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 ![[x]](/images/equations/Integer/Inline15.svg)
means "the nearest integer to 
" (nearest integer
function), i.e., nint(x) in computer parlance. The symbol ![[x]](/images/equations/Integer/Inline17.svg)
("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).
