystyle \left({\frac {a}{b}}\right)^{0}=1.}

Ifa ≠ 0, then

(ab)−n=bnan.{\displaystyle \left({\frac {a}{b}}\right)^{-n}={\frac {b^{n}}{a^{n}}}.}

If⁠ab{\displaystyle {\tfrac {a}{b}}}⁠ is in canonical form, the canonical form of the result is⁠bnan{\displaystyle {\tfrac {b^{n}}{a^{n}}}}⁠ ifa > 0 orn is even. Otherwise, the canonical form of the result is⁠−bn−an.{\displaystyle {\tfrac {-b^{n}}{-a^{n}}}.}⁠

Continued fraction representation

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Main article:Simple continued fraction

Afinite continued fraction is an expression such as

a0+1a1+1a2+1⋱+1an,{\displaystyle a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\ddots +{\cfrac {1}{a_{n}}}}}}}}},}

wherean are integers. Every rational number⁠ab{\displaystyle {\tfrac {a}{b}}}⁠ can be represented as a finite continued fraction, whosecoefficientsan can be determined by applying theEuclidean algorithm to(a, b).

Other representations

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are different ways to represent the same rational value.

Formal construction

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A diagram showing a representation of the equivalent classes of pairs of integers

The rational numbers may be built asequivalence classes ofordered pairs ofintegers.[13][14]

More precisely, let⁠(Z×(Z∖{0})){\displaystyle (\mathbb {Z} \times (\mathbb {Z} \setminus \{0\}))}⁠ be the set of the pairs(m, n) of integers suchn ≠ 0. Anequivalence relation is defined on this set by

(m1,n1)∼(m2,n2)⟺m1n2=m2n1.{\displaystyle (m_{1},n_{1})\sim (m_{2},n_{2})\iff m_{1}n_{2}=m_{2}n_{1}.}[13][14]

Addition and multiplication can be defined by the following rules:

(m1,n1)+(m2,n2)≡(m1n2+n1m2,n1n2),{\displaystyle (m_{1},n_{1})+(m_{2},n_{2})\equiv (m_{1}n_{2}+n_{1}m_{2},n_{1}n_{2}),}
(m1,n1)×(m2,n2)≡(m1m2,n1n2).{\displaystyle (m_{1},n_{1})\times (m_{2},n_{2})\equiv (m_{1}m_{2},n_{1}n_{2}).}[13]

This equivalence relation is acongruence relation, which means that it is compatible with the addition and multiplication defined above; the set of rational numbers⁠Q{\displaystyle \mathbb {Q} }⁠ is the defined as thequotient set by this equivalence relation,⁠(Z×(Z∖{0}))/∼,{\displaystyle (\mathbb {Z} \times (\mathbb {Z} \backslash \{0\}))/\sim ,}⁠ equipped with the addition and the multiplication induced by the above operations. (This construction can be carried out with anyintegral domain and produces itsfield of fractions.)[13]

The equivalence class of a pair(m, n) is denoted⁠mn.{\displaystyle {\tfrac {m}{n}}.}⁠ Two pairs(m1,n1) and(m2,n2) belong to the same equivalence class (that is are equivalent) if and only if

m1n2=m2n1.{\displaystyle m_{1}n_{2}=m_{2}n_{1}.}

This means that

m1n1=m2n2{\displaystyle {\frac {m_{1}}{n_{1}}}={\frac {m_{2}}{n_{2}}}}

if and only if[13][14]

m1n2=m2n1.{\displaystyle m_{1}n_{2}=m_{2}n_{1}.}

Every equivalence class⁠mn{\displaystyle {\tfrac {m}{n}}}⁠ may be represented by infinitely many pairs, since

⋯=−2m−2n=−m−n=mn=2m2n=⋯.{\displaystyle \cdots ={\frac {-2m}{-2n}}={\frac {-m}{-n}}={\frac {m}{n}}={\frac {2m}{2n}}=\cdots .}

Each equivalence class contains a uniquecanonical representative element. The canonical representative is the unique pair(m, n) in the equivalence class such thatm andn arecoprime, andn > 0. It is called therepresentation in lowest terms of the rational number.

