Rational number

Arithmetic

Irreducible fraction

Every rational number may be expressed in a unique way as an irreducible fraction ⁠${\displaystyle \frac{a}{b},}$⁠ 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 ⁠${\displaystyle \frac{a}{b},}$⁠ its canonical form may be obtained by dividing both 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 ⁠${\displaystyle \frac{n}{1},}$⁠ which is its canonical form as a rational number.

Equality

${\displaystyle \frac{a}{b}=\frac{c}{d}}$ if and only if ${\displaystyle ad=bc}$

If both fractions are in canonical form, then:

${\displaystyle \frac{a}{b}=\frac{c}{d}}$ if and only if ${\displaystyle a=c}$ and ${\displaystyle b=d}$

Ordering

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

${\displaystyle \frac{a}{b}<\frac{c}{d}}$ if and only if ${\displaystyle ad<bc.}$

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:

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

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

Subtraction

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

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

Multiplication

The rule for multiplication is:

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

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

Inverse

Every rational number ⁠${\displaystyle \frac{a}{b}}$⁠ has an additive inverse, often called its opposite,

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

If ⁠${\displaystyle \frac{a}{b}}$⁠ is in canonical form, the same is true for its opposite.

A nonzero rational number ⁠${\displaystyle \frac{a}{b}}$⁠ has a multiplicative inverse, also called its reciprocal,

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

If ⁠${\displaystyle \frac{a}{b}}$⁠ is in canonical form, then the canonical form of its reciprocal is either ⁠${\displaystyle \frac{b}{a}}$⁠ or ⁠${\displaystyle \frac{-b}{-a},}$⁠ depending on the sign of a.

Division

If b, c, d are nonzero, the division rule is

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

Thus, dividing ⁠${\displaystyle \frac{a}{b}}$⁠ by ⁠${\displaystyle \frac{c}{d}}$⁠ is equivalent to multiplying ⁠${\displaystyle \frac{a}{b}}$⁠ by the reciprocal of ⁠${\displaystyle \frac{c}{d}:}$⁠

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

Exponentiation to integer power

If n is a non-negative integer, then

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

The result is in canonical form if the same is true for ⁠${\displaystyle \frac{a}{b}.}$⁠ In particular,

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

If a ≠ 0, then

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

If ⁠${\displaystyle \frac{a}{b}}$⁠ is in canonical form, the canonical form of the result is ⁠${\displaystyle \frac{b^n}{a^n}}$⁠ if a > 0 or n is even. Otherwise, the canonical form of the result is ⁠${\displaystyle \frac{-b^n}{-a^n}.}$⁠

Continued fraction representation

A finite continued fraction is an expression such as

${\displaystyle a_0+\frac{1}{a_1+\frac{1}{a_2+\frac{1}{\ddots +\frac{1}{a_n}}}},}$

where a_n are integers. Every rational number ⁠${\displaystyle \frac{a}{b}}$⁠ can be represented as a finite continued fraction, whose coefficients a_n can be determined by applying the Euclidean algorithm to (a, b).

Other representations

• common fraction: ⁠${\displaystyle \frac{8}{3}}$⁠
• mixed numeral: ⁠${\displaystyle 2\frac{2}{3}}$⁠
• repeating decimal using a vinculum: ${\displaystyle 2.\overline{6}}$
• repeating decimal using parentheses: ${\displaystyle 2.(6)}$
• continued fraction using traditional typography: ${\displaystyle 2+\frac{1}{1+\frac{1}{2}}}$
• continued fraction in abbreviated notation: ${\displaystyle [2;1,2]}$
• Egyptian fraction: ${\displaystyle 2+\frac{1}{2}+\frac{1}{6}}$
• prime power decomposition: ${\displaystyle 2^3\times 3^{-1}}$
• quote notation: ${\displaystyle 3^{\prime }6}$

are different ways to represent the same rational value.

Formal construction

The rational numbers may be built as equivalence classes of ordered pairs of integers.

