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:

$+x := x \in \Z_{\ge 0}$.

Definition 1

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

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

if and only if:

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

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): $2$