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 can be made into an algebraic structure by defining addition and multiplication in a natural way:

The order AutoMorphisms? of Z turn out to be order IsoMorphic? to Z itself under the induced ordering. Thus we can take their composition to define a group operation (addition) on Z, unique up to choice of zero. Order AutoMorphisms? take the form f(x)=x+z for some z in Z.

The group EndoMorphisms? of Z again turn out to be group IsoMorphic? to Z itself under induced addition, so we can take their multiplication to define a ring multiplication thereupon. This is unique up to choice of ordering; by convention 0 < 1. Group EndoMorphisms? take the form f(x)=x*z=z*x for some z in Z.

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.