Definition:Negative

Definition
Let $$\left({R, +, \circ, \le}\right)$$ be an ordered ring whose zero is $$0_R$$.

Then $$x \in R$$ is negative iff $$x \le 0_R$$.

The set of all negative elements of $$R$$ is denoted:


 * $$R_- \ \stackrel {\mathbf {def}} {=\!=} \ \left\{{x \in R: x \le 0_R}\right\}$$

Integers
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 Unique Minus 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 Unique Minus 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 $$\Z_-$$.

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


 * $$-x \in \Z \iff x \in \Z_-$$.

Ordering on Integers
The integers are ordered on the relation $$<$$ as follows:


 * $$\forall x, y \in \Z: y - x \in \Z_- \iff x > y$$

That is, $$x$$ is greater than $$y$$ iff $$y - x$$ is negative.