Rational number


In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a nonzero denominator. For example, is a rational number, as is every integer, or eventually begins to repeat the same finite sequence of digits over and over. This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones.
A real number that is not rational is called irrational. Irrational numbers include the square root of 2 Pi|,, and the golden ratio. Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational.
The field of rational numbers is the unique field that contains the integers, 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, 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.

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 ; a rational matrix is a matrix of rational numbers, though it sometimes also refers to a matrix whose entries are rational functions; 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". 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 ".
This unusual history originated in the fact that ancient Greeks "avoided heresy by forbidding themselves from thinking of those lengths as numbers". So such lengths were irrational, in the sense of illogical, that is "not to be spoken about".

Arithmetic

Irreducible fraction

Every rational number may be expressed in a unique way as an irreducible fraction where and are coprime integers 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 both 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

If both fractions are in canonical form, then:

Ordering

If both denominators are positive :
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 are coprime integers.

Subtraction

If both fractions are in canonical form, the result is in canonical form if and only if are coprime integers.

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 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:
  • mixed numeral:
  • repeating decimal using a vinculum:
  • repeating decimal using parentheses:
  • continued fraction 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 classes of ordered pairs of integers.
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 by this equivalence relation, equipped with the addition and the multiplication induced by the above operations.
The equivalence class of a pair is denoted
Two pairs and belong to the same equivalence class if and only if
This means that
if and only if
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, 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 other than 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 of the integers 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
, we have
Any totally ordered set which is countable, dense, 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 of fractions from which one can remove the reducible fractions, obtaining 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.
As the set of all rational numbers is countable, and the set of all real 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 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 nor a closed set.
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 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 above.