# Definition:Strictly Positive

## Definition

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

Then $x \in R$ is **strictly positive** if and only if $0_R \le x$ and $x \ne 0_R$.

One may write (more conveniently) that $0_R < x$ or $x > 0_R$ to express that $x$ is **strictly positive**.

Thus, the set of all **strictly positive** elements of $R$ can be denoted:

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

## Numbers

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

- $(1): \quad$ The natural numbers $\N$
- $(2): \quad$ The integers $\Z$
- $(3): \quad$ The rational numbers $\Q$
- $(4): \quad$ The real numbers $\R$

### Natural Numbers

The **strictly positive natural numbers** are the set defined as:

- $\N_{>0} := \set {x \in \N: x > 0}$

That is, all the natural numbers that are strictly greater than zero:

- $\N_{>0} := \set {1, 2, 3, \ldots}$

### Integers

The **strictly positive integers** are the set defined as:

- $\Z_{> 0} := \set {x \in \Z: x > 0}$

That is, all the integers that are strictly greater than zero:

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

### Rational Numbers

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

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

That is, all the rational numbers that are strictly greater than zero.

### Real Numbers

The **strictly positive real numbers** are the set defined as:

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

That is, all the real numbers that are strictly greater than zero.

## Also known as

The notation $R_+^*$ is frequently seen for $R_{>0_R}$, that is for $\set {x \in R: 0_R < x}$.

However, the notation $R_+$ and $R^+$ are also frequently seen for both $\set {x \in R: 0_R \le x}$ and $\set {x \in R: 0_R < x}$, and so suffer badly from ambiguity.

## Also defined as

Some treatments of this subject reserve the term define **positive** to define $x \in R$ where $0_R < x$, using the term **non-negative** for what is defined on this website as 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

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $2$: Ordered and Well-Ordered Integral Domains: $\S 7$. Order