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 c ...
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 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 pro ...
A rational number is a
real number. The real numbers that are rational are those whose
decimal expansion either terminates after a finite number of
digits (example: ), or eventually begins to
repeat the same finite
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
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 numer ...
, but also in every other integer
base, such as the
binary
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that ta ...
and
hexadecimal 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 numbers ...
, 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 num ...
,
almost all real numbers are irrational.
Rational numbers can be
formally defined as
equivalence classes of pairs of integers with , using the
equivalence relation defined as follows:
:
The fraction then denotes the equivalence class of .
Rational numbers together with
addition and
multiplication 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 ide ...
, 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 ide ...
if and only if it contains the rational numbers as a subfield. Finite
extensions of are called
algebraic number fields, and the
algebraic closure
In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics.
Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ( ...
of is the field of
algebraic numbers.
In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, 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, the ra ...
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 numbe ...
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 r ...
s, or infinite
decimals (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
coefficients 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 (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" (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 example ...
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 c ...
''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 proc ...
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 cult ...
"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 equivale ...
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 ...
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
:
if and only if
If both fractions are in canonical form, then:
:
if and only if
and
Ordering
If both denominators are positive (particularly if both fractions are in canonical form):
:
if and only if
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:
:
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 equivale ...
.
Subtraction
:
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 equivale ...
.
Multiplication
The rule for multiplication is:
:
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, often called its ''opposite'',
:
If is in canonical form, the same is true for its opposite.
A nonzero rational number has a
multiplicative inverse, also called its ''reciprocal'',
:
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
:
Thus, dividing by is equivalent to multiplying by the
reciprocal of :
:
Exponentiation to integer power
If is a non-negative integer, then
:
The result is in canonical form if the same is true for . In particular,
:
If , then
:
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
:
where are integers. Every rational number can be represented as a finite continued fraction, whose
coefficients can be determined by applying the
Euclidean algorithm 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 a ...
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 integer ...
using traditional typography:
* continued fraction in abbreviated notation:
*
Egyptian fraction:
*
prime power decomposition:
*
quote notation
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension ...
:
are different ways to represent the same rational value.
Formal construction
The rational numbers may be built as
equivalence classes of
ordered pairs 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 is defined on this set by
:
Addition and multiplication can be defined by the following rules:
:
:
This equivalence relation is a
congruence relation, which means that it is compatible with the addition and multiplication defined above; the set of rational numbers 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 s ...
and produces its
field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
.)
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
:
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 may be defined on the rational numbers, that extends the natural order of the integers. One has
:
if
:
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 ide ...
, 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 fiel ...
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 to .
is the
field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
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
In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics.
Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ( ...
of , i.e. the field of roots of rational polynomials, is the field of
algebraic numbers.
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
:
(where
are positive), we have
:
Any
totally ordered 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 numbers ...
, 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 as in a
Cartesian coordinate system, 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, that is,
almost all real numbers are irrational, in the sense of
Lebesgue measure.
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, the ra ...
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 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 or marke ...
expansions as
regular continued fractions.
In the usual
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
of the real numbers, the rationals are neither an
open set nor a
closed set.
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 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 w ...
. The rational numbers are an important example of a space which is not
locally compact. 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 numbers ...
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 point
]
In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equiva ...
s. The space is also
totally disconnected space, totally disconnected. 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 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 a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a 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 because the only ways ...
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 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 on the rational numbers \Q is equivalent to either the usual real absolute value or a -adic absolute value.
Definitions
Raisi ...
states that any non-trivial
absolute value on the rational numbers is equivalent to either the usual real absolute value or a
-adic absolute value.
See also
*
Dyadic rational
*
Floating point
*
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 b ...
*
Naive height—height of a rational number in lowest term
*
Niven's theorem
In mathematics, Niven's theorem, named after Ivan Niven, states that the only rational values of ''θ'' in the interval 0° ≤ ''θ'' ≤ 90° for which the sine of ''θ'' degrees is also a rational number ...
*
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)
Sets of real numbers