In

integer
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...

s . The

"Rational Number" From MathWorld – A Wolfram Web Resource

{{Authority control Elementary mathematics Field (mathematics) Fractions (mathematics)

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, a rational number is a number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

that can be expressed as the quotient
In arithmetic
Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne' ...

or fraction
A fraction (from Latin ', "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths ...

of two integer
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...

s, a numerator
A fraction (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Rom ...

and a non-zero denominator
A fraction (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Rom ...

. For example, is a rational number, as is every integer (e.g. ). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by a boldface (or blackboard bold
Image:Blackboard bold.svg, 250px, An example of blackboard bold letters
Blackboard bold is a typeface style that is often used for certain symbols in mathematics, mathematical texts, in which certain lines of the symbol (usually vertical or near-v ...

$\backslash mathbb$, Unicode or ); it was thus denoted in 1895 by Giuseppe Peano
Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

after '' quoziente'', Italian for "quotient
In arithmetic
Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne' ...

", and first appeared in Bourbaki's ''Algèbre''.
The decimal expansion of a rational number either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

of digits over and over (example: ). Conversely, any repeating or terminating decimal represents a rational number. These statements are true in base 10
The decimal numeral system
A numeral system (or system of numeration) is a writing system
A writing system is a method of visually representing verbal communication
Communication (from Latin ''communicare'', meaning "to share") is t ...

, and in every other integer base
Base or BASE may refer to:
Brands and enterprises
* Base (mobile telephony provider), a Belgian mobile telecommunications operator
*Base CRM
Base CRM (originally Future Simple or PipeJump) is an enterprise software company based in Mountain Vie ...

(for example, binary
Binary may refer to:
Science and technology
Mathematics
* Binary number
In mathematics and digital electronics, a binary number is a number expressed in the base-2 numeral system or binary numeral system, which uses only two symbols: ty ...

or hexadecimal
In mathematics and computing, the hexadecimal (also base 16 or hex) numeral system is a Numeral system#Positional systems in detail, positional numeral system that represents numbers using a radix (base) of 16. Unlike the decimal system repres ...

).
A real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

that is not rational is called irrational
Irrationality is cognition
Cognition () refers to "the mental action or process of acquiring knowledge and understanding through thought, experience, and the senses". It encompasses many aspects of intellectual functions and processes such as: ...

. Irrational numbers include , , , and . The decimal expansion of an irrational number continues without repeating. Since the set of rational numbers is countable
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, and the set of real numbers is uncountable
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, almost all In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no genera ...

real numbers are irrational.
Rational numbers can be formally defined as equivalence class
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

es of pairs of integers with , using the equivalence relation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

defined as follows:
: $\backslash left(\; p\_1,\; q\_1\; \backslash right)\; \backslash sim\; \backslash left(\; p\_2,\; q\_2\; \backslash right)\; \backslash iff\; p\_1\; q\_2\; =\; p\_2\; q\_1.$
The fraction then denotes the equivalence class of .
Rational numbers together with addition
Addition (usually signified by the plus symbol
The plus and minus signs, and , are mathematical symbol
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object
A mathematical object is an ...

and multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

form a field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

which contains the integer
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...

s, and is contained in any field containing the integers. In other words, the field of rational numbers is a prime field
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, and a field has characteristic zero
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

if and only if it contains the rational numbers as a subfield. Finite extensions
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of ...

of are called algebraic number field
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s, and the algebraic closure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of is the field of algebraic number
An algebraic number is any complex number
In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, ev ...

s.
In mathematical analysis
Analysis is the branch of mathematics dealing with Limit (mathematics), limits
and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...

, the rational numbers form a dense subset
In topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric obje ...

of the real numbers. The real numbers can be constructed from the rational numbers by completion, using Cauchy sequence
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s, Dedekind cut
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s, or infinite decimal
The decimal numeral system
A numeral system (or system of numeration) is a writing system
A writing system is a method of visually representing verbal communication
Communication (from Latin ''communicare'', meaning "to share") is t ...

s (for more, see Construction of the real numbers
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

).
Terminology

The term ''rational'' in reference to the set refers to the fact that a rational number represents a ''ratio
In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8∶6, which is equivalent to ...

