In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numeratorp and a non-zero denominatorq.[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
The term rational in reference to the set 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 curveis 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,[8] while the use of rational for qualifying numbers appeared almost a century earlier, in 1570.[9] 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)".[10][11]
This unusual history originated in the fact that ancient Greeks "avoided heresy by forbidding themselves from thinking of those [irrational] lengths as numbers".[12] So such lengths were irrational, in the sense of illogical, that is "not to be spoken about" (ἄλογος in Greek).[13]
Starting from a rational number 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.
Embedding of integers
Any integer n can be expressed as the rational number which is its canonical form as a rational number.
If both denominators are positive (particularly if both fractions are in canonical form):
if and only if
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.[6]
Addition
Two fractions are added as follows:
If both fractions are in canonical form, the result is in canonical form if and only if b, d are coprime integers.[6][14]
Subtraction
If both fractions are in canonical form, the result is in canonical form if and only if b, d are coprime integers.[14]
Multiplication
The rule for multiplication is:
where the result may be a reducible fraction—even if both original fractions are in canonical form.[6][14]
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 finite continued fraction is an expression such as
where an are integers. Every rational number can be represented as a finite continued fraction, whose coefficientsan can be determined by applying the Euclidean algorithm to (a, b).
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. (This construction can be carried out with any integral domain and produces its field of fractions.)[6]
The equivalence class of a pair (m, n) is denoted
Two pairs (m1, n1) and (m2, n2) belong to the same equivalence class (that is are equivalent) if and only if
The integers may be considered to be rational numbers identifying the integer n 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.[6]
has no field automorphism other than the identity. (A field automorphism must fix 0 and 1; as it must fix the sum and the difference of two fixed elements, it must fix every integer; as it must fix the quotient of two fixed elements, it must fix every rational number, and is thus the identity.)
is a prime field, which is a field that has no subfield other than itself.[15] 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[14] 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
The rationals are a densely ordered set: between any two rationals, there sits another one, and, therefore, infinitely many other ones.[6] 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.[17]
Countability
The set of all rational numbers is countable, as is illustrated in the figure to the right. As a rational number can be expressed as a ratio of two integers, it is possible to assign two integers to any point on a square lattice as in a Cartesian coordinate system, such that any grid point corresponds to a rational number. This method, however, exhibits a form of redundancy, as several different grid points will correspond to the same rational number; these are highlighted in red on the provided graphic. An obvious example can be seen in the line going diagonally towards the bottom right; such ratios will always equal 1, as any non-zero number divided by itself will always equal one.
It is possible to generate all of the rational numbers without such redundancies: examples include the Calkin–Wilf tree and Stern–Brocot tree.
As the set of all rational numbers is countable, and the set of all real numbers (as well as the set of irrational 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.
The metric space is not complete, and its completion is the p-adic number fieldOstrowski's theorem states that any non-trivial absolute value on the rational numbers is equivalent to either the usual real absolute value or a p-adic absolute value.