Rational number

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In mathematics, a rational number is a
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers ca ...
that can be expressed as the
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
or
fraction A fraction (from la, fractus, "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 ...
of two
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s, a numerator and a non-zero
denominator A fraction (from la, fractus, "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 ...
. 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 boldface , or
blackboard bold Blackboard bold is a typeface style that is often used for certain symbols in mathematical texts, in which certain lines of the symbol (usually vertical or near-vertical lines) are doubled. The symbols usually denote number sets. One way of pr ...
$\mathbb.$ A rational number is a real number. The real numbers that are rational are those whose
decimal expansion A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator: r = b_k b_\ldots b_0.a_1a_2\ldots Here is the decimal separator, is ...
either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in
base 10 The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
, but also in every other integer base, such as the binary and
hexadecimal In mathematics and computing, the hexadecimal (also base-16 or simply hex) numeral system is a positional numeral system that represents numbers using a radix (base) of 16. Unlike the decimal system representing numbers using 10 symbols, hexad ...
ones (see ). A real number that is not rational is called
irrational Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...
. Irrational numbers include , , , and . Since the set of rational numbers is
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
, and the set of real numbers is
uncountable In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal nu ...
,
almost all In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the mathe ...
real numbers are irrational. Rational numbers can be formally defined as
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es of pairs of integers with , using the
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation ...
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 ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' of ...
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 mathematical operations of arithmetic, with the other ones being addi ...
form a field which contains the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s, and is contained in any field containing the integers. In other words, the field of rational numbers is a
prime field In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive iden ...
, and a field has
characteristic zero In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive id ...
if and only if it contains the rational numbers as a subfield. Finite extensions of are called
algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...
s, and the algebraic closure of is the field of
algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
s. In
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied in th ...
, the rational numbers form a
dense subset In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, th ...
of the real numbers. The real numbers can be constructed from the rational numbers by completion, using
Cauchy sequence In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numb ...
s,
Dedekind cut In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind but previously considered by Joseph Bertrand, are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of the rat ...
s, or infinite
decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
s (see
Construction of the real numbers In mathematics, there are several equivalent ways of defining the real numbers. One of them is that they form a complete ordered field that does not contain any smaller complete ordered field. Such a definition does not prove that such a complete ...
).

# Terminology

The term ''rational'' in reference to the set refers to the fact that a rational number represents a ''
ratio In mathematics, a ratio shows 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 the ...
'' 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, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves v ...
s are rational numbers. For example, a
rational point In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the fiel ...
is a point with rational
coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is signi ...
(i.e., a point whose coordinates are rational numbers); a ''rational matrix'' is a matrix 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, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ra ...
" (a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An examp ...
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 ...
''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: Language * Morphological derivation, a word-formation process * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars Law * Derivative work, in copyright law * Derivation proceeding, a pro ...
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 northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of cu ...
"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 integers that have no other common divisors than 1 (and −1, when negative numbers are considered). I ...
, where and are
coprime integers In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiva ...
and . This is often called the
canonical form In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an obj ...
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 mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiva ...
.

## 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 mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiva ...
.

## 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 is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (op ...
, 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 In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/''b ...
, 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 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, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves v ...
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 ...
to .

# Other representations

*
common fraction A fraction (from la, fractus, "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 ...
: * mixed numeral: *
repeating decimal A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. It can be shown that a number is rational if an ...
using a vinculum: * repeating decimal using
parentheses A bracket is either of two tall fore- or back-facing punctuation marks commonly used to isolate a segment of text or data from its surroundings. Typically deployed in symmetric pairs, an individual bracket may be identified as a 'left' or 'r ...
: *
continued fraction In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its intege ...
using traditional typography: * continued fraction in abbreviated notation: * Egyptian fraction: * prime power decomposition: * quote notation: are different ways to represent the same rational value.