'' of two integers. In mathematics, "rational" is often used as a noun abbreviating "rational number". The adjective ''rational'' sometimes means that the coefficient
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s are rational numbers. For example, a rational point
In number theory and algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, ...

is a point with rational coordinates
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

(i.e., a point whose coordinates are rational numbers); a ''rational matrix'' is a matrix
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics), a rectangular array of numbers, symbols, or expressions
* Matrix (logic), part of a formula in prenex normal form
* Matrix (biology), the material in between a eukaryoti ...

of rational numbers; a ''rational 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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

" (a polynomial
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

is a rational expression and defines a rational function, even if its coefficients are not rational numbers). However, a rational curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...

''is not'' a curve defined over the rationals, but a curve which can be parameterized by rational functions.
Etymology

Although nowadays ''rational numbers'' are defined in terms of ''ratios'', the term ''rational'' is not aderivation
Derivation may refer to:
* Derivation (differential algebra), a unary function satisfying the Leibniz product law
* Derivation (linguistics)
* Formal proof or derivation, a sequence of sentences each of which is an axiom or follows from the precedi ...

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, while the use of ''rational'' for qualifying numbers appeared almost a century earlier, in 1570. 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 )".
This unusual history originated in the fact that ancient Greeks
Ancient Greece ( el, Ἑλλάς, Hellás) was a civilization belonging to a period of History of Greece, Greek history from the Greek Dark Ages of the 12th–9th centuries BC to the end of Classical Antiquity, antiquity ( AD 600). This era was ...

"avoided heresy by forbidding themselves from thinking of those rrationallengths as numbers". So such lengths were ''irrational'', in the sense of ''illogical'', that is "not to be spoken about" ( in Greek).
This etymology is similar to that of ''imaginary'' numbers and ''real'' numbers.
Arithmetic

Irreducible fraction

Every rational number may be expressed in a unique way as anirreducible fraction
An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction in which the numerator and denominator are integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquial ...

, where and are coprime integers
In number theory, two integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...

and . This is often called the canonical form
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

of the rational number.
Starting from a rational number , its canonical form may be obtained by dividing and by their greatest common divisor
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...

, and, if , changing the sign of the resulting numerator and denominator.
Embedding of integers

Any integer can be expressed as the rational number , which is its canonical form as a rational number.Equality

:$\backslash frac\; =\; \backslash frac$ if and only if $ad\; =\; bc$ If both fractions are in canonical form, then: :$\backslash frac\; =\; \backslash frac$ if and only if $a\; =\; c$ and $b\; =\; d$Ordering

If both denominators are positive (particularly if both fractions are in canonical form): :$\backslash frac\; <\; \backslash frac$ if and only if $ad\; <\; bc.$ 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.Addition

Two fractions are added as follows: :$\backslash frac\; +\; \backslash frac\; =\; \backslash frac.$ If both fractions are in canonical form, the result is in canonical form if and only if and arecoprime integers
In number theory, two integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...

.
Subtraction

:$\backslash frac\; -\; \backslash frac\; =\; \backslash frac.$ If both fractions are in canonical form, the result is in canonical form if and only if and arecoprime integers
In number theory, two integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...

.
Multiplication

The rule for multiplication is: :$\backslash frac\; \backslash cdot\backslash frac\; =\; \backslash frac.$ where the result may be a reducible fraction—even if both original fractions are in canonical form.Inverse

Every rational number has anadditive inverse
In mathematics, the additive inverse of a number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be ...

, often called its ''opposite'',
:$-\; \backslash left(\; \backslash frac\; \backslash right)\; =\; \backslash frac.$
If is in canonical form, the same is true for its opposite.
A nonzero rational number has a multiplicative inverse
Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

, also called its ''reciprocal'',
:$\backslash left(\backslash frac\backslash right)^\; =\; \backslash frac.$
If is in canonical form, then the canonical form of its reciprocal is either or , depending on the sign of .
Division

If , , and are nonzero, the division rule is :$\backslash frac\; =\; \backslash frac.$ Thus, dividing by is equivalent to multiplying by thereciprocal
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another poly ...

of :
:$\backslash frac\; =\; \backslash frac\; \backslash cdot\; \backslash frac.$
Exponentiation to integer power

If is a non-negative integer, then :$\backslash left(\backslash frac\backslash right)^n\; =\; \backslash frac.$ The result is in canonical form if the same is true for . In particular, :$\backslash left(\backslash frac\backslash right)^0\; =\; 1.$ If , then :$\backslash left(\backslash frac\backslash right)^\; =\; \backslash frac.$ If is in canonical form, the canonical form of the result is if or is even. Otherwise, the canonical form of the result is .Continued fraction representation

A finite continued fraction is an expression such as :$a\_0\; +\; \backslash cfrac,$ where are integers. Every rational number can be represented as a finite continued fraction, whosecoefficient
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s can be determined by applying the Euclidean algorithm
In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an effi ...

to .
Other representations

*common fraction
A fraction (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Rom ...

