Definition:Positive

Definition

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

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

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

$R_{\ge 0_R} := \left\{{x \in R: 0_R \le x}\right\}$

Ordered Integral Domain

An ordered integral domain is an integral domain $\left({D, +, \times}\right)$ with a property $P$ such that:

$(1): \quad \forall a, b \in D: P \left({a}\right) \land P \left({b}\right) \implies P \left({a + b}\right)$

$(2): \quad \forall a, b \in D: P \left({a}\right) \land P \left({b}\right) \implies P \left({a \times b}\right)$

$(3): \quad \forall a \in D: P \left({a}\right) \lor P \left({-a}\right) \lor a = 0_D$

For condition $(3)$, exactly one of the conditions applies for every element of $D$.

An ordered integral domain can be denoted:

$\left({D, +, \times \le}\right)$

where $\le$ is the ordering induced by the positivity property.

Positivity

The property $P$ is called the positivity property.

As its name implies, it is identified with the property of being positive.

Hence the above conditions can be written in natural language as:

$(1): \quad$ The sum of any two positive elements is also positive.

$(2): \quad$ The product of any two positive elements is also positive.

$(3): \quad$ Every element is either positive, or negative, or zero.

Trichotomy Law

The property:

$\forall a \in D: P \left({a}\right) \lor P \left({-a}\right) \lor a = 0_D$

is known as the trichotomy law.

Numbers

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 can not be ordered, 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:

$\Z_{\ge 0} = \left\{{0, 1, 2, 3, \ldots}\right\}$

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 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 Unique Minus congruence classes, positive 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 positive iff $b \le a$.

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:

$+x \in \Z \iff x \in \Z_{\ge 0}$.

Ordering on Integers

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

$\forall x, y \in \Z: y - x \in \Z_{\ge 0} \iff x \le y$

That is, $x$ is less than or equal to $y$ iff $y - x$ is positive.

Rational Numbers

The positive rational numbers are the set defined as:

$\Q_{\ge 0} := \left\{{x \in \Q: x \ge 0}\right\}$

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

Real Numbers

The positive real numbers are the set:

$\R_{\ge 0} = \left\{{x \in \R: x \ge 0}\right\}$

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

Also known as

The notations $R_+$ and $R^+$ are frequently seen for $\left\{{x \in R: 0_R \le x}\right\}$.

However, these notations are also used for $\left\{{x \in R: 0_R < x}\right\}$, that is, $R_{> 0_R}$, and so suffer from being ambiguous.

Also defined as

Some treatments of this subject use the term define non-negative to define $x \in R$ where $0_R \le x$, reserving the term positive for what is defined on this website as strictly positive.

With the conveniently unambiguous notation that has been adopted on this site, the distinction between the terms loses its importance, as the symbology removes the confusion.

Also see

• Strictly Positive
• Negative
• Strictly Negative

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 23$