 - A natural number is a number that occurs commonly and obviously in
nature. As such, it is a whole, non-negative number. The set of natural
numbers, denoted N, can be defined in either of two ways:
N = {0, 1, 2, 3, ...}
N = (1, 2, 3, 4, ...}
In mathematical equations, unknown or unspecified natural numbers are represented by
lowercase, italicized letters from the middle of the alphabet. The most common is n,
followed by m, p, and q. In subscripts, the lowercase i
is sometimes used to represent a non-specific natural number when denoting the elements in
a sequence or series. However, i is more often used to represent the
positive square root of -1, the unit imaginary
number.
The set N, whether or not it includes zero, is a denumerable
set. Denumerability refers to the fact that, even though there might be an infinite
number of elements in a set, those elements can be denoted by a list that implies the
identity of every element in the set. For example, it is intuitive from either the
list {1, 2, 3, 4, ...} or the list {0, 1, 2, 3, ...} that 356,804,251 is a natural number,
but 356,804,251.5, 2/3, and -23 are not.
Both of the sets of natural numbers defined above are denumerable. They are also
exactly the same size. It's not difficult to prove this; their elements can be
paired off one-to-one, with no elements being left out of either set. In infinite
sets, the existence of a one-to-one correspondence is the litmus test for determining cardinality,
or size. The set of integers and the set
of rational numbers has the same
cardinality as N. However, the sets of real numbers, imaginary numbers, and complex numbers have cardinality larger
than that of N.
See also: integer, rational number, real number, imaginary number, complex number
| LAST UPDATED: |
31 Jan 2001
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