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

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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

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

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8 Answers

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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.

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Addendum:

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$$ \mathbb{R}_{>0}^\ast = \mathbb{R}_{\neq 1} = \left\{ x \in \mathbb{R}_{>0} \mid x \neq 1 \right\} \,. $$

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

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

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I'd completely avoid using $\mathbb{R}^+$ since people won't know if $0$ is included or not. So $\mathbb{R}_0^+$ would be a possibility, but then how would you denote $\{x\in\mathbb{R}:x>0\}$? Again, with $\mathbb{R}^+$ people won't know that $0$ isn't included. Personally, I prefer writing $[0,\infty)$ and $(0,\infty)$ when it's clear from the context that an interval in $\mathbb{R}$ is meant.

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Some of my profs use $\mathbb{R^{\ge 0}}$. I like to add whatever to the top so $\mathbb{R^{\le a}}$ just means all reals less than $a$.

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The following is also pretty common notation for the non-negative reals: $\mathbb{R}_{\geq 0}$ or $\mathbb{R}_{+}$.

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I've learned in elementary school that $\mathbb{R}_{*}$ means the set without the zero, so $\mathbb{R}^{+}=[0,\infty)$ and $\mathbb{R}^{+}_{*}=(0,\infty)$.

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$\mathbb{R}^+$ includes $0$ in Probability Tutorials. $\mathbb{R}^+_0$ is more clear though, so I've used it in the exercises.

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I find $\mathbb R_{\geq 0}$ clumsy (I would never write this on a board when working and I don't often see papers with functions $f:\mathbb R_{\geq 0}\rightarrow \mathbb R_{\geq 0}$).

$\mathbb R^+$ seems restrictive, not least if you wish to consider higher dimensions.

I like $[0,\infty)$, but it too can be awkward in many settings.

Since $\mathbb R_{+}$ is well-established as the nonnegative reals, I prefer some version of the "doubleplus" notation such as $\mathbb R_{\mkern.5pt+\mkern-1pt+}$ for the (strictly) positive reals.

This satisfies the blackboard test, and it also looks reasonably tidy: $f: \mathbb R_{\mkern1mu+\mkern-1mu+} \rightarrow \mathbb R_{\mkern.5pt+\mkern-1mu+}$.

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Having struggled with this, due to notational issues in a paper, in the end I've concluded that any of the above, including $\mathbb R_{\geq 0}$ or $\mathbb R_{+}$ for the nonnegative reals and $\mathbb R_{>0}$ or $\mathbb R_{*+}$ for the positives is fine provided you can make the subscripts small enough (in Latex).

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