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 | \(\ds \forall x, y \in \R_{>0}:\) | \(\ds x + y \in \R_{>0} \) | |||||
| \((\R_{>0} 2)\) | $:$ | Closure under multiplication | \(\ds \forall x, y \in \R_{>0}:\) | \(\ds xy \in \R_{>0} \) | |||||
| \((\R_{>0} 3)\) | $:$ | Trichotomy | \(\ds \forall x \in \R:\) | \(\ds x \in \R_{>0} \lor x = 0 \lor -x \in \R_{>0} \) |
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.