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The integers consist of the numbers {..., -3, -2, -1, 0, 1, 2, 3, ...} |
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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"). |
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The integers form a MathematicalGroup? under addition. The group of its self-homomorphisms turns out to be isomorphic to the integers themselves, so their composition can be used to define multiplication thereupon and turn the integers into a ring. |
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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. |
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.
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