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,

**is a**

*Q**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

*= {1, 2, 3, ...}..*

**N**For the set of natural numbers * N* and the set of
integers

*, neither of which are "dense," denumeration lists are straightforward.For*

**Z***, it is less obvious how such a list might be constructed.An example appears below.The matrix includes all possible numbers of the form*

**Q***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

*, 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*

**N***has exactly the same number of elements as the set*

**Q***.Some people find this hard to believe, but the logic is sound.*

**N**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!

*This was last updated in*September 2005

*Posted by:*Margaret Rouse

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