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!