An integer (pronounced IN-tuh-jer) is a whole number (not a fractional number) that can be positive, negative, or zero.

Examples of integers are: -5, 1, 5, 8, 97, and 3,043.

Examples of numbers that are not integers are: -1.43, 1 3/4, 3.14, .09, and 5,643.1.

The set of integers, denoted ** Z**, is formally defined as
follows:

* Z* = {..., -3, -2, -1, 0, 1, 2, 3, ...}

In mathematical equations, unknown or unspecified integers are represented by
lowercase, italicized letters from the "late middle" of the alphabet. The
most common are *p*, *q*, *r*, and *s*.

The set ** Z** 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 the list {..., -3, -2, -1, 0, 1, 2, 3, ...} that 356,804,251 and -67,332 are integers, but 356,804,251.5, -67,332.89, -4/3, and 0.232323 ... are not.

The elements of ** Z** can be paired off one-to-one with the
elements of

*, the set of natural numbers, with no elements being left out of either set. Let*

**N***= {1, 2, 3, ...}. Then the pairing can proceed in this way:*

**N**In infinite sets, the existence of a one-to-one correspondence is the litmus test for
determining cardinality, or size. The set of natural numbers and the set of
rational numbers> have the same
cardinality as * Z*. However, the sets of real numbers, imaginary numbers, and complex numbers have cardinality larger
than that of

*.*

**Z***This was last updated in*September 2005

*Posted by:*Margaret Rouse

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