# Definition:Addition/Integers

## Definition

The **addition operation** in the domain of integers $\Z$ is written $+$.

We have that the set of integers is the Inverse Completion of Natural Numbers.

Thus it follows that elements of $\Z$ are the isomorphic images of the elements of equivalence classes of $\N \times \N$ where two tuples are equivalent if the difference between the two elements of each tuple is the same.

Thus addition can be formally defined on $\Z$ as the operation induced on those equivalence classes as specified in the definition of integers.

That is, the integers being defined as all the difference congruence classes, integer addition can be defined directly as the operation induced by natural number addition on these congruence classes:

- $\forall \left({a, b}\right), \left({c, d}\right) \in \N \times \N: \left[\!\left[{a, b}\right]\!\right]_\boxminus + \left[\!\left[{c, d}\right]\!\right]_\boxminus = \left[\!\left[{a + c, b + d}\right]\!\right]_\boxminus$

## Sources

- 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 4$: Number systems $\text{I}$: The rational numbers - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 2.1$: The integers