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The Integers

I've been reviewing some basic number theory, so I tried to look up the defining properties of the integers in Eric Weisstein's World of Mathematics (MathWorld). Much to my surprise, I didn't find a definition in the sense that I was looking for. I wanted a list of defining properties. Feel free to look at the MathWorld entry for Integer.

The natural numbers are a natural consequence of the fact that we can count the members of a set in any order and we always get the same answer. Consider a set A that you want to count, and the set B which are the natural numbers from one to the size of A. No matter how you count A, you always use the elements of B once each. Using mathematical language: If you have two finite sets A and B, and there exists a one-to-one correspondence between A and B, then there is no one-to-one correspondence between A and any proper subset of B. Counting is a wonderful thing, and much of mathematics can be derived as a consequence of it.

The integers Z are just the generalization of the natural numbers: Z = {..., –3, –2, –1, 0, 1, 2, 3, ...}. So we certainly know what the integers are. But how do we define them?

Well, my search through MathWorld led me back to the definition of an Integral Domain. Any set S together with two binary operators + and · is an Integral Domain if it has the following properties for all a, b and c in S:

  • Ring Properties:
    • Addition is associative:  (a + b) + c  =  a + (b + c).
    • Addition is commutative:  a + b  =  b + a.
    • Additive identity:  There exists an element 0, such that 0 + a  =  a + 0  =  a.
    • Additive inverse: For every a there exists –a in S such that a + (–a)  =  (–a) + a  =  0.
    • Multiplication is associative: (a · b) · c  =  a · (b · c).
    • Distributivity: a · (b + c)  =  (a · b) + (a · c)   and   (b + c) · a  =  (b · a) + (c · a).
  • Multiplication is commutative: a · b  =  b · a.
  • Multiplicative identity: There exists an element 1, such that 1 · a  =  a · 1  =  a.
  • Zero has no divisors: If a · b  =  0,  then  a  = 0,  or  b  =  0,  or both.

Now, the integers are not the only integral domain. There are others, most notably congruence arithmetic modulo-p, where p is prime.

This month's question: What defining properties do the Integers have that distinguish them from other integral domains? This month's topic has been well-covered historically, and you can just look it up—but where's the fun in that? The fun is that there's more than one way to get the job done.

Send all responses to my email address is mathrec at this domain.