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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 Mathematica, 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 the Mathematica
data type of an integer.
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]](/web/20140329004118im_/http://mathworld.wolfram.com/images/equations/Integer/Inline15.gif)
means "the
nearest integer to 
" (nearest
integer function), i.e., nint(x) in computer parlance. The symbol ![[x]](/web/20140329004118im_/http://mathworld.wolfram.com/images/equations/Integer/Inline17.gif)
("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).
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