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
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, 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 N, 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:
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.