The integers may be considered to be rational numbers identifying the integern with the rational number⁠n1.{\displaystyle {\tfrac {n}{1}}.}⁠

Atotal order may be defined on the rational numbers, that extends the natural order of the integers. One has

m1n1≤m2n2{\displaystyle {\frac {m_{1}}{n_{1}}}\leq {\frac {m_{2}}{n_{2}}}}

If

(n1n2>0andm1n2≤n1m2)or(n1n2<0andm1n2≥n1m2).{\displaystyle {\begin{aligned}&(n_{1}n_{2}>0\quad {\text{and}}\quad m_{1}n_{2}\leq n_{1}m_{2})\\&\qquad {\text{or}}\\&(n_{1}n_{2}<0\quad {\text{and}}\quad m_{1}n_{2}\geq n_{1}m_{2}).\end{aligned}}}

Properties

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The set⁠Q{\displaystyle \mathbb {Q} }⁠ of all rational numbers, together with the addition and multiplication operations shown above, forms afield.[13]

⁠Q{\displaystyle \mathbb {Q} }⁠ has nofield automorphism other than the identity. (A field automorphism must fix 0 and 1; as it must fix the sum and the difference of two fixed elements, it must fix every integer; as it must fix the quotient of two fixed elements, it must fix every rational number, and is thus the identity.)

⁠Q{\displaystyle \mathbb {Q} }⁠ is aprime field, which is a field that has no subfield other than itself.[15] The rationals are the smallest field withcharacteristic zero. Every field of characteristic zero contains a unique subfield isomorphic to⁠Q.{\displaystyle \mathbb {Q} .}⁠

With the order defined above,⁠Q{\displaystyle \mathbb {Q} }⁠ is anordered field[14] that has no subfield other than itself, and is the smallest ordered field, in the sense that every ordered field contains a unique subfieldisomorphic to⁠Q.{\displaystyle \mathbb {Q} .}⁠

⁠Q{\displaystyle \mathbb {Q} }⁠ is thefield of fractions of theintegers⁠Z.{\displaystyle \mathbb {Z} .}⁠[16] Thealgebraic closure of⁠Q,{\displaystyle \mathbb {Q} ,}⁠ i.e. the field of roots of rational polynomials, is the field ofalgebraic numbers.

The rationals are adensely ordered set: between any two rationals, there sits another one, and, therefore, infinitely many other ones.[13] For example, for any two fractions such that

ab<cd{\displaystyle {\frac {a}{b}}<{\frac {c}{d}}}

(whereb,d{\displaystyle b,d} are positive), we have

ab<a+cb+d<cd.{\displaystyle {\frac {a}{b}}<{\frac {a+c}{b+d}}<{\frac {c}{d}}.}

Anytotally ordered set which is countable, dense (in the above sense), and has no least or greatest element isorder isomorphic to the rational numbers.[17]

Countability

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Illustration of the countability of the positive rationals

The set of positive rational numbers iscountable, as is illustrated in the figure.

More precisely, one can sort the fractions by increasing values of the sum of the numerator and the denominator, and, for equal sums, by increasing numerator or denominator. This produces asequence of fractions from which one can remove the reducible fractions (in red on the figure), obtaining a sequence that contains each rational number exactly once. This establishes a bijection between the rational numbers and the natural numbers, which maps each rational number to its rank in the sequence.

A similar method can be used for numbering all rational numbers (positive and negative).

As the set of all rational numbers is countable, and the set of all real numbers (as well as the set of irrational numbers) is uncountable, the set of rational numbers is anull set, that is,almost all real numbers are irrational, in the sense ofLebesgue measure.[18]

Real numbers and topological properties

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The rationals are adense subset of thereal numbers; every real number has rational numbers arbitrarily close to it.[13] A related property is that rational numbers are the only numbers withfinite expansions asregular continued fractions.[19]

In the usualtopology of the real numbers, the rationals are neither anopen set nor aclosed set.[20]

By virtue of their order, the rationals carry anorder topology. The rational numbers, as a subspace of the real numbers, also carry asubspace topology. The rational numbers form ametric space by using theabsolute difference metricd(x,y)=|x−y|,{\displaystyle d(x,y)=|x-y|,} and this yields a third topology on⁠Q.{\displaystyle \mathbb {Q} .}⁠ All three topologies coincide and turn the rationals into atopological field. The rational numbers are an important example of a space which is notlocally compact. The rationals are characterized topologically as the uniquecountablemetrizable space withoutisolated points. The space is alsototally disconnected. The rational numbers do not form acomplete metric space, and thereal numbers are the completion of⁠Q{\displaystyle \mathbb {Q} }⁠ under the metricd(x,y)=|x−y|{\displaystyle d(x,y)=|x-y|} above.[14]

p-adic numbers

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Main article:p-adic number

In addition to the absolute value metric mentioned above, there are other metrics which turn⁠Q{\displaystyle \mathbb {Q} }⁠ into a topological field:

Letp be aprime number and for any non-zero integera, let|a|p=p−n,{\displaystyle |a|_{p}=p^{-n},} wherepn is the highest power ofpdividinga.

