How does one denote the set of all positive real numbers?

What is the "standard" way to denote all positive (or non-negative) real numbers? I'd think

$$ \mathbb R^+ $$

but I believe that that is usually used to denote "all real numbers including infinity".

So is there a standard way to denote the set

$$ \{x \in \mathbb R : x \geq 0\} \; ?$$

Solution 1:

The unambiguous notations are: for the positive-real numbers $$ \mathbb{R}_{>0} = \left\{ x \in \mathbb{R} \mid x > 0 \right\} \,, $$ and for the non-negative-real numbers $$ \mathbb{R}_{\geq 0} = \left\{ x \in \mathbb{R} \mid x \geq 0 \right\} \,. $$ Notations such as $\mathbb{R}_{+}$ or $\mathbb{R}^{+}$ are non-standard and should be avoided, becuase it is not clear whether zero is included. Furthermore, the subscripted version has the advantage, that $n$-dimensional spaces can be properly expressed. For example, $\mathbb{R}_{>0}^{3}$ denotes the positive-real three-space, which would read $\mathbb{R}^{+,3}$ in non-standard notation.

In Algebra one may come across the symbol $\mathbb{R}^\ast$, which refers to the multiplicative units of the field $\big( \mathbb{R}, +, \cdot \big)$. Since all real numbers except $0$ are multiplicative units, we have $$ \mathbb{R}^\ast = \mathbb{R}_{\neq 0} = \left\{ x \in \mathbb{R} \mid x \neq 0 \right\} \,. $$ But caution! The positive-real numbers can also form a field, $\big( \mathbb{R}_{>0}, \cdot, \star \big)$, with the operation $x \star y = \mathrm{e}^{ \ln(x) \cdot \ln(y) }$ for all $x,y \in \mathbb{R}_{>0}$. Here, all positive-real numbers except $1$ are the "multiplicative" units, and thus $$ \mathbb{R}_{>0}^\ast = \mathbb{R}_{\neq 1} = \left\{ x \in \mathbb{R}_{>0} \mid x \neq 1 \right\} \,. $$

Solution 2:

Not that I knew of. There are many, e.g.

• $\mathbb{R^+_0}$,
• $\mathbb{R^+}$ and
• $[0, \infty)$.