Real Numbers:Rational

The rational numbers (

\mathbb {Q}

) are included in the real numbers (

\mathbb {R}

), while themselves including the integers (

\mathbb {Z}

), which in turn include the natural numbers (

\mathbb {N}

)

In mathematics, a rational number is a number that can be expressed as the quotient or fraction $Template:Sfrac$ of two integers, a numerator $p$ and a non-zero denominator $q$ .^[1] For example, $Template:Sfrac$ is a rational number, as is every integer (e.g. $5 Template:= Template:Sfrac$ ). 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 a boldface $Q$ (or blackboard bold $\mathbb {Q}$ , Unicode Template:Unichar or Template:Unichar);^[4] it was thus denoted in 1895 by Giuseppe Peano after quoziente, Italian for "quotient",Template:Citation needed and first appeared in Bourbaki's Algèbre.^[5]

The decimal expansion of a rational number either terminates after a finite number of digits (example: $Template:Sfrac Template:= 0.75$ ), or eventually begins to repeat the same finite sequence of digits over and over (example: $Template:Sfrac Template:= 0.20454545...$ ).^[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).Template:Citation needed

A real number that is not rational is called irrational.^[5] Irrational numbers include $[[square root of 2| Template:Sqrt]]$ , [[Pi|Template:Pi]], $e$ , 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 $q \neq 0$ , using the equivalence relation defined as follows:

\left(p_{1},q_{1}\right)\sim \left(p_{2},q_{2}\right)\iff p_{1}q_{2}=p_{2}q_{1}.

The fraction $Template:Sfrac$ then denotes the equivalence class of $(p, q)$ .^[7]

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).Template:Citation needed

Arithmetic

Template:See also

Irreducible fraction

Every rational number may be expressed in a unique way as an irreducible fraction $Template:Sfrac$ , 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 $Template:Sfrac$ , 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.Template:Citation needed

Embedding of integers

Any integer $n$ can be expressed as the rational number $Template:Sfrac$ , which is its canonical form as a rational number.Template:Citation needed

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

^[7]

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.^[7]

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 $b$ and $d$ are coprime integers.^[7]^[9]

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 $b$ and $d$ are coprime integers.^[9]Template:Verify source

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.^[7]^[9]

Inverse

Every rational number $Template:Sfrac$ has an additive inverse, often called its opposite,

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

If $Template:Sfrac$ is in canonical form, the same is true for its opposite.

A nonzero rational number $Template:Sfrac$ has a multiplicative inverse, also called its reciprocal,

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

If $Template:Sfrac$ is in canonical form, then the canonical form of its reciprocal is either $Template:Sfrac$ or $Template:Sfrac$ , depending on the sign of $a$ .Template:Citation needed

Division

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

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

Thus, dividing $Template:Sfrac$ by $Template:Sfrac$ is equivalent to multiplying $Template:Sfrac$ by the reciprocal of $Template:Sfrac$ :

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

^[9]Template:Verify source

Exponentiation to integer power

If $n$ 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 $Template:Sfrac$ . In particular,

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

If $a \neq 0$ , then

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

If $Template:Sfrac$ is in canonical form, the canonical form of the result is $Template:Sfrac$ if $a > 0$ or $n$ is even. Otherwise, the canonical form of the result is $Template:Sfrac$ .Template:Citation needed

Formal construction

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]^[9]

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

\left(m_{1},n_{1}\right)\sim \left(m_{2},n_{2}\right)\iff m_{1}n_{2}=m_{2}n_{1}.

^[7]^[9]

Addition and multiplication can be defined by the following rules:

\left(m_{1},n_{1}\right)+\left(m_{2},n_{2}\right)\equiv \left(m_{1}n_{2}+n_{1}m_{2},n_{1}n_{2}\right),

\left(m_{1},n_{1}\right)\times \left(m_{2},n_{2}\right)\equiv \left(m_{1}m_{2},n_{1}n_{2}\right).

^[7]

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 $Q$ is the defined as the quotient set by this equivalence relation, $(Z × (Z \ {0})) / ~$ , 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.)^[7]

The equivalence class of a pair $(m, n)$ is denoted $Template:Sfrac$ . Two pairs $(m 1, n 1)$ and $(m 2, n 2)$ belong to the same equivalence class (that is are equivalent) if and only if $m 1 n 2 Template:= m 2 n 1$ . This means that $Template:Sfrac Template:= Template:Sfrac$ if and only $m 1 n 2 Template:= m 2 n 1$ .^[7]^[9]

Every equivalence class $Template:Sfrac$ 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 $(m, n)$ 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 $Template:Sfrac$ .

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}}}\leq {\frac {m_{2}}{n_{2}}}

if

(n_{1}n_{2}>0\quad {\text{and}}\quad m_{1}n_{2}\leq n_{1}m_{2})\qquad {\text{or}}\qquad (n_{1}n_{2}<0\quad {\text{and}}\quad m_{1}n_{2}\geq n_{1}m_{2}).

Properties

Illustration of the countability of the positive rationals

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

$Q$ has no field automorphism other than the identity.Template:Citation needed

With the order defined above, $Q$ is an ordered field^[9] 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 $Q$ .Template:Citation needed

$Q$ is a prime field, which is a field that has no subfield other than itself.^[10] The rationals are the smallest field with characteristic zero. Every field of characteristic zero contains a unique subfield isomorphic to $Q$ .Template:Citation needed

$Q$ is the field of fractions of the integers $Z$ .^[11] The algebraic closure of $Q$ , i.e. the field of roots of rational polynomials, is the field of algebraic numbers.Template: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.Template: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

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

(where $b,d$ are positive), we have

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

Template:Citation needed

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.^[12]

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.^[7] A related property is that rational numbers are the only numbers with finite expansions as regular continued fractions.Template: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 $d (x, y) Template:= Template:Abs$ , 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 Template:Citation needed; the real numbers are the completion of $Q$ under the metric $d (x, y) Template:= Template:Abs$ above.^[9]