Rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator p and a non-zero denominator q.[1] For example, is a rational number, as is every integer (e.g., ). The set of all rational numbers, also referred to as "the rationals",[2] the field of rationals[3] or the field of rational numbers is usually denoted by boldface Q, or blackboard bold
"Rationals" redirects here. For other uses, see Rational (disambiguation).
A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: 3/4 = 0.75), or eventually begins to repeat the same finite sequence of digits over and over (example: 9/44 = 0.20454545...).[4] This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see Repeating decimal § Extension to other bases).
A real number that is not rational is called irrational.[5] Irrational numbers include the square root of 2 (), π, e, and the golden ratio (φ). Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational.[1]
Rational numbers can be formally defined as equivalence classes of pairs of integers (p, q) with q ≠ 0, using the equivalence relation defined as follows:
The fraction then denotes the equivalence class of (p, q).[6]
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.[7]
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 (see Construction of the real numbers).
:
common fraction
:
mixed numeral
repeating decimal using :
parentheses
using traditional typography:
continued fraction
continued fraction in abbreviated notation:
![{\displaystyle [2;1,2]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1887ebaaed6f055311780a6b5c6eb3c2d43481e7)
:
Egyptian fraction
:
prime power decomposition
:
quote notation
are different ways to represent the same rational value.
Real numbers and topological properties[edit]
The rationals are a dense subset of the real numbers; every real number has rational numbers arbitrarily close to it.[6] A related property is that rational numbers are the only numbers with finite expansions as regular continued fractions.[18]
In the usual topology of the real numbers, the rationals are neither an open set nor a closed set.[19]
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.[14]
