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

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Math Simplified

5 min readFeb 12, 2022

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!

Stacked children’s numbered blocks — Photo by Susan Holt Simpson on Unsplash

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,

S equals, open curly braces, Elon Musk, Cat, USA, Table, close curly braces.

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.

“Empty set” over mathematical notation. Notation reads: slashed circle equals, open curly braces, close curly braces.

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.

Picture has 3 rows. On the first row to the left, mathematical notation reads: A equals, open curly braces, Cyan, Yellow, close curly braces. On the first row to the right, it reads: B equals, open curly braces, Magenta, Black, close curly braces. On the second row, notation reads: A union B equals, open curly braces, Cyan, Yellow, Magenta, Black, close curly braces. On the third row, it reads: Obs: If an element is in A or B, then it’s in A union B!

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

Mathematical notation reads: 0 equals empty set.

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:

Example of natural numbers as sets containing four lines of notation. First line reads: 1 = 0 union {0} = {0} = {Empty Set}. Second line reads: 2 = 1 union {1} = {0,1} = { Empty Set, { Empty Set } }. Third line reads: 3 = 2 union {2} = {0,1,2} = { Empty Set, { Empty Set }, { Empty Set, { Empty Set } } }. Four line has 3 vertically aligned dots under the equal sign, followed by a note on the right that reads: “Notice a pattern here?”.

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

Math notation reads: x plus 1, equals, SUCCESSOR of x, equals, x union set containing only x.

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

Definition of the ADD function, taking two inputs. First line reads: ADD of 0 and 1 equals 1. Second line reads: ADD of x plus 1 and 1, equals, ADD of x and 1, plus 1

For example,

Example of ADD function calculation containing four lines. First line reads: ADD of 3 and 1, equals, ADD of 2 and 1, plus 1. Second line reads: equals, ADD of 1 and 1, plus 1 plus 1. Third line reads: equals, ADD of 0 and 1, plus 1 plus 1 plus 1. Four line reads: equals, 1 plus 1 plus 1 plus 1, equals 4 (diagonal lines giving emphasis to four).

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

Definition of natural number addition. First line reads: ADD of 0 and y equals y. Second line reads: ADD of x plus 1 and y, equals, ADD of x and y, plus 1. To the top right, a note reads: “Obs: Only operation being used is +1, which is just set unions!”.

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!