Integers
An integer is the number zero (0), a positive natural number (1, 2, 3, . . .), or the negation of a positive natural number (−1, −2, −3, . . .). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set of all integers is often denoted by the boldface Z or blackboard bold
Z
{\displaystyle \mathbb {Z} }
.
The set of natural numbers
N
{\displaystyle \mathbb {N} }
is a subset of
Z
{\displaystyle \mathbb {Z} }
, which in turn is a subset of the set of all rational numbers
Q
{\displaystyle \mathbb {Q} }
, itself a subset of the real numbers
R
{\displaystyle \mathbb {R} }
. Like the set of natural numbers, the set of integers
Z
{\displaystyle \mathbb {Z} }
is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5+1/2, 5/4 and √2 are not.
The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers. In fact, (rational) integers are algebraic integers that are also rational numbers.
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