:
* mixed numeral:
* repeating decimal
A repeating decimal or recurring decimal is decimal representation of a number whose Numerical digit, digits are periodic function, periodic (repeating its values at regular intervals) and the infinity, infinitely repeated portion is not zero. It c ...

using a vinculum
Vinculum may refer to:
* Vinculum (ligament), a band of connective tissue, similar to a ligament, that connects a flexor tendon to a phalanx bone
* Vinculum (symbol), a horizontal line used in mathematical notation for a specific purpose
* Vinculum ...

:
* repeating decimal using parentheses
A bracket is either of two tall fore- or back-facing punctuation
Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding ...

:
* continued fraction
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

using traditional typography:
* continued fraction in abbreviated notation:
* Egyptian fraction
An Egyptian fraction is a finite sum of distinct unit fractionA unit fraction is a rational number written as a fraction where the numerator
A fraction (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, ...

:
* prime power decomposition:
* quote notationQuote notation is a representation of the rational numbers based on Kurt Hensel
Kurt Wilhelm Sebastian Hensel (29 December 1861 – 1 June 1941) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathemati ...

:
are different ways to represent the same rational value.
Formal construction

The rational numbers may be built asequivalence class
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

es of ordered pair
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s of integer
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...

s.
More precisely, let be the set of the pairs of integers such . An equivalence relation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

is defined on this set by
: $\backslash left(m\_1,\; n\_1\; \backslash right)\; \backslash sim\; \backslash left(m\_2,\; n\_2\; \backslash right)\; \backslash iff\; m\_1\; n\_2\; =\; m\_2\; n\_1.$
Addition and multiplication can be defined by the following rules:
:$\backslash left(m\_1,\; n\_1\backslash right)\; +\; \backslash left(m\_2,\; n\_2\backslash right)\; \backslash equiv\; \backslash left(m\_1n\_2\; +\; n\_1m\_2,\; n\_1n\_2\backslash right),$
:$\backslash left(m\_1,\; n\_1\backslash right)\; \backslash times\; \backslash left(m\_2,\; n\_2\backslash right)\; \backslash equiv\; \backslash left(m\_1m\_2,\; n\_1n\_2\backslash right).$
This equivalence relation is a congruence relation
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics) ...

, which means that it is compatible with the addition and multiplication defined above; the set of rational numbers is the defined as the quotient set
Set, The Set, or SET may refer to:
Science, technology, and mathematics Mathematics
* Set (mathematics), a collection of distinct elements or members
* Category of sets, the category whose objects and morphisms are sets and total functions, respe ...

by this equivalence relation, , equipped with the addition and the multiplication induced by the above operations. (This construction can be carried out with any integral domain
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

and produces its field of fractions
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics) ...

.)
The equivalence class of a pair is denoted .
Two pairs and belong to the same equivalence class (that is are equivalent) if and only if . This means that if and only .
Every equivalence class may be represented by infinitely many pairs, since
:$\backslash cdots\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash cdots.$
Each equivalence class contains a unique '' canonical representative element''. The canonical representative is the unique pair in the equivalence class such that and are coprime
In number theory, two integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...

