Rational number In mathematics, a rational number is any number
that can be expressed as the quotient or fraction p/q of two integers, with the denominator
q not equal to zero. Since q may be equal to 1, every integer is a rational number.
The set of all rational numbers is usually denoted by a boldface Q ; it was thus named
in 1895 by Peano after quoziente, Italian for "quotient".
The decimal expansion of a rational number always either terminates after a finite number
of digits 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 base 10, but also for binary, hexadecimal, or any other integer base.
A real number that is not rational is called irrational. Irrational numbers include √2,
π, e, and φ. The decimal expansion of an irrational number continues without repeating.
Since the set of rational numbers is countable, and the set of real numbers is uncountable,
almost all real numbers are irrational. The rational numbers can be formally defined
as the equivalence classes of the quotient set (Z × (Z \ {0})) / ~, where the cartesian
product Z × (Z \ {0}) is the set of all ordered pairs (m,n) where m and n are integers, n
is not 0 (n ≠ 0), and "~" is the equivalence relation defined by (m1,n1) ~ (m2,n2) if,
and only if, m1n2 − m2n1 = 0. In abstract algebra, the rational numbers
together with certain operations of addition and multiplication form a field. This is the
archetypical field of characteristic zero, and is the field of fractions for the ring
of integers. Finite extensions of Q are called algebraic number fields, and the algebraic
closure of Q is the field of algebraic numbers. In mathematical analysis, 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.
Zero divided by any other integer equals zero, therefore zero is a rational number (but division
by zero is undefined). 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, the adjective rational often means that the
underlying field considered is the field Q of rational numbers. Rational polynomial usually,
and most correctly, means a polynomial with rational coefficients, also called a "polynomial
over the rationals". However, rational function does not mean the underlying field is the
rational numbers, and a rational algebraic curve is not an algebraic curve with rational
coefficients. Arithmetic
Embedding of integers Any integer n can be expressed as the rational
number n/1. Ordering
Where both denominators are positive: If either denominator is negative, the fractions
must first be converted into equivalent forms with positive denominators, through the equations:
and Addition
Two fractions are added as follows: Multiplication
The rule for multiplication is: Division
Where c ≠ 0: Note that division is equivalent to multiplying
by the reciprocal of the divisor fraction: Inverse
Additive and multiplicative inverses exist in the rational numbers:
Exponentiation to integer power If n is a non-negative integer, then
and (if a ≠ 0): Continued fraction representation
A finite continued fraction is an expression such as
where an are integers. Every rational number a/b has two closely related expressions as
a finite continued fraction, whose coefficients an can be determined by applying the Euclidean
algorithm to (a,b). Formal construction
Mathematically we may construct the rational numbers as equivalence classes of ordered
pairs of integers (m,n), with n ≠ 0. This space of equivalence classes is the quotient
space (Z × (Z \ {0})) / ~, where (m1,n1) ~ (m2,n2) if, and only if, m1n2 − m2n1 = 0.
We can define addition and multiplication of these pairs with the following rules:
and, if m2 ≠ 0, division by The equivalence relation (m1,n1) ~ (m2,n2)
if, and only if, m1n2 − m2n1 = 0 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 (Z × (Z \ {0}))
/ ~, i.e. we identify two pairs (m1,n1) and (m2,n2) 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 the equivalence class containing (m1,n1). If (m1,n1) ~ (m2,n2) then, by definition,
(m1,n1) belongs to and (m2,n2) belongs to; in this case we can write =. Given any equivalence
class there are a countably infinite number of representation, since
The canonical choice for is chosen so that gcd(m,n) = 1, i.e. m and n share no common
factors, i.e. m and n are coprime. For example, we would write instead of or, even though
= =. We can also define a total order on Q. Let
∧ be the and-symbol and ∨ be the or-symbol. We say that ≤ if:
The integers may be considered to be rational numbers by the embedding that maps m to.
Properties The set Q, together with the addition and
multiplication operations shown above, forms a field, the field of fractions of the integers
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 algebraic
numbers. 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
(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. 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.
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 d(x,y) = |x − y|, 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 isolated points. 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 d(x,y) = |x − y|, above.
p-adic 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 non-zero integer a, let |a|p = p−n, where pn is the
highest power of p dividing a. In addition set |0|p = 0. For any rational
number a/b, we set |a/b|p = |a|p / |b|p. Then dp(x,y) = |x − y|p defines a metric
on Q. The metric space (Q,dp) is not complete, and
its completion is the p-adic number field Qp. Ostrowski's theorem states that any non-trivial
absolute value on the rational numbers Q is equivalent to either the usual real absolute
value or a p-adic absolute value.