# 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 non-zero 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 ; 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 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 any other integer base.

A real number that is not rational is called irrational. Irrational numbers include square root of 2|, Pi|, E |, and Golden ratio|. 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.

Rational numbers can be formally defined as equivalence classes of pairs of integers such that, for the equivalence relation defined by if, and only if. With this formal definition, the fraction becomes the standard notation for the equivalence class of.

Rational numbers together with addition and multiplication form a field which contains the integers and is contained in any field containing the integers. In other words, the field of rational numbers is a prime field, and 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

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 coefficients are rational numbers. For example, a rational point is a point with rational coordinates ; 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". However, a rational curve

*is not*a curve defined over the rationals, but a curve which can be parameterized by rational functions.

## 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.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

If both fractions are in canonical form then### Ordering

If both denominators are positive, and, in particular, if both fractions are in canonical form,If either denominator is negative, 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.

### Subtraction

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

The rule for multiplication is:Even if both fractions are in canonical form, the result may be a reducible fraction.

### 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 isThus, dividing by is equivalent to multiplying by the reciprocal of :

### Exponentiation to integer power

If is a non-negative integer, thenThe 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 either or is even. Otherwise, the canonical form of the result is

## Continued fraction representation

A**finite continued fraction**is an expression such as

where

*a*are integers. Every rational number

_{n}*a*/

*b*can be represented as a finite continued fraction, whose coefficients

*a*can be determined by applying the Euclidean algorithm to.

_{n}## 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:
**3!6**

## 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

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**Q**of all rational numbers, together with the addition and multiplication operations shown above, forms a field.

**Q**has no field automorphism other than the identity.

With the order defined above,

**Q**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

**Q**.

**Q**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

**Q**.

**Q**is the field of fractions of the integers

**Z**. The algebraic closure of

**Q**, i.e. the field of roots of rational polynomials, is the field of algebraic numbers.

The set of all rational numbers is countable, while the set of all real numbers is uncountable. Being countable, the set of rational numbers is a null set, that is, almost all real numbers are irrational, in the sense of Lebesgue measure.

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.

## 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 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 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, where

*p*is the highest power of

^{n}*p*dividing

*a*.

In addition set For any rational number

*a*/

*b*, we set

Then defines a metric on

**Q**.

The metric space is not complete, and its completion is the

*p*-adic number field

**Q**

_{p}. 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.