Just what are numbers, really? (Part I)
You use them everyday to count and measure, but could you explain what their fundamental nature is? In Part I, we’ll expose the Natural Numbers
A screwdriver and a drill can both be used to drive a screw into place and build your favourite IKEA furniture, their function and application (in this case) are the same, but does that mean they are the same thing? A tool’s application tells you about its utility, but not the nature of the tool itself. Likewise, you use numbers to count and measure things, but have you ever asked yourself what they really are under the hood? If that’s a question in your mind, then come along and I’ll try to show you how modern mathematicians understand numbers, without the jargon! No prior knowledge required either!
Disclaimer: This article series was made using Mathoma’s excellent series on Set Theory as a guide. Their blog can be found here.
The Basic Tools: Sets and Unions
The modern concept of numbers is built using set theory, a.k.a. the study of collections of things. Sets are simply collections of objects, and the objects can be literally anything. For example,
is a perfectly valid set. Our objective throughout this series will be to describe every number as a set. A uniquely important set is the empty set, which (as the name suggests) is the set with nothing inside.
The other tool we need is the ability to join two sets together into a set with everything in both of them, called a set union. For example, if A = { Cyan, Yellow } and B = { Magenta, Black }, then the set union of A and B, denoted A U B, contains all colors in both sets.
This is all we’ll need to uncover the hidden nature of numbers! If these ideas sound fair enough to you, then let’s continue to our fist set of numbers.
The Natural Numbers
These are the first numbers we encounter as kids and the ones people are most familiar with as a tool for counting. Turns out we can express them purely in terms of sets, as well as all their arithmetic, beginning with 0. If we didn’t know any numbers existed, we would at least know the empty set existed, so we define
The next natural numbers are built by simply joining (a.k.a taking the union of) the previous number (a set) with the set containing the previous number, thus resulting in a bigger set. Visually, we see this recursive definition leads an elegant representation:
Simply put, a natural number is the set containing all natural numbers lesser than itself, with 0 being the empty set! Also note that a natural number’s set has exactly “that many” elements in it!
Canonically, mathematicians say that given a natural number x, then x+1 is given by the successor function of x, denoted as
You might then ask, “ok, but how could this relate to how I use these numbers to, you know, add and multiply?”. Let’s begin with addition.
Natural Number Addition
We already know how to get a number’s successor, i.e. how to add 1, so let’s shorthand the successor of x as x+1. We can express this neatly and recursively by the function Add( x, 1), which will add 1 to x, as
For example,
To make the addition of a natural number y to x, we need only change the base rule of addition to 0 to the following: Add( 0, y ) = y. Thus we can express natural number addition as
I encourage you to carry out some calculations and see how this formula checks out with the arithmetic you already know. Next, let’s see how multiplication works.
Natural Number Multiplication
Let’s not stray too far from what we already know, multiplication is just repeated addition (for the Naturals). We can apply the same strategy we used for addition and define a function Multiply(x,y) which gives the multiplication result of x and y. From our knowledge, this function must imply that 1*y=y, as well as (x+1)*y=x*y+y. As such we establish the following recursive rules:
Once again, feel free to test this formula out and confirm this works exactly how the multiplication you already know, including that x*0 = 0 for any x.
So far, we’ve defined the natural numbers and their arithmetic rules using only sets and set operations under the hood. Every x and y we’ve discussed are simply sets, and every + and * are nothing more than a bunch of set unions!
A Few Remarks Before Part II…
Before we move on to the integer numbers in Part II, I’d like you to stop and ponder on how we’ve built the natural numbers using nothing more than three simple beliefs:
- Collections of things (sets) exist
- We can join two collections by considering the collection of everything in both (set union).
- The collection with nothing in it (empty set) exists.
I think you could justify these axioms to any child, yet they are all we need to conceptualize all number systems and their arithmetic rules! Pretty cool right?
Secondly, you may have noticed I haven’t mentioned subtraction or division. The reason behind that is the issue that subtracting or dividing two natural numbers may not result in a natural number. Because of this, we can only effectively define subtraction using the integers, and division using the rationals.
If you’re curious about how we can built these complicated number systems and arithmetics using only sets, then stick around for Part II!