Integer

An integer is any whole number, positive, negative, or 0. Starting at 1 and going up are the counting numbers {1, 2, 3, 4, ...}, sometimes called "natural numbers".

More precisely, the set of all integers consists of all natural numbers {1, 2, 3, 4, ...}, their negatives {-1, -2, -3, -4, ...} and 0. A formal definition is that it is the only integral domain whose positive elements are well ordered and in which order is preserved by addition.

An integer may be even (divisible by two) or odd (not divisible by two), positive (more than zero) or negative (less than zero), whole (undivided) or composite (divisible into other integers), and various other classifications, such as prime (only divisible by itself and one).

Every integer larger than 1 has a unique prime factorization.

Some examples of integers: 1, 10/5, 98058493, -87, -3/3, both square roots of 9, and 0.

Likewise, the following numbers are not integers: 5/10, the square root of -9, 8.75, and pi.

See also: algebraic numbers

Generalizations

The set of integers form what is called in abstract algebra a ring. A ring is a set equipped with operations + and x with the usual properties learned in high-school algebra (commutativity, distributivity, linearity, and associativity), identities for both operations (additive and multiplicative), and inversion (subtraction). Other objects such as matrixes, polynomials, quaternions, and algebraic integers also form rings, and can therefore be viewed as generalizations of the integers.