# Definition:Strictly Positive/Real Number

## Definition

### Definition 1

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.

### Definition 2

The **strictly positive real numbers**, written $R_{>0}$, is the subset of $\R$ that satisfies the following:

\((\R_{>0} 1)\) | $:$ | Closure under addition | \(\displaystyle \forall x, y \in \R_{>0}:\) | \(\displaystyle x + y \in \R_{>0} \) | ||||

\((\R_{>0} 2)\) | $:$ | Closure under multiplication | \(\displaystyle \forall x, y \in \R_{>0}:\) | \(\displaystyle xy \in \R_{>0} \) | ||||

\((\R_{>0} 3)\) | $:$ | Trichotomy | \(\displaystyle \forall x \in \R:\) | \(\displaystyle x \in \R_{>0} \lor x = 0 \lor -x \in \R_{>0} \) |

## 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 merely refer to this as **positive**, as their treatments do not accept $0$ as being either **positive** or negative.

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