³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980240057³2ip³2129.97.182.28³2data³2newauthor³31³3minor³30³3text³3The integers consist of the numbers {..., -3, -2, -1, 0, 1, 2, 3, ...}

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.
³3summary³3³2tscreate³2980240057³2keepts³2980245146³2id³21076³2host³2cnts2p28.uwaterloo.ca³2revision³21³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980245146³2ip³2129.97.130.2³2data³2text³3The 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''.

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 ModularArithmetic''''s 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.
³3minor³30³3newauthor³30³3summary³3Added notes on ordering³2tscreate³2980240057³2keepts³2980245189³2id³21076³2host³2cnts3p02.uwaterloo.ca³2revision³22³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980245189³2ip³2129.97.130.2³2data³2newauthor³30³3minor³31³3text³3The 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''.

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 ModularArithmetic''''s 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.
³3summary³3³2tscreate³2980240057³2keepts³2980245280³2id³21076³2host³2cnts3p02.uwaterloo.ca³2revision³23³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980245280³2ip³2129.97.130.2³2data³2text³3The 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 ModularArithmetic''''s 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.
³3minor³31³3newauthor³30³3summary³3³2tscreate³2980240057³2keepts³2980272019³2id³21076³2host³2cnts3p02.uwaterloo.ca³2revision³24³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980272019³2ip³2129.97.112.43³2data³2newauthor³30³3minor³31³3text³3The 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 ModularArithmetic''''s 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.
³3summary³3³2tscreate³2980240057³2keepts³2980640227³2id³21076³2host³2cnts1p43.uwaterloo.ca³2revision³25³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980640227³2ip³2129.97.182.2³2data³2text³3The 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 ModularArithmetic''''s 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.
³3minor³31³3newauthor³30³3summary³3*³2tscreate³2980240057³2keepts³2980640241³2id³21076³2host³2cnts2p02.uwaterloo.ca³2revision³26³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980640241³2ip³2129.97.182.2³2data³2newauthor³30³3minor³31³3text³3The 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 ModularArithmetic''''s 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.
³3summary³3*³2tscreate³2980240057³2keepts³2980722099³2id³21076³2host³2cnts2p02.uwaterloo.ca³2revision³27