IntegerNumbers

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 EndoMorphisms, 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.