# Even and odd

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In the previous session we collected the rules of computation with numbers. We call them [axioms](https://hackmd.io/@alexhkurz/BJw5gSMi8) because that is, for now, our working definition of what a system of numbers is.

We will see later that this basic definition allows for variations. To distinguish this particular list of axioms, mathematicican call a structure satsifying these axioms a **field**.

We already know two fields. The field of rational number $\mathbb Q$ for calculations with fractions, and the field of reald numbers $\mathbb R$, which adds to the fractions numbers such as $\sqrt 2$ or $\pi$.

Do we know any other fields?

When you start learning about addition and multiplication, it is very useful to know some rules about even and odd numbers. Like

even + even = even
...
odd * odd = odd

For example, if you have to compute
$$7+8$$

and you know that the result must be odd, you can use this to check whether your calculation is correct. Similarly for multiplications. Many mistakes of students beginning to learn time tables can be caught like this.

This idea can also be extended to find errors in data transmission on, say, mobile phone networks, or even to design codes that correct errors automatically.

**Activity:** How should we fill in the following two tables?

|+|even| odd|
|:---:|:---:|:---:|
| even | | |
| odd | | |

and

|$\cdot$|even| odd|
|:---:|:---:|:---:|
| even | | |
| odd | | |

**Activity/Homework:** Show that with the operations of $+$ and $\cdot$ you defined in the two tables above, the set $\{even,odd\}$ is a field.