# Category:Definitions/Negative Numbers

This category contains definitions related to Negative Numbers.

Related results can be found in Category:Negative Numbers.

The concept of negative can be applied to the following sets of numbers:

- $(1): \quad$ The integers $\Z$
- $(2): \quad$ The rational numbers $\Q$
- $(3): \quad$ The real numbers $\R$

The Complex Numbers cannot be Ordered Compatibly with Ring Structure, so there is no such concept as a negative complex number.

### Integers

The **negative integers** comprise the set:

- $\set {0, -1, -2, -3, \ldots}$

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:

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

### Rational Numbers

The **negative rational numbers** are the set defined as:

- $\Q_{\le 0} := \set {x \in \Q: x \le 0}$

That is, all the rational numbers that are less than or equal to zero.

### Real Numbers

The **negative real numbers** are the set defined as:

- $\R_{\le 0} := \set {x \in \R: x \le 0}$

That is, all the real numbers that are less than or equal to zero.

### Complex Numbers

As the Complex Numbers cannot be Ordered Compatibly with Ring Structure, the concept of a **negative complex number**, relative to a specified zero, is not defined.

However, the **negative** of a complex number is defined as follows:

Let $z = a + i b$ be a complex number.

Then the **negative of $z$** is defined as:

- $-z = -a - i b$

## Pages in category "Definitions/Negative Numbers"

The following 2 pages are in this category, out of 2 total.