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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
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
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)
where we used the geometric series sum formula
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
):
thus
is the following rational number
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
thus
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 [[pi approximations|
approximations]], and the one and only optimal
approximation, the [[pi convergents|
convergents]] (the partial [[Continued fractions for pi|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
and
-
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