Pure Math

The integers

Construction of the Integers

You may have heard of the integers before. Namely, it’s the set of numbers

    \[\dots, -2, -1, 0, 1, 2,\dots\]

stretching on infinitely. But what fundamental properties does this amazing set satisfy? How much can we derive based on these axioms? Are the theorems we see and take for granted really trivial?

The integers, commonly denoted as \Z, are governed by two operations: addition (+) andĀ multiplication (*). Specifically, we haveĀ additive and multiplicative closure, which means

    \[(a+b),\, (a*b)\in\Z\ \forall\, a,b\in \Z.\]

This means that if we take two integers a and b, their sum a+b and their product a*b are integers as well. We also quickly see that the integers form a ring, as for all a,b,c\in \Z we have the following properties:

    \begin{align*} \textbf{Commutativity:}\ & a+b = b+a \\ & a*b = b*a \\ \textbf{Associativity:}\ & (a+b)+c = a+(b+c) \\ & (a*b)*c = a*(b*c) \\ \textbf{Distributivity:}\ & a*(b+c) = (a*b)+(a*c) \\ \textbf{Zero:}\ & \text{There exists an element 0 such that}\ a+0=a \\ \textbf{One:}\ & \text{There exists an element 1 such that}\ a*1=a \\ \textbf{Negativity:}\ & \text{There exists an element}\ a^\prime\ \text{such that}\ a+a^\prime = 0 \end{align*}

With the integers defined, we now look at one of its special subsets: the positive integers. How do we define positivity? We cannot simply say an integer is positive when it’s greater than 0, as we have not defined the notion of order in the integers, and thus cannot say when integers are greater or less than each other.

Exercise: Is there a way to define “greater than” or “less than” with our above axioms?

Let S be a subset of the integers that is closed under addition and multiplication. Remember that this means if we have a,b\in S, then a+b and a*b must also be in S. But this is not enough to construct the positive integers; the integers is a subset of the integers satisfying additive and multiplicative closure. Thus, we stipulate further that the identity 0 cannot be in S, and that for any integer a, only one of a and -a lie in S (here, -a refers to the additive inverse guaranteed by Negativity).

Exercise: Does the positive integers satisfy these two properties?
Exercise: How do we know that 1, -2 are not both in S?

S can either be the positive integers or the negative integers. To simplify things, we call the positive integers \Z^+ and the negative integers \Z^-. Now we can define the symbols > and <. Let a,b,p be integers where a+p=b. We say a<b if p\in \Z^+, and a>b if p\in \Z^-.