# Definition:Strictly Positive Real Number/Definition 1

## Definition

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 denoted as

The $\mathsf{Pr} \infty \mathsf{fWiki}$-specific notation $\R_{> 0}$ is actually non-standard.

The conventional symbol to denote this concept is $\R_+^*$.

Note that $\R^+$ is also seen sometimes, but this is usually interpreted as the set $\set {x \in \R: x \ge 0}$.

## Also known as

Throughout Euclid's *The Elements*, the term **magnitude** is universally used for this concept.

It must of course be borne in mind that at that stage in the development of mathematics, neither of the concepts **real number** nor **positive** were fully understood except intuitively.

Some sources refer to this just as **positive**, as their treatments do not accept $0$ as being either **positive** or **negative**.

## Also see

- Results about
**strictly positive real numbers**can be found**here**.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.1$: Set Notation - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**positive number** - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**positive number**