More precisely, let ⁠${\displaystyle (\mathbb{Z}\times (\mathbb{Z}\setminus \{0\}))}$⁠ be the set of the pairs (m, n) of integers such n ≠ 0. An equivalence relation is defined on this set by

${\displaystyle (m_1,n_1)\sim (m_2,n_2)⟺m_1{n}_2=m_2{n}_1.}$

Addition and multiplication can be defined by the following rules:

${\displaystyle (m_1,n_1)+(m_2,n_2)\equiv (m_1{n}_2+n_1{m}_2,n_1{n}_2),}$

${\displaystyle (m_1,n_1)\times (m_2,n_2)\equiv (m_1{m}_2,n_1{n}_2).}$

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 ⁠${\displaystyle \mathbb{Q}}$⁠ is the defined as the quotient set by this equivalence relation, ⁠${\displaystyle (\mathbb{Z}\times (\mathbb{Z}\mathrm{\setminus }\{0\}))/\sim ,}$⁠ 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 (m, n) is denoted ⁠${\displaystyle \frac{m}{n}.}$⁠ Two pairs (m₁, n₁) and (m₂, n₂) belong to the same equivalence class (that is are equivalent) if and only if

${\displaystyle m_1{n}_2=m_2{n}_1.}$

This means that

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

if and only if

${\displaystyle m_1{n}_2=m_2{n}_1.}$

Every equivalence class ⁠${\displaystyle \frac{m}{n}}$⁠ may be represented by infinitely many pairs, since

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

Each equivalence class contains a unique canonical representative element. The canonical representative is the unique pair (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 ⁠${\displaystyle \frac{n}{1}.}$⁠

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

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

If

${\displaystyle \begin{array}{cc}& (n_1{n}_2>0\text{and}{m}_1{n}_2\le n_1{m}_2)\\ & \text{or}\\ & (n_1{n}_2<0\text{and}{m}_1{n}_2\ge n_1{m}_2).\end{array}}$

Properties

The set ⁠${\displaystyle \mathbb{Q}}$⁠ of all rational numbers, together with the addition and multiplication operations shown above, forms a field.

⁠${\displaystyle \mathbb{Q}}$⁠ 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.)

⁠${\displaystyle \mathbb{Q}}$⁠ 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 ⁠${\displaystyle \mathbb{Q}.}$⁠

With the order defined above, ⁠${\displaystyle \mathbb{Q}}$⁠ 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 ⁠${\displaystyle \mathbb{Q}.}$⁠

⁠${\displaystyle \mathbb{Q}}$⁠ is the field of fractions of the integers ⁠${\displaystyle \mathbb{Z}.}$⁠ The algebraic closure of ⁠${\displaystyle \mathbb{Q},}$⁠ i.e. the field of roots of rational polynomials, is the field of algebraic numbers.

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

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

(where ${\displaystyle b,d}$ are positive), we have

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

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.

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.

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

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 ${\displaystyle d(x,y)=|x-y|,}$ and this yields a third topology on ⁠${\displaystyle \mathbb{Q}.}$⁠ 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, and the real numbers are the completion of ⁠${\displaystyle \mathbb{Q}}$⁠ under the metric ${\displaystyle d(x,y)=|x-y|}$ above.

p-adic numbers

In addition to the absolute value metric mentioned above, there are other metrics which turn ⁠${\displaystyle \mathbb{Q}}$⁠ into a topological field:

Let p be a prime number and for any nonzero integer a, let ${\displaystyle |a{|}_p=p^{-n},}$ where pⁿ is the highest power of p dividing a.

In addition set ${\displaystyle |0{|}_p=0.}$ For any rational number ⁠${\displaystyle \frac{a}{b},}$⁠ we set

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

Then

${\displaystyle d_p(x,y)=|x-y{|}_p}$

defines a metric on ⁠${\displaystyle \mathbb{Q}.}$⁠

The metric space ⁠${\displaystyle (\mathbb{Q},d_p)}$⁠ is not complete, and its completion is the p-adic number field ⁠${\displaystyle {\mathbb{Q}}_p.}$⁠ Ostrowski's theorem states that any non-trivial absolute value on the rational numbers ⁠${\displaystyle \mathbb{Q}}$⁠ is equivalent to either the usual real absolute value or a p-adic absolute value.