Introduction
This blog post is about a way to define addition and multiplication on a number line using the geometry of the plane that surrounds the line.
Recently I talked about how numbers have multiple purposes and one of those purposes is locating. When you draw a number line, that’s pretty much what you’re doing – saying that the numbers are exactly locations on that line. There is an issue with this idea, which is this: how do you add or multiply locations?
The usual way of dealing with this is to think of numbers as not just locations but journeys too, so that when we see something like 10+3, the first number is a location and the second one is a journey and the answer 13 is the location we arrive at by beginning at 10 and travelling onwards 3. This is fine and I like it very much, actually, but there is still the huge problem of how to multiply locations or indeed how to multiply journeys. You can say that 3×10 is three lots of a journey of 10, which makes sense, but now the 3 is neither a journey nor a location but a literal amount and we have three different things so far that the numbers mean. Is there a way that preserves the location-ness of everything?
Yes, there is.
There is indeed a way to define addition and multiplication of locations on a number line that preserves their fundamental location-ness, by looking outwards to the other lines of the plane your line is part of. The system doesn’t directly use lengths or angles, but only uses the most fundamental geometrical actions of drawing lines through two points, finding where two lines meet, and drawing lines parallel to other lines. This is my favourite thing about it, that it’s fully based on the relationships between the points and the lines as objects. As a pure mathematician and a finite geometer specifically, geometry isn’t really about measurements at all but is all about relationships, so something that focuses on the relationships deeply appeals to me.
I created this method in October 2020, heavily based on the method invented by Marshall Hall Jr in the late 1950s. Hall’s method works by adding coordinates to all the points in the plane including the ones on your number line, making equations for the lines, and referring to the coordinates of specific points on specific lines to define the arithmetic. I studied his method in 2001 while doing the honours year in my undergraduate maths degree, but it wasn’t until 19 years later that I realised there was a way to do it without referring to coordinates, by focusing my attention on the number line itself rather than the coordinate axes. My method for multiplication also has striking similarities to René Descartes’ original definition for the multiplication of lengths published in the 1630s, even though I didn’t mean it to and only found this out later. An important difference between them is that his method has the two factors on different sides of a triangle instead of along the same line. I find it very interesting that Hall’s book is called “The Theory of Groups” and Descartes’ book is called “Geometry”, highlighting the deep connection between geometry and algebra which all three methods point to.
Anyway, enough historical notes. Let’s get to it.
How to do geometric arithmetic
You can watch me doing both processes live in a video here, or you can read a text description and see screenshots from the video below.
Setting up
To do parallel line arithmetic, you need to set up a few things.
First, choose a line to be your number line. The points on the number line are your numbers.
Next, you’ll need to choose two different points on the line to call 0 and 1. The point 0 will be important for defining addition and both 0 and 1 will be important for defining multiplication.
Finally, you’ll need to choose two lines other than the number line itself that are not parallel to the number line and not parallel to each other. It doesn’t really matter where they are because the specific lines themselves aren’t important, only their directions, since during the constructions for addition and multiplication, you won’t be making them intersect with any lines. Instead you will draw several lines parallel to each of these lines. I call one direction “home” and the other “away”. (The reason why I chose these words specifically rather than “in” and “out” for example is because I am Australian and “Home and Away” means something to me.) To make it easier to focus when I’m drawing diagrams for the arithmetic, I usually put my home and away lines a bit away from the part of my number line I drew.
Addition
To add two numbers geometrically, you follow this process.
First, you need to have your two points a and b, and the point 0 on your number line. You don’t need a and b to be different from each other or different from 0, but I’ve drawn them as different to make it easier to show how the process works.
Create a journey from 0 to a first following the away direction and then following the home direction. That is, draw a line through 0 parallel to the away line.
Then draw a line parallel to the home line that passes through a.
Find the point where these two lines meet. If you follow the journey from 0 to a first along the away direction and then along the home direction, this point is where the journey turns from going away to going home.
Now draw a line parallel to the number line, through this turning point. This is the turning line for all additions that go some number plus a.
Now you are ready to do b+a.
Draw a line through b parallel to the away line and find where it meets the turning line. This is where the journey from b will turn and return home to the number line.
Now draw a line through this turning point parallel to the home line, and find where it meets the number line.
This point is the point b+a.
Multiplication
To add two numbers geometrically, you follow this process, which is similar to the process for addition, but with two very important differences.
First, you need to have your two points a and b, and the two points 0 and 1 on your number line. Just like before, you don’t need a and b to be different from each other or different from 0 or 1, but I have chosen them that way to make it easier to show how the process works.
Create a journey from 1 to a first following the away direction and then following the home direction. (This is the first point of difference between multiplication and addition, that the journey begins at 1 and not 0.)
That is, draw a line through 1 parallel to the away line.
Then draw a line parallel to the home line that passes through a.
Find the point where these two lines meet. If you follow the journey from 1 to a first along the away direction and then along the home direction, this point is where the journey turns from going away to going home.
Now draw a line that passes through both this turning point and 0. This is the turning line for all multiplications that go some number times a. (This is the second point of difference between multiplication and addition, that the turning line passes through 0 rather than being parallel to the number line.)
Now you are ready to do b×a.
Draw a line through b parallel to the away line and find where it meets the turning line. This is where the journey from b will turn and return home to the number line.
Now draw a line through this turning point parallel to the home line, and find where it meets the number line.
This point is the point b×a.
Thoughts
So that’s David Butler’s methods of geometric addition and multiplication. I love it so much, especially the multiplication. Addition is a little more complicated than you’d expect if you are used to adding numbers by joining journeys head to tail, but multiplication somehow feels so much simpler than it has a right to be.
My favourite thing to do is to convince myself that various algebraic properties of numbers have to be true using these two definitions of addition and multiplication. For example, this diagram shows that a+b = b+a (at least for positive numbers).
And this diagram shows that b+b = b×2 (at least for numbers more than 1).
And this diagram shows that when you multiply two negative numbers, you get a positive number. (I know those numbers are negative because they’re on the opposite side of 0 from 1.)
Yeah none of them are formal proofs, but I still love them.
The truly remarkable thing is that it doesn’t matter how you choose your home and away directions, it will still work! For example, here are three diagrams showing 1+1=2 and 2×2=4.
The geometry of the real plane is so neatly structured that it works every time. If our number line was in a plane with a different structure, this might not work the same every time. Indeed, you may end up with entirely different rules for how addition or multiplication work, such as multiplication not being associative.(That is, (a×b)×c not being the same as a×(b×c), which is very annoying!)
I may do future blog posts about some of that other stuff, but for now, I’m enjoying just revelling in the coolness that is the existence of a method for multiplying locations, and the even higher coolness of watching geometry cause the algebraic properties of numbers. I hope you enjoyed it too.