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.