# The field of two numbers

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In the previous session, we have seen that we can define an addition and a multiplication on `even` and `odd` so that they satisfy the [field axioms](https://hackmd.io/@alexhkurz/BJw5gSMi8).

If you look at these axioms, you will see that we require
- a so-called neutral element (aka identity) of addition denoted $0$
- a neutral element (aka identity) of multiplication denoted $1$.

**Exercise:**
- What are $0$ and $1$ in the field $\{even,odd\}$?
- Rewrite the addition and multiplciation table for $even$ and $odd$ in terms of $0,1$:

|+|0| 1|
|:---:|:---:|:---:|
| 0 | | |
| 1 | | |

and

|$\cdot$|0| 1|
|:---:|:---:|:---:|
| 0 | | |
| 1 | | |

**Question:** What do you get for $1+1$? Discuss.

**Question:** In any field, for any element $x$, there must be an element $-x$ such that $-x+x=0$. Given the tables you filled in above, which of the two numbers $0$ or $1$ is $-1$?

**Activity:** Are there any other ways to fill in the two tables above without violating the field axioms?

**Exercise:** If you have programming experience, you can write a program that takes as input the two tables above and checks whether all the axioms are satisfied. This could be helpful, in particular if we get to bigger tables later on. But while useful, it is not a requirement for these investigations.

## Takeaways

1. It always pays in mathematics to think about what the **axioms** are. If you know the axioms, use them. (If you are still exploring and you didn't decide yet what axioms you want, keep your eyes open.)

In our example, because
- $0$ is the neutral element wrt to addition and $1$ is the neutral element wrt multiplication, we can immediately fill parts of the two tables;
- the axioms imply that $0\cdot x= 0$ for all numbers $x$, the rest of the multiplication table follows;
- for every number $x$ there must be a negative number $-x$, the rest of the addition table follows.

2. While most maths is done without computers, it can help to use **computer programs**.