TheInfoList

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 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 ...

$\mathbb$, 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: : $\left\left( p_1, q_1 \right\right) \sim \left\left( p_2, q_2 \right\right) \iff p_1 q_2 = p_2 q_1.$ 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 a
derivation 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 an
irreducible 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

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

## Ordering

If both denominators are positive (particularly if both fractions are in canonical form): :$\frac < \frac$ 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.

Two fractions are added as follows: :$\frac + \frac = \frac.$ If both fractions are in canonical form, the result is in canonical form if and only if 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, ...
.

## Subtraction

:$\frac - \frac = \frac.$ If both fractions are in canonical form, the result is in canonical form if and only if 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, ...
.

## Multiplication

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

## Inverse

Every rational number has an
additive 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'', :$- \left\left( \frac \right\right) = \frac.$ 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'', :$\left\left(\frac\right\right)^ = \frac.$ 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 :$\frac = \frac.$ Thus, dividing by is equivalent to multiplying by the
reciprocal 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 : :$\frac = \frac \cdot \frac.$

## Exponentiation to integer power

If is a non-negative integer, then :$\left\left(\frac\right\right)^n = \frac.$ The result is in canonical form if the same is true for . In particular, :$\left\left(\frac\right\right)^0 = 1.$ If , then :$\left\left(\frac\right\right)^ = \frac.$ 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 :$a_0 + \cfrac,$ where are integers. Every rational number can be represented as a finite continued fraction, whose
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 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 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
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 : $\left\left(m_1, n_1 \right\right) \sim \left\left(m_2, n_2 \right\right) \iff m_1 n_2 = m_2 n_1.$ Addition and multiplication can be defined by the following rules: :$\left\left(m_1, n_1\right\right) + \left\left(m_2, n_2\right\right) \equiv \left\left(m_1n_2 + n_1m_2, n_1n_2\right\right),$ :$\left\left(m_1, n_1\right\right) \times \left\left(m_2, n_2\right\right) \equiv \left\left(m_1m_2, n_1n_2\right\right).$ 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 :$\cdots = \frac = \frac = \frac = \frac = \cdots.$ 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 :$\frac \le \frac$ if :$\left(n_1n_2 > 0 \quad \text \quad m_1n_2 \le n_1m_2\right)\qquad \text\qquad \left(n_1n_2 < 0 \quad \text \quad m_1n_2 \ge n_1m_2\right).$

# Properties

The set of all rational numbers, together with the addition and multiplication operations shown above, forms 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 ...
. 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
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
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 :$\frac < \frac$ (where $b,d$ are positive), we have :$\frac < \frac < \frac.$ 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 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: 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.

In addition to the absolute value metric mentioned above, there are other metrics which turn into a topological field: Let be a
prime 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.

*
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''