# 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 $z \in \Z: z = \eqclass {\tuple {a, b} } \boxminus$ is **negative** if and only if $b > a$.

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:

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

## Also see

## Sources

- 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: The Real Number System: $2$