In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two , a numerator and a nonzero denominator .
Since may be equal to 1, every integer is a rational number. The set of all rational numbers, often referred to as "
the rationals", the
field of rationals or the
field of rational numbers is usually denoted by a boldface (or
blackboard bold $\backslash mathbb\{Q\}$, Unicode ℚ);
it was thus denoted in 1895 by
Giuseppe Peano after
, Italian for "
quotient".
The decimal expansion of a rational number always either terminates after a finite number of numerical digit or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for decimal, but also for any other integer radix (e.g. binary, hexadecimal).
A real number that is not rational is called irrational. Irrational numbers include , , , and Golden ratio. The decimal expansion of an irrational number continues without repeating. Since the set of rational numbers is countable set, and the set of real numbers is uncountable set, almost all real numbers are irrational.[
]
Rational numbers can be formally defined as equivalence classes of pairs of integers such that , for the equivalence relation defined by iff . With this formal definition, the fraction becomes the standard notation for the equivalence class of .
The rational numbers together with addition and multiplication form a field which contains the and is contained in any field containing the integers. Finite field extension of are called algebraic number fields, and the algebraic closure of is the field of .
(2018). 053440264X, Thomson Brooks/Cole. 053440264X
In mathematical analysis, the rational numbers form a dense set of the real numbers. The real numbers can be constructed from the rational numbers by completion, using , , or infinite .
Terminology
The term
rational in reference to the set
Q refers to the fact that a rational number represents a
ratio of two integers. In mathematics, "rational" is often used as a noun abbreviating "rational number". The adjective
rational sometimes means that the
are rational numbers. For example, a
rational point is a point with rational
coordinates (that is 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, for avoiding confusion with "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.
Arithmetic
Embedding of integers
Any integer
n can be expressed as the rational number
n/1.
Equality
 $\backslash frac\{a\}\{b\}\; =\; \backslash frac\{c\}\{d\}$ if and only if $ad\; =\; bc.$
Ordering
Where both denominators are positive:
 $\backslash frac\{a\}\{b\}\; <\; \backslash frac\{c\}\{d\}$ if and only if $ad\; <\; bc.$
If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations:
 $\backslash frac\{a\}\{b\}\; =\; \backslash frac\{a\}\{b\}$
and
 $\backslash frac\{a\}\{b\}\; =\; \backslash frac\{a\}\{b\}.$
Addition
Two fractions are added as follows:
 $\backslash frac\{a\}\{b\}\; +\; \backslash frac\{c\}\{d\}\; =\; \backslash frac\{ad+bc\}\{bd\}.$
Subtraction
 $\backslash frac\{a\}\{b\}\; \; \backslash frac\{c\}\{d\}\; =\; \backslash frac\{adbc\}\{bd\}.$
Multiplication
The rule for multiplication is:
 $\backslash frac\{a\}\{b\}\; \backslash cdot\backslash frac\{c\}\{d\}\; =\; \backslash frac\{ac\}\{bd\}.$
Division
Where
c ≠ 0 and
b ≠ 0:
 $\backslash frac\{a\}\{b\}\; \backslash div\; \backslash frac\{c\}\{d\}\; =\; \backslash frac\{ad\}\{bc\}.$
Note that division is equivalent to multiplying by the reciprocal of the divisor fraction:
 $\backslash frac\{ad\}\{bc\}\; =\; \backslash frac\{a\}\{b\}\; \backslash times\; \backslash frac\{d\}\{c\}.$
Inverse
Additive inverse and multiplicative inverses exist in the rational numbers:
 $\; \backslash left(\; \backslash frac\{a\}\{b\}\; \backslash right)\; =\; \backslash frac\{a\}\{b\}\; =\; \backslash frac\{a\}\{b\}\; \backslash quad\backslash mbox\{and\}\backslash quad$
\left(\frac{a}{b}\right)^{1} = \frac{b}{a} \mbox{ if } a \neq 0.
Exponentiation to integer power
If
n is a nonnegative integer, then
 $\backslash left(\backslash frac\{a\}\{b\}\backslash right)^n\; =\; \backslash frac\{a^n\}\{b^n\}$
and (if
a ≠ 0):
 $\backslash left(\backslash frac\{a\}\{b\}\backslash right)^\{n\}\; =\; \backslash frac\{b^n\}\{a^n\}.$
Continued fraction representation
A
finite continued fraction is an expression such as
 $a\_0\; +\; \backslash cfrac\{1\}\{a\_1\; +\; \backslash cfrac\{1\}\{a\_2\; +\; \backslash cfrac\{1\}\{\; \backslash ddots\; +\; \backslash cfrac\{1\}\{a\_n\}\; \}\}\},$
where
a_{n} are integers. Every rational number
a/
b can be represented as a finite continued fraction, whose
a_{n} can be determined by applying the Euclidean algorithm to (
a,
b).
Other representations

common fraction: $\backslash frac\{8\}\{3\}$

mixed numeral: $2\backslash tfrac\{2\}\{3\}$

repeating decimal using a vinculum: $2.\backslash overline\{6\}$

repeating decimal using parentheses: $2.(6)$

continued fraction using traditional typography: $2\; +\; \backslash cfrac\{1\}\{1\; +\; \backslash cfrac\{1\}\{2\}\; \}$

