Just what are numbers, really? (Part II)

In Part I we exposed the natural numbers as just sets. Now we’ll use them as a foundation to uncover the Integer Numbers.

This is the second article in my “Just what are numbers, really?” series, where we explore what fundamentally makes a number in the view of modern mathematics. No prior math knowledge is required! No jargon here either!

Photo by Jarosław Kwoczała on Unsplash

I highly recommend you read Part I, where we introduce sets, set unions and the natural numbers as sets, along with all their arithmetic. If that’s what brought you here, then come along and let’s discover the integer numbers!

Disclaimer: This article series was made using Mathoma’s excellent series on Set Theory as a guide. Their blog can be found here.

More Basic Tools: Equivalences and Pairs

An important tool we’ll use to build the integers are equivalence relations, which are (unsurprisingly) a binary criterion that tells us wether two objects are “equivalent” or not. For instance, if x and y are two animals, we can define them to be equivalent, shorthanded as x ~ y, if they have the same number of legs. If they do, the relation is True, if they don’t, the relation is False. That would mean

We can use an equivalence relation to partition a set according to which set elements are equivalent. For example, consider our previous leg number equivalence relation and the following set S:

If we take S and group all elements which satisfy x ~ y into their own sets, the set of these sets is called the quotient set, shorthanded as S / ~, in this case

Remember, the slash has nothing to do with numerical division, but we are dividing the set into distinct parts with equivalent elements. Sometimes we will also refer to all set elements equivalent to some x in the set, shorthanded as [x]. For example:

Another tool we need is the ability to pair up elements of different sets into a set of pairs. To illustrate this, say A is the set of mammals in S and B is the set of insects in S, i.e.

Imagine taking an animal from A and another from B and pairing them up in a set, such as { Cow, Ant } or { Dog, Bee }. Now make a set consisting of every such pair combination. Mathematicians refer to this operation as the Cartesian Product of A and B, denoted A x B, a fancy way of saying “the set of all pairing combinations”. In our case,

That’s all the tools we’ll need for now. Sound fair enough? Then let’s build the Integers!

The Integer Numbers

Here’s the key insight that will unfold into the integer numbers: Every integer number can be expressed as the difference of two natural numbers. Sounds simple enough right?

We can say 5=6–1, or -4=1–5, the only hurdle is that there are infinitely many ways of expressing an integer as the difference of naturals. Since an integer cannot be described by a single pair of naturals, what about a set of pairs? Here’s where equivalence relations come in.

Consider two pairs of naturals {a, b} and {c,d}. We can define them to be equivalent if the difference of the pair elements is the same, i.e. a-b=c-d. If we frame this relation instead as

then the only operation we need is natural number addition, which we’ve already defined in Part I using only set unions!

Now imagine taking every pair of Naturals and group them according to this equivalence relation, a.k.a consider the quotient set N x N / ~. Every set in this set has the unique property that its pairs have the same difference, with the difference corresponding to an integer! For example:

And so the hidden nature of the integers unfolds: Any integer number x is the set of all natural number pairs {a,b} where x=a-b. The Set of Integers is given by

Note how the natural number 2 is not the same as the integer number 2!

If you’re not yet convinced the Integers correspond to these sets, let’s see how they can fully express integer number arithmetic.

Integer Addition (and Subtraction)

Let’s consider two integers x and y, where x=a-b and y=c-d. We know from school math that x+y=a-b+c-d, which we can express as the difference

This is the difference of two naturals, which corresponds to the integer [ {a+c, b+d}]!

From this simple observation, we can define the arithmetic rule for integer addition: If x=[{a,b}] and y=[{c,d}], then x+y=[{a+c,b+d}]. For example:

You may have already guessed the rule for integer subtraction: x-y=x+(-y).

As always, I recommend you play around with this to get familiar with the integers as sets of equivalent pairs.

Integer Multiplication

Take x=a-b and y=c-d like before and let’s lean on our school math knowledge once again. We know that multiplication must satisfy

Eureka! This the difference of two naturals once again, and since we know how to multiply Naturals from Part I, we need no other tools to find x*y. We can thus express integer multiplication as follows:

If x=[{a,b}] and y=[{c,d}], then x*y=[{ac+bd,ad+bc}]

For example:

Experiment with this rule and see how it expresses all the integer multiplication rules you’re already familiar with!

With the integers and their arithmetic now in our grasp, let’s continue to the objects that the Pythagoreans believed to contain the secrets of nature, the Rational Numbers!

A few remarks before Part III…

If you’ve gotten this far, then congrats! You’ve gotten through all the set theory fundamentals for this article series, so the road only gets easier from now on.

I encourage you to also meditate on what we’ve accomplished so far. We’ve built the Integer Arithmetic only using sets and Natural Arithmetic, which in turn we’ve built only using sets in Part I. Diophantus of Alexandria considered this set of numbers to be inexistent, yet we’ve constructed them using nothing but sets, which, if you dig deep enough, have nothing inside them! That, in my opinion, gives us a glimpse at the beauty of set theory.