In addition set|0|p=0.{\displaystyle |0|_{p}=0.} For any rational number⁠ab,{\displaystyle {\frac {a}{b}},}⁠ we set

|ab|p=|a|p|b|p.{\displaystyle \left|{\frac {a}{b}}\right|_{p}={\frac {|a|_{p}}{|b|_{p}}}.}

Then

dp(x,y)=|x−y|p{\displaystyle d_{p}(x,y)=|x-y|_{p}}

defines ametric on⁠Q.{\displaystyle \mathbb {Q} .}⁠[21]

The metric space⁠(Q,dp){\displaystyle (\mathbb {Q} ,d_{p})}⁠ is not complete, and its completion is thep-adic number field⁠Qp.{\displaystyle \mathbb {Q} _{p}.}⁠Ostrowski's theorem states that any non-trivialabsolute value on the rational numbers⁠Q{\displaystyle \mathbb {Q} }⁠ is equivalent to either the usual real absolute value or ap-adic absolute value.

See also

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Set inclusions between thenatural numbers (ℕ), theintegers (ℤ), therational numbers (ℚ), thereal numbers (ℝ), and thecomplex numbers (ℂ)


References

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  1. ^abRosen, Kenneth (2007).Discrete Mathematics and its Applications (6th ed.). New York, NY: McGraw-Hill. pp. 105,158–160.ISBN 978-0-07-288008-3.
  2. ^Lass, Harry (2009).Elements of Pure and Applied Mathematics (illustrated ed.). Courier Corporation. p. 382.ISBN 978-0-486-47186-0.Extract of page 382
  3. ^Robinson, Julia (1996).The Collected Works of Julia Robinson. American Mathematical Soc. p. 104.ISBN 978-0-8218-0575-6.Extract of page 104
  4. ^"Rational number".Encyclopedia Britannica. Retrieved2020-08-11.
  5. ^Weisstein, Eric W."Rational Number".Wolfram MathWorld. Retrieved2020-08-11.
  6. ^Gilbert, Jimmie;Linda, Gilbert (2005).Elements of Modern Algebra (6th ed.). Belmont, CA: Thomson Brooks/Cole. pp. 243–244.ISBN 0-534-40264-X.
  7. ^Oxford English Dictionary (2nd ed.). Oxford University Press. 1989. Entryratio,n., sense 2.a.
  8. ^Oxford English Dictionary (2nd ed.). Oxford University Press. 1989. Entryrational,a. (adv.) andn.1, sense 5.a.
  9. ^Oxford English Dictionary (2nd ed.). Oxford University Press. 1989. Entryirrational,a. andn., sense 3.
  10. ^Shor, Peter (2017-05-09)."Does rational come from ratio or ratio come from rational".Stack Exchange. Retrieved2021-03-19.
  11. ^Coolman, Robert (2016-01-29)."How a Mathematical Superstition Stultified Algebra for Over a Thousand Years". Retrieved2021-03-20.
  12. ^Kramer, Edna (1983).The Nature and Growth of Modern Mathematics. Princeton University Press. p. 28.
  13. ^abcdefghijklBiggs, Norman L. (2002).Discrete Mathematics. India: Oxford University Press. pp. 75–78.ISBN 978-0-19-871369-2.
  14. ^abcdefghi"Fraction - Encyclopedia of Mathematics".encyclopediaofmath.org. Retrieved2021-08-17.
  15. ^Sūgakkai, Nihon (1993).Encyclopedic Dictionary of Mathematics, Volume 1. London, England: MIT Press. p. 578.ISBN 0-2625-9020-4.
  16. ^Bourbaki, N. (2003).Algebra II: Chapters 4 - 7. Springer Science & Business Media. p. A.VII.5.
  17. ^Giese, Martin; Schönegge, Arno (December 1995).Any two countable densely ordered sets without endpoints are isomorphic - a formal proof with KIV(PDF) (Technical report). Retrieved17 August 2021.
  18. ^Royden, Halsey; Fitzpatrick, Patrick (2017-02-13).Real Analysis (4th ed.). Pearson. pp. 7–54.ISBN 9780134689494.
  19. ^Anthony Vazzana; David Garth (2015).Introduction to Number Theory (2nd, revised ed.). CRC Press. p. 1.ISBN 978-1-4987-1752-6.Extract of page 1
  20. ^Richard A. Holmgren (2012).A First Course in Discrete Dynamical Systems (2nd, illustrated ed.). Springer Science & Business Media. p. 26.ISBN 978-1-4419-8732-7.Extract of page 26
  21. ^Weisstein, Eric W."p-adic Number".Wolfram MathWorld. Retrieved2021-08-17.

Notes

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External links

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