The integers are the unique smallest TotalOrderedSet that has neither an upper nor a lower bound. There are a countable number of them, and each has both a predecessor and successor. As a set they are usually denoted Z, for *Zahlen* (Ger. "number").

The integers turn out to be order isomorphic to the set of their own order automorphisms, allowing us to take their compositions to form a group operation (addition), and in turn to the set of its own group EndoMorphism*s*, allowing us to take *their* compositions to define a ring multiplication. Thus Z has a natural RinG structure, unique up to choice of zero and whether 0<1 or 1<0 (former by convention).

Labelled as such, Z = {... < -2 < -1 < 0 < 1 < 2 < ...}. This is the unique infinite cyclic group / RinG, and more over the finite ones are precisely the homomorphic images of Z, i.e. the ModularArithmetics Z/(p*Z) for some p in Z.

The integers are the archetypical example of an IntegralDomain?. The units are 1 and -1, and every nonzero nonunit can be expressed uniquely in terms of a product of primes. There are an infinite number of PrimeNumbers? in Z and their distribution is pretty irregular. The unique smallest field containing a copy of Z is the RationalNumbers.