rational number
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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a rational number is a
number A number is a mathematical object used to count, measure, and label. The most basic 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 can ...
that can be expressed as the quotient or
fraction A fraction (from , "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, thre ...
of two
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all rational numbers is often referred to as "the rationals", and is closed under
addition Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol, +) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication, and Division (mathematics), divis ...
, subtraction,
multiplication Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division (mathematics), division. The result of a multiplication operation is called a ''Product (mathem ...
, and division by a nonzero rational number. It is a field under these operations and therefore also called the field of rationals or the field of rational numbers. It is usually denoted by boldface , or blackboard bold 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_\cdots b_0.a_1a_2\cdots Here is the decimal separator ...
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 cal ...
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 (''decimal fractions'') of t ...
, but also in every other integer base, such as the binary and
hexadecimal Hexadecimal (also known as base-16 or simply hex) is a Numeral system#Positional systems in detail, positional numeral system that represents numbers using a radix (base) of sixteen. Unlike the decimal system representing numbers using ten symbo ...
ones (see ). A real number that is not rational is called irrational. Irrational numbers include the
square root of 2 The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2. It may be written as \sqrt or 2^. It is an algebraic number, and therefore not a transcendental number. Te ...
, , and the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...
(). Since the set of rational numbers is countable, and the set of real numbers is uncountable,
almost all In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if X is a set (mathematics), set, "almost all elements of X" means "all elements of X but those in a negligible set, negligible subset of X". The meaning o ...
real numbers are irrational. The field of rational numbers is the unique field that contains the
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s, and is contained in any field containing the integers. In other words, the field of rational numbers is a prime field. A field has characteristic zero if and only if it contains the rational numbers as a subfield. Finite extensions of are called algebraic number fields, and the algebraic closure 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 ( ...
, the rational numbers form a dense subset of the real numbers. The real numbers can be constructed from the rational numbers by completion, using Cauchy sequences, Dedekind cuts, or infinite decimals (see Construction of the real numbers).


Terminology

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 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 is a rational expression and defines a rational function, even if its coefficients are not rational numbers). However, a rational curve ''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 of ''ratio''. On the contrary, 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 "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).


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). ...
where and are
coprime integers In number theory, 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 of the rational number. Starting from a rational number its canonical form may be obtained by dividing and by their greatest common divisor, 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.


Addition

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 are
coprime integers In number theory, 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 are
coprime integers In number theory, 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 an element , denoted , is the element that when added to , yields the additive identity, 0 (zero). In the most familiar cases, this is the number 0, but it can also refer to a more generalized zero el ...
, often called its ''opposite'', : - \left( \frac \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 Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a ra ...
, also called its ''reciprocal'', : \left(\frac\right)^ = \frac. If is in canonical form, then the canonical form of its reciprocal is either or depending on the sign of .


Division

If 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(\frac\right)^n = \frac. The result is in canonical form if the same is true for In particular, :\left(\frac\right)^0 = 1. If , then :\left(\frac\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 coefficients can be determined by applying the Euclidean algorithm to .


Other representations

* common fraction: * mixed numeral: *
repeating decimal A repeating decimal or recurring decimal is a decimal representation of a number whose digits are eventually periodic (that is, after some place, the same sequence of digits is repeated forever); if this sequence consists only of zeros (that i ...
using a vinculum: 2.\overline 6 * repeating decimal using parentheses: 2.(6) * continued fraction using traditional typography: 2 + \tfrac 1 * continued fraction in abbreviated notation: ; 1, 2/math> * Egyptian fraction: 2 + \tfrac 1 2 + \tfrac 1 6 * prime power decomposition: 2^3 \times 3^ * quote notation: 3'6 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 (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s. More precisely, let be the set of the pairs of integers such . An equivalence relation is defined on this set by : (m_1, n_1) \sim (m_2, n_2) \iff m_1 n_2 = m_2 n_1. Addition and multiplication can be defined by the following rules: :(m_1, n_1) + (m_2, n_2) \equiv (m_1n_2 + n_1m_2, n_1n_2), :(m_1, n_1) \times (m_2, n_2) \equiv (m_1m_2, n_1n_2). 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 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 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 fie ...
.) 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 :m_1n_2 = m_2n_1. This means that :\frac = \frac if and only if :m_1n_2 = m_2n_1. 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, 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 :\frac \le \frac If :\begin & (n_1n_2 > 0 \quad \text \quad m_1n_2 \le n_1m_2) \\ & \qquad \text \\ & (n_1n_2 < 0 \quad \text \quad m_1n_2 \ge n_1m_2). \end


Properties

The set of all rational numbers, together with the addition and multiplication operations shown above, forms a field. has no field automorphism other than the identity. (A field automorphism must fix 0 and 1; as it must fix the sum and the difference of two fixed elements, it must fix every integer; as it must fix the quotient of two fixed elements, it must fix every rational number, and is thus the identity.) is a prime field, 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 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 fie ...
of the
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s The algebraic closure 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 :\frac < \frac (where b,d are positive), we have :\frac < \frac < \frac. 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 positive rational numbers is countable, as is illustrated in the figure. More precisely, one can sort the fractions by increasing values of the sum of the numerator and the denominator, and, for equal sums, by increasing numerator or denominator. This produces a
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 cal ...
of fractions, from which one can remove the reducible fractions (in red on the figure), for getting a sequence that contains each rational number exactly once. This establishes a bijection between the rational numbers and the natural numbers, which maps each rational number to its rank in the sequence. A similar method can be used for numbering all rational numbers (positive and negative). As the set of all rational numbers is countable, and the set of all real numbers (as well as the set of irrational numbers) is uncountable, the set of rational numbers is a null set, that is,
almost all In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if X is a set (mathematics), set, "almost all elements of X" means "all elements of X but those in a negligible set, negligible subset of X". The meaning o ...
real numbers are irrational, in the sense of Lebesgue measure.


Real numbers and topological properties

The rationals are a dense subset 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 expansions as regular continued fractions. In the usual topology of the real numbers, the rationals are neither an
open set In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
nor a
closed set In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its lim ...
. By virtue of their order, the rationals carry an order topology. 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'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology ...
. The rational numbers form a metric space by using the absolute difference metric d(x,y)=, x-y, , and this yields a third topology on All three topologies coincide and turn the rationals into a topological field. The rational numbers are an important example of a space which is not locally compact. The rationals are characterized topologically as the unique countable metrizable space without isolated points. The space is also totally disconnected. The rational numbers do not form a complete metric space, and the real numbers are the completion of under the metric d(x,y)=, x-y, above.


''p''-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 (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 , a, _p = p^, where is the highest power of dividing . In addition set , 0, _p = 0. For any rational number we set :\left, \frac\_p = \frac. Then :d_p(x,y) =, x-y, _p defines a metric 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 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 circles * Gaussian rational * Naive height—height of a rational number in lowest term * Niven's theorem * Rational data type


References


Notes


External links

*
"Rational Number" From MathWorld – A Wolfram Web Resource
{{Authority control Elementary mathematics Field (mathematics) Fractions (mathematics) Sets of real numbers