The set of integers

in in |

In this set, we can define an addition and a multiplication, as follows:

- ;
- .

- The mapping is an injection.
- For any two natural numbers and , .
- For any two natural numbers and , .

- Let and be two natural numbers such that
. We have:

- For any two natural numbers and , we have:

- For any two natural numbers and , we have:

- If , then , i.e. .
- If , then . In other words, for any , .
- If , then , i.e. .

With the identification determined by the mapping , we can write . The sign rules are given by the following proposition:

The other equation is obtained now by the commutative law of the addition in . Assume now that an integer has two opposites and . Then we have:

by Prop 1.15 | ||