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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?

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.

Order

The missing properties are all related to order. The complex numbers, for example are an integral domain, but they do not have the order properties that we associate with Integers, Rational Numbers and Real Numbers. To be fair, we could define an order for Complex Numbers. For example, we could order any two complex numbers according to their real part, then according to their imaginary part, if the real parts are equal. More on that later.

The integers are a totally ordered set. This statement can be distilled down to two properties:

  • Transitivity: a < b and b < c implies that a < c
  • Trichotomy (strong version): For any two numbers a and b, exactly one of the following is true
    • a < b
    • a = b
    • a > b.

Of course, the same set of numbers can have different order properties for two different definitions of >. So, at some point, we have to define this relation.

Peano's Axioms

So, what are the order properties that are missing? Let's start by looking at the historical approach. The integers were traditionally defined through Peano's Axioms. There are a few different versions, all of which are equivalent. I'll use Eric Weisstein's statement of Peano's Axioms:

  1. Zero is a number
  2. If a is a number, the successor of a is a number
  3. zero is not the successor of a number
  4. Two numbers of which the successors are equal are themselves equal
  5. (induction axiom.) If a set S of numbers contains zero and also the successor of every number in S, then every number is in S

These axioms define the non-negative integers (whole numbers), rather than the full set of integers. We identify the successor of a as a + 1. For every integral domain, zero is a number, and a + 1 is a number. The third axiom eliminates all negative integers from the set, since there is some number of times that you can add one to a negative integer and reach zero. So this property imposes some order on the set, since it separates the positive integers from the negative integers. Let's look at the fourth axiom. It states that if a + 1 = b + 1, then a = b. This axiom is also true in any integral domain, if we identify the successor of a as a + 1. Finally, the last axiom indicates that the non-negative integers are the smallest set satisfying the other axioms.

So, does all of the order come from the statement that there is no a such that a + 1 = 0? Well, not quite. Let S be a subset of an integral domain, such that S satisfies axioms one through four. This set includes the positive integers (i.e., the successors of zero), but it can also contain all sorts of other numbers, since it can contain any number of infinite sequences. For example, if a is in the set, then π + n is also in the set, where n is a positive integer. a might not be the first member of the sequence. There might not even be a first member of the sequence if S contains a + n for any integer n. So the third axiom separates the negative integers from the positive integers, and the fifth axiom eliminates all sequences of nonintegers.

In addition to separating the positive integers from the negative integers, the third axiom eliminates all of the finite rings. For example, {0, 1, 2} form a small finite ring, where a + b is defined by "normal" arithmetic if a + b would be less than three, and is otherwise three less than the "normal" result. This is Conruence Arithmetic modulo 3. Any prime ring—that is, a ring obeying congruence arithmetic modulo p, where p is prime—is an integral domain, but (p – 1) + 1 = 0 in this ring. So prime rings do not satsify the third axiom.

Ulimately, Peano's axioms establish the non-negative integers by induction as the successors of zero. The ordering of the non-negative integers follows trivially from the way they are defined, since the successor relationship defines a total order.

Order Under Addition

Once we have established the non-negative integers, we can build rational numbers and real numbers, but the order properties need to be transferred. We define the comparison between two positive rational numbers m/n > p/q if and only if mq > np.

This is certainly not the only way to define a total order on the rational numbers. For example, we could define that m/n < p/q if m + n < p + q or if m + n = p + q and m < p, where the rational numbers m/n and p/q are in lowest terms (that is, m and n have no common divisors and p and q have no common divisors). This ordering is even useful, since it allows us to demonstrate that the rational numbers are countable—that is, they can be put into a one-to-one correspondence with the positive integers.

The normal order relation for rational numbers is chosen because it preserves order under addition. That is, if a < b, then a + c < b + c for all c. In fact, this is the only ordering which preserves the order of the integers within the rational numbers (that is, the positive rational numbers m/1 < n/1 if and only if the integers m < n) and preserves order under addition.

It is very interesting to consider what we have after we require an integral domain to be totally ordered under addition. There are a number of interesting systems with this property. The integers, real numbers and rational numbers are the most obvious. The total order that I defined for complex numbers has this property. So do polynomials with real coefficients (ordered according to the sign of the leading coefficient, then by degree, then by the magnitude of the leading coefficient.)

The integers are fundamentally different from the other integral domains totally ordered under addition, because they are the only one that consists of a single ordered sequence. Well, OK, it's not really a "sequence", since a sequence has a first element whereas the integers start at –∞, but the positive (or the non-negative) integers are a single ordered sequence. This is the final property. The formal way to state this is that the integers are the only integral domain whose positive elements are well ordered, and which has order preserved under addition. A set is said to be well ordered if and only if every subset has a least element.

In contrast, the real numbers have a different property. The real numbers are different from the other integral domains totally ordered under addition, because they are complete. That is, every bounded ordered sequence in the set of real numbers has a limit which is also in the set of real numbers.


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

Thanks,
Steve