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Rational numbers

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Rational numbers are numbers that can be expressed as the ratio of two integers. If
 a
and
 b, b   ≠   0,
are both integers, then their ratio, denoted as
 a  ⁄  b
or
 a b
, is a rational number. For example, the fraction
−
 125 37
and the integer
 74
are both rational numbers.
 π
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
 a1, a0 ∈ ℤ, a1   ≥   1.
The rational numbers, designated by
 ℚ
, 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
 numerator denominator
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
 a0 ∈ ℤ
. They are the ordinary integers (i.e. members of
 ℤ
).

Base b expansions of rational numbers

The base
 b
expansions of rational numbers are eventually periodic, for example (see
 π
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
 n = a.bcccccc…
with an eventually periodic representation, where
 a.b
is the pre-periodic prefix and
 c
is the periodic pattern, is rational. For example, in base 10 (the same principle works in any fixed base
 b
):
${\begin{array}{l}\displaystyle {90n=100n-10n={\mbox{abc.ccccc...}}-{\mbox{ab.cccccc...}}={\mbox{abc}}-{\mbox{ab}},}\end{array}}$ thus
 n
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
 b
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
 b
). Considering
 n = 0.9999999…
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
 b
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.

 π
, approximately
 3.1415926535897932384626433832795…
, is not a rational number and hence is irrational. But there are a plethora of rational
 π
approximations
, and the one and only optimal
 π
approximation, the
 π
convergents
(the partial continued fractions for
 π
).

Continued fractions for rational numbers

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

The rational numbers (in reduced form)
 a b
, a ∈ ℤ , b ∈ ℤ+ ,
may be sorted with a graded ordering, where we first order by increasing sum
 | a | + | b |
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
 | a |
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
 p
and
 q
prime.

Rational numbers among the algebraic numbers

1. Rational numbers: algebraic numbers of degree one (rational integers: algebraic integers of degree one)
2. Quadratic numbers: algebraic numbers of degree two (quadratic integers: algebraic integers of degree two)
3. Cubic numbers: algebraic numbers of degree three (cubic integers: algebraic integers of degree three)
4. Quartic numbers: algebraic numbers of degree four (quartic integers: algebraic integers of degree four)
5. Quintic numbers: algebraic numbers of degree five (quintic integers: algebraic integers of degree five)
6. ...