# 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

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

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

In Algebra one may come across the symbol $$\mathbb{R}^\ast\mathbb{R}^\ast$$, which refers to the multiplicative units of the field $$\big( \mathbb{R}, +, \cdot \big)\big( \mathbb{R}, +, \cdot \big)$$. Since all real numbers except $$00$$ are multiplicative units, we have
$$\mathbb{R}^\ast = \mathbb{R}_{\neq 0} = \left\{ x \in \mathbb{R} \mid x \neq 0 \right\} \,. \mathbb{R}^\ast = \mathbb{R}_{\neq 0} = \left\{ x \in \mathbb{R} \mid x \neq 0 \right\} \,.$$
But caution! The positive-real numbers can also form a field, $$\big( \mathbb{R}_{>0}, \cdot, \star \big)\big( \mathbb{R}_{>0}, \cdot, \star \big)$$, with the operation $$x \star y = \mathrm{e}^{ \ln(x) \cdot \ln(y) }x \star y = \mathrm{e}^{ \ln(x) \cdot \ln(y) }$$ for all $$x,y \in \mathbb{R}_{>0}x,y \in \mathbb{R}_{>0}$$. Here, all positive-real numbers except $$11$$ are the “multiplicative” units, and thus
$$\mathbb{R}_{>0}^\ast = \mathbb{R}_{\neq 1} = \left\{ x \in \mathbb{R}_{>0} \mid x \neq 1 \right\} \,. \mathbb{R}_{>0}^\ast = \mathbb{R}_{\neq 1} = \left\{ x \in \mathbb{R}_{>0} \mid x \neq 1 \right\} \,.$$