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# natural number

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.

This was last updated in September 2005

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