Rational number

The decimal expansion of a rational number 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...).[6] Conversely, any repeating or terminating decimal represents a rational number. These statements are true in base 10, and in every other integer base (for example, binary or hexadecimal).[citation needed]

A real number that is not rational is called irrational.[5] 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.[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:

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 Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers.[8]

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 (for more, see Construction of the real numbers).[citation needed]

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 (i.e., 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, to avoid confusion between "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.[citation needed]

Although nowadays rational numbers are defined in terms of ratios, the term rational is not a derivation of ratio. On the opposite, it is ratio that is derived from rational: the first use of ratio with its modern meaning was attested in English about 1660,[9] while the use of rational for qualifying numbers appeared almost a century earlier, in 1570.[10] 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 (following his peculiar use of ἄλογος)".[11][12]

This unusual history originated in the fact that ancient Greeks "avoided heresy by forbidding themselves from thinking of those [irrational] lengths as numbers".[13] So such lengths were irrational, in the sense of illogical, that is "not to be spoken about" (ἄλογος in Greek).[14]

This etymology is similar to that of imaginary numbers and real numbers.

Every rational number may be expressed in a unique way as an irreducible fraction a/b, where a and b are coprime integers and b > 0. This is often called the canonical form of the rational number.

Starting from a rational number a/b, its canonical form may be obtained by dividing a and b by their greatest common divisor, and, if b < 0, changing the sign of the resulting numerator and denominator.[citation needed]

Any integer n can be expressed as the rational number n/1, which is its canonical form as a rational number.[citation needed]

If both denominators are positive (particularly if both fractions are in canonical form):

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.[7]

If both fractions are in canonical form, the result is in canonical form if and only if b and d are coprime integers.[7][15]

If both fractions are in canonical form, the result is in canonical form if and only if b and d are coprime integers.[15][verification needed]

where the result may be a reducible fraction—even if both original fractions are in canonical form.[7][15]

Every rational number a/b has an additive inverse, often called its opposite,

A nonzero rational number a/b has a multiplicative inverse, also called its reciprocal,

If a/b is in canonical form, then the canonical form of its reciprocal is either b/a or b/a, depending on the sign of a.[citation needed]

Thus, dividing a/b by c/d is equivalent to multiplying a/b by the reciprocal of c/d:

The result is in canonical form if the same is true for a/b. In particular,

If a/b is in canonical form, the canonical form of the result is bn/an if a > 0 or n is even. Otherwise, the canonical form of the result is bn/an.[citation needed]

where an are integers. Every rational number a/b can be represented as a finite continued fraction, whose coefficients an can be determined by applying the Euclidean algorithm to (a, b).

A diagram showing a representation of the equivalent classes of pairs of integers

The rational numbers may be built as equivalence classes of ordered pairs of integers.[7][15]

The equivalence class of a pair (m, n) is denoted m/n. Two pairs (m1, n1) and (m2, n2) belong to the same equivalence class (that is are equivalent) if and only if m1n2 = m2n1. This means that m1/n1 = m2/n2 if and only m1n2 = m2n1.[7][15]

Every equivalence class m/n may be represented by infinitely many pairs, since

Each equivalence class contains a unique canonical representative element. The canonical representative is the unique pair (m, n) in the equivalence class such that m and n are coprime, and n > 0. It is called the representation in lowest terms of the rational number.

The integers may be considered to be rational numbers identifying the integer n with the rational number n/1.

A total order may be defined on the rational numbers, that extends the natural order of the integers. One has

The set Q of all rational numbers, together with the addition and multiplication operations shown above, forms a field.[7]

With the order defined above, Q is an ordered field[15] 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.[citation needed]

Q is a prime field, which is a field that has no subfield other than itself.[16] The rationals are the smallest field with characteristic zero. Every field of characteristic zero contains a unique subfield isomorphic to Q.[citation needed]

Q is the field of fractions of the integers Z.[17] The algebraic closure of Q, i.e. the field of roots of rational polynomials, is the field of algebraic numbers.[citation needed]

The set of all rational numbers is countable (see the figure), while the set of all real numbers (as well as the set of irrational 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.[citation needed]

The rationals are a densely ordered set: between any two rationals, there sits another one, and, therefore, infinitely many other ones.[7] For example, for any two fractions such that

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.[18]

The rationals are a dense subset of the real numbers: every real number has rational numbers arbitrarily close to it.[7] A related property is that rational numbers are the only numbers with finite expansions as regular continued fractions.[citation needed]

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) = |xy|, 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[citation needed]; the real numbers are the completion of Q under the metric d(x, y) = |xy| above.[15]

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 = pn, 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.

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.[citation needed]