# Systems of 4 numbers

([home](https://github.com/alexhkurz/introduction-to-numbers/blob/master/README.md) ... [previous](https://hackmd.io/@alexhkurz/B1UDo-VyD) ... [next](https://hackmd.io/@alexhkurz/r1Gdg_EoU))

## Introduction

We are still on our way exploring what different systems of numbers make sense.

So far we found that there is exactly one field of 2 numbers and also exactly one with 3 numbers.

If the number of elements is small, like 2 or 3, we can find these results just by trying all possiblities by hand.

But as the number of elements gets larger, this method quickly becomes infeasable. Already 4 elements call for new methods.

## Addition modulo 4

To simplify the task, let us first focus only at addition.

If we decide to name the 4 numbers as $0,1,2,3$, there is an obvious candidate for an operation that we would want to call addition:

|+|0| 1| 2| 3|
|:---:|:---:|:---:|:---:|:---:|
| 0 | 0 |1 | 2|3
| 1 | 1| 2| 3 |
| 2 | 2| 3| |
| 3 | 3| | |

If we start this way, we have the familiar $1+1=2$ and $1+2=3$.

**Activity:** How many ways are there to fill in the rest of the table?

This should work using the same methods that we have seen already, namely calculations using the axioms and exclusion of the impossible via deriving contradictions.

**Exercise:** How long does it take you to check that the table we obtained in the previous activity is actually associative?

## More operations on 4 numbers

**Question:** Are other ways to define an "addition" on $\{0,1,2,3\}$? What if we start with the following?

I put addition in double quotes, because the tables you will come up with will look even more different from our familiar addition than the one above.

|+|0| 1| 2| 3|
|:---:|:---:|:---:|:---:|:---:|
| 0 | 0 |1 | 2|3
| 1 | 1| 3| |
| 2 | 2|| |
| 3 | 3| | |

or

|+|0| 1| 2| 3|
|:---:|:---:|:---:|:---:|:---:|
| 0 | 0 |1 | 2|3
| 1 | 1| 0| |
| 2 | 2|| 0|
| 3 | 3| | |

or

|+|0| 1| 2| 3|
|:---:|:---:|:---:|:---:|:---:|
| 0 | 0 |1 | 2|3
| 1 | 1| 0| |
| 2 | 2|| 1|
| 3 | 3| | |

## Takeaway

...