Integer

One of the numbers ..., , , 0, 1, 2, .... The set of integers forms a ring that is denoted . A given integer  may be negative (), nonnegative (), zero (), or positive (). The set of integers is, not surprisingly, called Integers in the Wolfram Language, and a number  can be tested to see if it is a member of the integers using the command Element[x, Integers]. The command IntegerQ[x] returns True if  has function head Integer in the Wolfram Language.

Numbers that are integers are sometimes described as "integral" (instead of integer-valued), but this practice may lead to unnecessary confusions with the integrals of integral calculus.

The ring  of integers has cardinal number of aleph0. The generating function for the nonnegative integers is

There are several symbols used to perform operations having to do with conversion between real numbers and integers. The symbol  ("floor ") means "the largest integer not greater than ," i.e., int(x) in computer parlance. The symbol  means "the nearest integer to " (nearest integer function), i.e., nint(x) in computer parlance. The symbol  ("ceiling ") means "the smallest integer not smaller than ," or -int(-x), where int(x) is the integer part of .

The German mathematician and logician Kronecker vociferously opposed the work of Georg Cantor on infinite sets and summarized his view that arithmetic and analysis should be based on whole numbers only by saying, "God made the natural numbers; all else is the work of man" (Bell 1986, p. 477).

REFERENCES:

Bell, E. T. Men of Mathematics. New York: Simon and Schuster, 1986.

Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1168, 2002.