continued fraction in abbreviated notation: 2;

egyptian fraction: $2\; +\; \backslash frac\{1\}\{2\}\; +\; \backslash frac\{1\}\{6\}$

prime power decomposition: $\backslash frac\{117\}\{1000\}\; =\; 2^\{3\}\backslash times3^2\backslash times5^\{3\}\backslash times13$

quote notation: 3!6
are different ways to represent the same rational value.
Formal construction
Mathematically we may construct the rational numbers as equivalence classes of
of
(
m,
n), with . This space of equivalence classes is the quotient space where if, and only if, We can define addition and multiplication of these pairs with the following rules:
 $\backslash left(m\_1,\; n\_1\backslash right)\; +\; \backslash left(m\_2,\; n\_2\backslash right)\; \backslash equiv\; \backslash left(m\_1n\_2\; +\; n\_1m\_2,\; n\_1n\_2\backslash right)$
 $\backslash left(m\_1,\; n\_1\backslash right)\; \backslash times\; \backslash left(m\_2,\; n\_2\backslash right)\; \backslash equiv\; \backslash left(m\_1m\_2,\; n\_1n\_2\backslash right)$
and, if
m_{2} ≠ 0, division by
 $\backslash frac\{\backslash left(m\_1,\; n\_1\backslash right)\}\; \{\backslash left(m\_2,\; n\_2\backslash right)\}\; \backslash equiv\; \backslash left(m\_1n\_2,\; n\_1m\_2\backslash right).$
The equivalence relation ( m_{1}, n_{1}) ~ ( m_{2}, n_{2}) if, and only if, is a congruence relation, i.e. it is compatible with the addition and multiplication defined above, and we may define Q to be the quotient set i.e. we identify two pairs ( m_{1}, n_{1}) and ( m_{2}, n_{2}) if they are equivalent in the above sense. (This construction can be carried out in any integral domain: see field of fractions.) We denote by ( m_{1}, n_{1}) the equivalence class containing ( m_{1}, n_{1}). If ( m_{1}, n_{1}) ~ ( m_{2}, n_{2}) then, by definition, ( m_{1}, n_{1}) belongs to ( m_{2}, n_{2}) and ( m_{2}, n_{2}) belongs to ( m_{1}, n_{1}); in this case we can write . Given any equivalence class ( m, n) there are a countably infinite number of representation, since
 $\backslash cdots\; =\; (2m,2n)\; =\; (m,n)\; =\; (m,n)\; =\; (2m,2n)\; =\; \backslash cdots.$
The canonical choice for ( m, n) is chosen so that n is positive and , i.e. m and n share no common factors, i.e. m and n are coprime. For example, we would write (1,2) instead of (2,4) or (−12,−24), even though .
We can also define a total order on Q. Let ∧ be the andsymbol and ∨ be the orsymbol. We say that if:
 $(n\_1n\_2\; >\; 0\; \backslash \; \backslash and\; \backslash \; m\_1n\_2\; \backslash le\; n\_1m\_2)\; \backslash \; \backslash or\; \backslash \; (n\_1n\_2\; <\; 0\; \backslash \; \backslash and\; \backslash \; m\_1n\_2\; \backslash ge\; n\_1m\_2).$
The integers may be considered to be rational numbers by the embedding that maps m to ( m,1).
Properties
The set
Q, together with the addition and multiplication operations shown above, forms a field, the field of fractions of the
Z.
The rationals are the smallest field with characteristic zero: every other field of characteristic zero contains a copy of Q. The rational numbers are therefore the prime field for characteristic zero.
The algebraic closure of Q, i.e. the field of roots of rational polynomials, is the .
The set of all rational numbers is countable. Since the set of all real numbers is uncountable, we say that almost all real numbers are irrational, in the sense of Lebesgue measure, i.e. the set of rational numbers is a null set.
The rationals are a densely ordered set: between any two rationals, there sits another one, and, therefore, infinitely many other ones. For example, for any two fractions such that
 $\backslash frac\{a\}\{b\}\; <\; \backslash frac\{c\}\{d\}$
(where
$b,d$ are positive), we have
 $\backslash frac\{a\}\{b\}\; <\; \backslash frac\{ad\; +\; bc\}\{2bd\}\; <\; \backslash frac\{c\}\{d\}.$
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.
Real numbers and topological properties
The rationals are a
dense set 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 set expansions as regular continued fractions.
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. The rational numbers form a metric space by using the absolute difference metric and this yields a third topology on Q. 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 . The space is also totally disconnected. The rational numbers do not form a complete metric space; the real numbers are the completion of Q under the metric above.
padic numbers
In addition to the absolute value metric mentioned above, there are other metrics which turn
Q into a topological field:
Let p be a prime number and for any nonzero integer a, let , where p^{n} is the highest power of p divisor a.
In addition set For any rational number a/ b, we set
Then defines a metric space on Q.
The metric space ( Q, d_{p}) is not complete, and its completion is the padic number Q_{ p}. Ostrowski's theorem states that any nontrivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a padic number absolute value.
See also
External links