# Definition:Negative/Integer

## Definition

As the set of integers is the Inverse Completion of Natural Numbers, 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 **negative** can be formally defined on $\Z$ as a relation induced on those equivalence classes as specified in the definition of integers.

That is, the integers being defined as all the difference congruence classes, **negative** can be defined directly as the relation specified as follows:

The integer $z \in \Z: z = \left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus$ is **negative** iff $b > a$.

The set of negative integers is denoted $\Z_{\le 0}$.

An element of $\Z$ can be specifically indicated as being negative by prepending a $-$ sign:

- $-x \in \Z \iff x \in \Z_{\le 0}$

## Also see

## Sources

- 1964: Murray R. Spiegel:
*Theory and Problems of Complex Variables*... (previous) ... (next): $1$: Complex Numbers: The Real Number System: $2$