This problem forms the first part of an investigation into how to
represent numbers using geometric transformations that will
ultimately lead us to discover numbers that are not on the number
I'm going to assume that you are happy with the idea of numbers
that lie on the number line. Just to remind you, here is a picture
of a small part of the number line.
The plan of action is to develop pictures or geometric
representations of the numbers we know about already. We'll also
develop ways to add, subtract, multiply and divide using just those
pictures. Then we'll change the picture very slightly...
Here's my first attempt at a picture of the number 1. It's simply a
I don't think this is a good enough picture yet, but it's helpful
to think about its shortcomings.
Let's try to do arithmetic with these arrows. Try three simple
I tried these with three friends and here are the answers they gave
They all had good explanations for their answers. Can you
guess what they were?
My arrow picture obviously has some shortcomings! How would
you improve it?