# Formal construction

The rational numbers may be built as
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es of
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
s of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s. More precisely, let be the set of the pairs of integers such . An
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation ...
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, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done wit ...
, 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 In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
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, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural sett ...
and produces its
field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field (mathematics), field in which it can be Embedding, embedded. The construction of the field of fractions is modeled on the relationship between the integral do ...
.) 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 mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
, 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 X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive) ...
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. has no
field automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
other than the identity. (A field automophism 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.) is a
prime field In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive iden ...
, which is a field that has no subfield other than itself. The rationals are the smallest field with characteristic zero. Every field of characteristic zero contains a unique subfield isomorphic to . With the order defined above, is an
ordered field In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered fie ...
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, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word iso ...
to . is the
field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field (mathematics), field in which it can be Embedding, embedded. The construction of the field of fractions is modeled on the relationship between the integral do ...
of the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s . The algebraic closure of , i.e. the field of roots of rational polynomials, is the field of
algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
s. 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, 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 X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexiv ...
set which is countable, dense (in the above sense), and has no least or greatest element is order isomorphic to the rational numbers.

## Countability

The set of all rational numbers is
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
, as is illustrated in the figure to the right. As a rational number can be expressed as a ratio of two integers, it is possible to assign two integers to any point on a
square lattice In mathematics, the square lattice is a type of lattice in a two-dimensional Euclidean space. It is the two-dimensional version of the integer lattice, denoted as . It is one of the five types of two-dimensional lattices as classified by thei ...
as in a
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
, such that any grid point corresponds to a rational number. This method, however, exhibits a form of redundancy, as several different grid points will correspond to the same rational number; these are highlighted in red on the provided graphic. An obvious example can be seen in the line going diagonally towards the bottom right; such ratios will always equal 1, as any non-zero number divided by itself will always equal one. It is possible to generate all of the rational numbers without such redundancies: examples include the Calkin–Wilf tree and Stern–Brocot tree. 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 a
null set In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null ...
, that is,
almost all In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the mathe ...
real numbers are irrational, in the sense of
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wi ...
.

# Real numbers and topological properties

The rationals are a
dense subset In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, th ...
of the
real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
every real number has rational numbers arbitrarily close to it. A related property is that rational numbers are the only numbers with finite expansions as regular continued fractions. In the usual topology of the real numbers, the rationals are neither an
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
nor a
closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a clo ...
. By virtue of their order, the rationals carry an
order topology In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If ''X'' is a totally ordered set, th ...
. The rational numbers, as a subspace of the real numbers, also carry a
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
. The rational numbers form a
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
by using the
absolute difference The absolute difference of two real numbers x and y is given by , x-y, , the absolute value of their difference. It describes the distance on the real line between the points corresponding to x and y. It is a special case of the Lp distance for ...
metric , and this yields a third topology on . All three topologies coincide and turn the rationals into a
topological field In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is wi ...
. The rational numbers are an important example of a space which is not
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
. The rationals are characterized topologically as the unique
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
metrizable space In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) s ...
without isolated points. The space is also
totally disconnected In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty se ...
. The rational numbers do not form a
complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
, and the
real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
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 and for any non-zero integer , let , where is the highest power of dividing . In addition set . For any rational number , we set . Then defines a
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathema ...
on . The metric space is not complete, and its completion is the -adic number field . Ostrowski's theorem states that any non-trivial
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), and ...
on the rational numbers is equivalent to either the usual real absolute value or a -adic absolute value.

Floating point In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be ...
*
Ford circle In mathematics, a Ford circle is a circle with center at (p/q,1/(2q^2)) and radius 1/(2q^2), where p/q is an irreducible fraction, i.e. p and q are coprime integers. Each Ford circle is tangent to the horizontal axis y=0, and any two Ford circles ...
s *
Gaussian rational In mathematics, a Gaussian rational number is a complex number of the form ''p'' + ''qi'', where ''p'' and ''q'' are both rational numbers. The set of all Gaussian rationals forms the Gaussian rational field, denoted Q(''i''), obtained ...
* Naive height—height of a rational number in lowest term * Niven's theorem * Rational data type *'' Divine Proportions: Rational Trigonometry to Universal Geometry''