 - A rational number is a number determined by the ratio of some integer p to some nonzero natural number q.
The set of rational numbers is denoted Q, and
represents the set of all possible integer-to-natural-number ratios
p/q.In mathematical expressions, unknown or unspecified
rational numbers are represented by lowercase, italicized letters from the late
middle or end of the alphabet, especially r, s, and
t, and occasionally u through z. Rational
numbers are primarily of interest to theoreticians.Theoretical mathematics has
potentially far-reaching applications in communications and computer science, especially
in data encryption and security.
If r and t are rational numbers such that r <
t, then there exists a rational number s such that r
< s < t. This is true no matter how small the
difference between r and t, as long as the two are not
equal.In this sense, the set Q is
"dense."Nevertheless, Q is a
denumerable set.Denumerability refers to the fact that, even
though a set might contain an infinite number of elements, and even though those
elements might be "densely packed," the elements can be defined by a list that
assigns them each a unique number in a sequence corresponding to the set of
natural numbers N = {1, 2, 3, ...}..
For the set of natural numbers N and the set of
integers Z, neither of which are "dense," denumeration
lists are straightforward.For Q, it is less
obvious how such a list might be constructed.An example appears
below.The matrix includes all possible numbers of the form
p/q, where p is an integer and q is a
nonzero natural number.Every possible rational number is represented in
the array.Following the pink line, think of 0 as the "first stop," 1/1 as
the "second stop," -1/1 as the "third stop," 1/2 as the "fourth stop," and so
on.This defines a sequential (although redundant) list of the rational
numbers.There is a one-to-one correspondence between the elements of the
array and the set of natural numbers N.
To demonstrate a true one-to-one correspondence between
Q and N, a modification must
be added to the algorithm shown in the illustration.Some of the elements
in the matrix are repetitions of previous numerical values.For example,
2/4 = 3/6 = 4/8 = 5/10, and so on.These redundancies can be eliminated by
imposing the constraint, "If a number represents a value previously encountered,
skip over it."In this manner, it can be rigorously proven that the set
Q has exactly the same number of elements as the set
N.Some people find this hard to believe, but
the logic is sound.
In contrast to the natural numbers, integers, and rational numbers, the sets
of irrational numbers, real numbers, imaginary numbers, and complex numbers are
non-denumerable. They have cardinality greater than that of the
set N.This leads to the conclusion that some
"infinities" are larger than others!
| LAST UPDATED: |
03 Dec 2000
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