Rational Numbers Definition

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Rational Numbers Definition
Rational Numbers Definition
n mathematics, a rational number is any number that can be expressed as the
quotient or fraction a/b of two integers, with the denominator b not equal to zero.
Since b may be equal to 1, every integer is a rational number. The set of all rational
numbers is usually denoted by a boldface Q (or blackboard bold , Unicode )
, which
stands for quotient.
The decimal expansion of a rational number always either terminates after a finite number
of digits or begins to repeat the same finite sequence of digits over and over. Moreover,
any repeating or terminating decimal represents a rational number.
These statements hold true not just for base 10, but also for binary, hexadecimal, or any
other integer base. A real number that is not rational is cal ed irrational. Irrational numbers
include 2, , and e. The decimal expansion of an irrational number continues forever
without repeating. Since the set of rational numbers is countable, and the set of real
numbers is uncountable, almost al real numbers are irrational.

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The rational numbers can be formal y defined as the equivalence classes of the quotient
set (Z x (Z
{0})) / ~, where the cartesian product Z x (Z
{0}) is the set of all ordered
pairs (m,n) where m and n are integers, n is not zero (n 0), and "~" is the equivalence
relation defined by (m1,n1) ~ (m2,n2) if, and only if, m1n2 - m2n1 = 0.
In abstract algebra, the rational numbers together with certain operations of addition and
multiplication form a field. This is the archetypical field of characteristic zero, and is the
field of fractions for the ring of integers.
Finite extensions of Q are cal ed algebraic number fields, and the algebraic closure of Q is
the field of algebraic numbers.
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.
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.
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) = |x - y|, 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 local y compact.

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The rationals are characterized topological y as the unique countable metrizable space
without isolated points. The space is also total y disconnected. The rational numbers do
not form a complete metric space; the real numbers are the completion of Q under the
metric d(x,y) = |x - y|, above..
Rational Numbers Properties
The set Q, together with the addition and multiplication operations shown above, forms a
field, the field of fractions of the integers Z.
The rationals are the smallest field with characteristic zero: every other field of
characteristic zero contains a copy of Q. The rational numbers are therefore the prime
field for characteristic zero.
The algebraic closure of Q, i.e. the field of roots of rational polynomials, is the algebraic
numbers. The set of al rational numbers is countable. Since the set of all real numbers is
uncountable, we say that almost al real numbers are irrational, in the sense of Lebesgue
measure, i.e. the set of rational numbers is a null set.
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

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