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**Rational numbers** are numbers that can be expressed as the ratio of two

integers. If

and

are both integers, then their ratio, denoted as

or

, is a rational number. For example, the fraction

and the integer

are both rational numbers.

on the other hand is not a rational number.

Rational numbers, being algebraic numbers of degree 1, are the roots of a nonconstant linear equation with integer coefficients

- ${\begin{array}{l}\displaystyle {a_{1}x+a_{0}=0,}\end{array}}$

where

The rational numbers, designated by

, are numbers which can be expressed, in

reduced form, as the

ratio of two

coprime integers, or more specifically as the

division of an

integer called the

numerator by a

positive integer called the

denominator. Given a

fraction, as

ratio, we can use the

Euclidean algorithm to obtain the

GCD and find whether the two numbers are coprime, and make them coprime otherwise.

## Rational integers

Rational integers (algebraic integers of degree 1) are the zeros of the monic linear polynomial with integer coefficients

- ${\begin{array}{l}\displaystyle {x+a_{0}{\!\,\!},}\end{array}}$

where

. They are the ordinary

integers (i.e. members of

).

## Base *b* expansions of rational numbers

The base

expansions of rational numbers are eventually periodic, for example (see

approximations)

- ${\begin{array}{l}\displaystyle {{\frac {22}{7}}=3+{\frac {1}{7}}=3+{\frac {142857}{999999}}=3+{\frac {142857}{1000000}}\left({\frac {1}{1-{\frac {1}{1000000}}}}\right)=3+142857\sum _{n=1}^{\infty }1000000^{-n}=3.142857142857142857142857142857\ldots ,}\end{array}}$

where we used the geometric series sum formula

- ${\begin{array}{l}\displaystyle {{\frac {1}{1-r}}=\sum _{n=0}^{\infty }r^{n},\quad \vert r\vert <1.}\end{array}}$

Long division gives the above decimal expansion, although without explicit emphasis on the geometric series involved in the decimal expansion.

Conversely, any number

with an eventually periodic representation, where

is the pre-periodic prefix and

is the periodic pattern, is rational.
For example, in base 10 (the same principle works in any fixed base

):

- ${\begin{array}{l}\displaystyle {90n=100n-10n={\mbox{abc.ccccc...}}-{\mbox{ab.cccccc...}}={\mbox{abc}}-{\mbox{ab}},}\end{array}}$

thus

is the following rational number

- ${\begin{array}{l}\displaystyle {n={\frac {{\mbox{abc}}-{\mbox{ab}}}{90}}.}\end{array}}$

### Dual representations and standard form

Any rational number whose

denominator is not coprime with the fixed base

used for the representation has two representations, due to the fact that

1 = 1.00000000… = 0.9999999999… |

in base 10 (or the equivalent in any base

). Considering

implies

- ${\begin{array}{l}\displaystyle {9n=10n-n=9.99999999\ldots -0.99999999\ldots =9,}\end{array}}$

thus

- ${\begin{array}{l}\displaystyle {n=1.}\end{array}}$

The

*standard form* base

expansions of rational numbers requires to keep only the repeating zeros representation (and to throw away the repeating nines representation).

## Base *b* expansions of irrational numbers

The expansions of irrational numbers are never periodic in any base.

, approximately

3.1415926535897932384626433832795… |

, is not a rational number and hence is

irrational. But there are a plethora of rational

approximations, and the one and only optimal

approximation, the

convergents (the partial

continued fractions for ).

## Continued fractions for rational numbers

All the continued fractions for rational numbers are finite (see Category:Continued fractions for rational numbers).

## Graded orderings of rational numbers

The rational numbers (in

reduced form)

may be sorted with a

graded ordering, where we first order by increasing sum

of

absolute values of

numerator and

denominator for all reduced form rational numbers, i.e. with

gcd(numerator, denominator) = 1 |

(first grading of the ordering), then by increasing absolute values of

numerators corresponding to that grade. This is the

Cantor ordering of rational numbers, giving a

one-to-one and onto mapping from the

natural numbers to the rational numbers, thus showing that the rational numbers are

countably infinite.

## Schinzel's conjecture

Assuming the Schinzel-Sierpinski conjecture, every positive rational number can be represented in an infinite number of ways as

- ${\begin{array}{l}\displaystyle {{\frac {a}{b}}={\frac {p+1}{q+1}}}\end{array}}$

and

- ${\begin{array}{l}\displaystyle {{\frac {a}{b}}={\frac {p-1}{q-1}},}\end{array}}$

with

and

prime.

## Rational numbers among the algebraic numbers

- Rational numbers: algebraic numbers of degree one (rational integers: algebraic integers of degree one)
- Quadratic numbers: algebraic numbers of degree two (quadratic integers: algebraic integers of degree two)
- Cubic numbers: algebraic numbers of degree three (cubic integers: algebraic integers of degree three)
- Quartic numbers: algebraic numbers of degree four (quartic integers: algebraic integers of degree four)
- Quintic numbers: algebraic numbers of degree five (quintic integers: algebraic integers of degree five)
- ...

## See also

## Notes