, and . It is called the representation in lowest terms of the rational number.
The integers may be considered to be rational numbers identifying the integer with the rational number .
A total order
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some Set (mathematics), set X, which satisfies the following for all a, b and c in X:
# a \ ...

may be defined on the rational numbers, that extends the natural order of the integers. One has
:$\backslash frac\; \backslash le\; \backslash frac$
if
:$(n\_1n\_2\; >\; 0\; \backslash quad\; \backslash text\; \backslash quad\; m\_1n\_2\; \backslash le\; n\_1m\_2)\backslash qquad\; \backslash text\backslash qquad\; (n\_1n\_2\; <\; 0\; \backslash quad\; \backslash text\; \backslash quad\; m\_1n\_2\; \backslash ge\; n\_1m\_2).$
Properties

The set of all rational numbers, together with the addition and multiplication operations shown above, forms afield
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

.
has no field automorphism
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

other than the identity.
With the order defined above, is an ordered fieldIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

that has no subfield other than itself, and is the smallest ordered field, in the sense that every ordered field contains a unique subfield isomorphic
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

to .
is a prime field
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, which is a field that has no subfield other than itself. The rationals are the smallest field with characteristic
Characteristic (from the Greek word for a property, attribute or trait
Trait may refer to:
* Phenotypic trait in biology, which involve genes and characteristics of organisms
* Trait (computer programming), a model for structuring object-oriented ...

zero. Every field of characteristic zero contains a unique subfield isomorphic to .
is the field of fractions
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics) ...

of the algebraic closure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of , i.e. the field of roots of rational polynomials, is the field of algebraic number
An algebraic number is any complex number
In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, ev ...

s.
The set of all rational numbers is countable
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

(see the figure), 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
In mathematical analysis, a null set N \subset \mathbb is a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null set in set theory
illustrating the intersection (set theory), interse ...

, that is, almost all In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no genera ...

real numbers are irrational, in the sense of Lebesgue measure In Measure (mathematics), measure theory, a branch of mathematics, the Lebesgue measure, named after france, French mathematician Henri Lebesgue, is the standard way of assigning a measure (mathematics), measure to subsets of ''n''-dimensional Eucli ...

.
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
:$\backslash frac\; <\; \backslash frac$
(where $b,d$ are positive), we have
:$\backslash frac\; <\; \backslash frac\; <\; \backslash frac.$
Any totally ordered
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

set which is countable, dense (in the above sense), and has no least or greatest element is order isomorphicIn the mathematical field of order theory
Order theory is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (g ...

to the rational numbers.
Real numbers and topological properties

The rationals are adense subset
In topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric obje ...

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
Finite is the opposite of Infinity, infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected ...

expansions as regular continued fractions.
By virtue of their order, the rationals carry an order topology
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. The rational numbers, as a subspace of the real numbers, also carry a subspace topologyIn topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a Topological_space#Definitions, topology induced from that of ''X'' called the subspace topology (or the relative ...

. The rational numbers form a metric space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

by using the absolute difference
The absolute difference of two real number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus ...

metric , and this yields a third topology on . All three topologies coincide and turn the rationals into a topological fieldIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

. The rational numbers are an important example of a space which is not locally compact In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

. The rationals are characterized topologically as the unique countable
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

metrizable space
In topology and related areas of mathematics, a metrizable space is a topological space that is Homeomorphism, homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a Metric (mathematics), m ...

without isolated point 400px, "0" is an isolated point of A = ∪ , 2In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus ...

s. The space is also totally disconnectedIn topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...

. The rational numbers do not form a complete metric space
In mathematical analysis
Analysis is the branch of mathematics dealing with Limit (mathematics), limits
and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathema ...

; the real numbers
Real may refer to:
* Reality, the state of things as they exist, rather than as they may appear or may be thought to be
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish col ...

are the completion of under the metric above.
-adic numbers

In addition to the absolute value metric mentioned above, there are other metrics which turn into a topological field: Let be aprime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

and for any non-zero integer , let , where is the highest power of .
In addition set . For any rational number , we set .
Then defines a metric
METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model
Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...

on .
The metric space is not complete, and its completion is the -adic number field . Ostrowski's theorem
In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value
of the absolute value function for real numbers
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the ...

states that any non-trivial absolute value
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

on the rational numbers is equivalent to either the usual real absolute value or a -adic absolute value.
See also

*Dyadic rational
In mathematics, a dyadic rational or binary rational is a number that can be expressed as a fraction
A fraction (from Latin ', "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday Engli ...

*Floating point
In computing
Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and soft ...

*Ford circle
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s
*Gaussian rationalIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

*
*Niven's theorem
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

* Rational data type
*'' Divine Proportions: Rational Trigonometry to Universal Geometry''
References

External links

*"Rational Number" From MathWorld – A Wolfram Web Resource

{{Authority control Elementary mathematics Field (mathematics) Fractions (mathematics)