Wiki: Rational number - upcScavenger

In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two , a numerator and a non-zero denominator .

Kenneth Rosen (2025). 9780072880083, McGraw-Hill. ISBN 9780072880083

For example, is a rational number, as is every integer (for example, The set of all rational numbers is often referred to as " the rationals",

(2025). 9780486471860, Courier Corporation. . ISBN 9780486471860

Extract of page 382 and is closed under addition, subtraction, multiplication, and division by a nonzero rational number. It is a field under these operations and therefore also called the field of rationals

(1996). 9780821805756, American Mathematical Soc. . ISBN 9780821805756

Extract of page 104 or the field of rational numbers. It is usually denoted by boldface , or blackboard bold

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 numerical digit (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in decimal, but also in every other integer radix, such as the binary and hexadecimal ones (see ).

A real number that is not rational is called irrational. Irrational numbers include the square root of 2 , , and the golden ratio (). Since the set of rational numbers is countable set, and the set of real numbers is uncountable set, almost all real numbers are irrational.

The field of rational numbers is the unique field that contains the , 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 field extension of are called algebraic number fields, and the algebraic closure of is the field of .

(2025). 053440264X, Thomson Brooks/Cole. 053440264X

In mathematical analysis, the rational numbers form a dense set of the real numbers. The real numbers can be constructed from the rational numbers by completion, using , , or infinite (see Construction of the real numbers).

Terminology

In mathematics, "rational" is often used as a noun abbreviating "rational number". The adjective rational sometimes means that the 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, 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" (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.

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, Entry ratio, n., sense 2.a. while the use of rational for qualifying numbers appeared almost a century earlier, in 1570. Entry rational, a. (adv.) and n., sense 5.a. 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)". Entry irrational, a. and n., sense 3.

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

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

\frac{a}{b} = \frac{c}{d}

if and only if

ad = bc

If both fractions are in canonical form, then:

\frac{a}{b} = \frac{c}{d}

if and only if

a = c

and

b = d

Norman L. Biggs (2025). 9780198713692, Oxford University Press. ISBN 9780198713692

Ordering

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

\frac{a}{b} < \frac{c}{d}

if and only if

ad < bc.

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:

\frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}.

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

Subtraction

\frac{a}{b} - \frac{c}{d} = \frac{ad-bc}{bd}.

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:

\frac{a}{b} \cdot\frac{c}{d} = \frac{ac}{bd}.

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,

- \left( \frac{a}{b} \right) = \frac{-a}{b}.

If is in canonical form, the same is true for its opposite.

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

\left(\frac{a}{b}\right)^{-1} = \frac{b}{a}.

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

\frac{\,\dfrac{a}{b}\,} {\dfrac{c}{d}} = \frac{ad}{bc}.

Thus, dividing by is equivalent to multiplying by the reciprocal of

\frac{ad}{bc} = \frac{a}{b} \cdot \frac{d}{c}.

Exponentiation to integer power

If is a non-negative integer, then

\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}.

The result is in canonical form if the same is true for In particular,

\left(\frac{a}{b}\right)^0 = 1.

If , then

\left(\frac{a}{b}\right)^{-n} = \frac{b^n}{a^n}.

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

a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{ \ddots + \cfrac{1}{a_n} }}},

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

Other representations

common fraction:
mixed numeral:
repeating decimal using a vinculum: $2.\overline 6$
repeating decimal using parentheses: $2.(6)$
continued fraction using traditional typography: $2 + \tfrac 1 {1 + \tfrac 1 2}$
continued fraction in abbreviated notation: $2;$
Egyptian fraction: $2 + \tfrac 1 2 + \tfrac 1 6$
prime power decomposition: $2^3 \times 3^{-1}$
quote notation: $3'6$

are different ways to represent the same rational value.

Formal construction

The rational numbers may be built as equivalence classes of of .

More precisely, let be the set of the pairs of integers such . An equivalence relation is defined on this set by

(m_1, n_1) \sim (m_2, n_2) \iff m_1 n_2 = m_2 n_1.

Addition and multiplication can be defined by the following rules:

(m_1, n_1) + (m_2, n_2) \equiv (m_1n_2 + n_1m_2, n_1n_2),

(m_1, n_1) \times (m_2, n_2) \equiv (m_1m_2, n_1n_2).

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.)

The equivalence class of a pair is denoted Two pairs and belong to the same equivalence class (that is are equivalent) if and only if

m_1n_2 = m_2n_1.

This means that

\frac{m_1}{n_1} = \frac{m_2}{n_2}

if and only if

m_1n_2 = m_2n_1.

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

\cdots  = \frac{-2m}{-2n} = \frac{-m}{-n} = \frac{m}{n} = \frac{2m}{2n} = \cdots.

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

\frac{m_1}{n_1} \le \frac{m_2}{n_2}

If

\begin{align}

& (n_1n_2 > 0 \quad \text{and} \quad m_1n_2 \le n_1m_2) \\ & \qquad \text{or} \\ & (n_1n_2 < 0 \quad \text{and} \quad m_1n_2 \ge n_1m_2). \end{align}

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. (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.Nihon Sūgakkai (1993). 9780262590204, MIT Press. . ISBN 9780262590204 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 isomorphism to

is the field of fractions of the [[integer]]s  The algebraic closure of  i.e. the field of roots of rational polynomials, is the field of [[algebraic number]]s.

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

\frac{a}{b} < \frac{c}{d}

(where

b,d

are positive), we have

\frac{a}{b} < \frac{a + c}{b + d} < \frac{c}{d}.

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.

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 (in red on the figure), 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 (positive and negative).

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.

(2017). 9780134689494, Pearson. ISBN 9780134689494

Real numbers and topological properties

The rationals are a dense set 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 set expansions as regular continued fractions.

(2025). 9781498717526, CRC Press. . ISBN 9781498717526

Extract of page 1

In the usual topology of the real numbers, the rationals are neither an open set nor a closed set.

(2025). 9781441987327, Springer Science & Business Media. . ISBN 9781441987327

Extract of page 26

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 $d(x,y)=|x-y|,$ 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 . 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 $d(x,y)=|x-y|$ above.

p-adic numbers

In addition to the absolute value metric mentioned above, there are other metrics which turn into a topological field:

Let be a prime number and for any non-zero integer , let $|a|_p = p^{-n},$ where is the highest power of divisor .

In addition set $|0|_p = 0.$ For any rational number we set

\left|\frac{a}{b}\right|_p = \frac{|a|_p}{|b|_p}.

Then

d_p(x,y) =|x-y|_p

defines a metric space on

The metric space is not complete, and its completion is the p-adic number Ostrowski'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 number absolute value.

Rational number

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