# Definition:Positive/Number

## Definition

The concept of positive 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 positive complex number.

As for the natural numbers, they are all positive by dint of their being the non-negative integers.

### Integers

Informally, the **positive integers** are the set:

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

As the set of integers $\Z$ 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 **positive** 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, **positive** can be defined directly as the relation specified as follows:

- The integer $z \in \Z: z = \eqclass {\tuple {a, b} } \boxminus$ is
**positive**if and only if $b \le a$.

The set of **positive integers** is denoted $\Z_{\ge 0}$.

An element of $\Z$ can be specifically indicated as being **positive** by prepending a $+$ sign:

- $+x := x \in \Z_{\ge 0}$.

### Ordering on Integers

### Definition 1

The integers are ordered on the relation $\le$ as follows:

- $\forall x, y \in \Z: x \le y$

- $\exists c \in P: x + c = y$

where $P$ is the set of positive integers.

That is, $x$ is **less than or equal** to $y$ if and only if $y - x$ is non-negative.

### Definition 2

The integers are ordered on the relation $\le$ as follows:

Let $x$ and $y$ be defined as from the formal definition of integers:

- $x = \eqclass {x_1, x_2} {}$ and $y = \eqclass {y_1, y_2} {}$ where $x_1, x_2, y_1, y_2 \in \N$.

Then:

- $x < y \iff x_1 + y_2 \le x_2 + y_1$

where:

- $+$ denotes natural number addition
- $\le$ denotes natural number ordering.

### Rational Numbers

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

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

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

### Real Numbers

The **positive real numbers** are the set:

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

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

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $2$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $2$