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

*This was last updated in*September 2005

*Posted by:*Margaret Rouse

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