# 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

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

• Note that $0$ is not positive. Mar 19, 2011 at 15:08
• Also, I wouldn't agree that $R_+$ usually includes $\infty$. The extended real line is used only in certain areas. Mar 19, 2011 at 15:09
• I removed the set theory tag since this isn't a set theory question. Mar 19, 2011 at 15:09
• $[0,\infty)$ or if you want to work with the extended real line, $[0, +\infty]$. Mar 19, 2011 at 15:12
• @YuvalFilmus Do not forget that this is just an english convention. In France for example, we usually say that 0 is both positive and negative. I have often seen $\mathbb{R}^+$ for all positive/null numbers and $\mathbb{R}^{\ast +}$ for all strictly positive numbers. Jun 17, 2014 at 9:53

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

• The last objection makes no sense since one could simply use $\mathbb R_+^3$.
– Did
Oct 25, 2015 at 10:51
• Actually for $\mathbb R^+\times\mathbb R^+\times\mathbb R^+$ I'd write $(\mathbb R^+)^3$. The notation with comma doesn't look right to me. Jun 13, 2017 at 6:08
• Would that itwere so simple. In Probability with Martingales Williams tells me "Everyone is agreed that $\mathbb{R}^+$ is $[0,\infty)$. Oct 17, 2017 at 19:38

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

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

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.

• All the mathematicians I ever met , ( a lot), understood that $R^+$ meant the positive reals. Aug 25, 2015 at 20:20
• @user254665: Well, certainly it means the positive reals, but now ask them what they mean by "positive" :-) Seriously, I know mathematicians who mean "$\ge0$" and other who mean "$>0$". Aug 28, 2015 at 17:26
• Edit: I think that $\mathbb{R}^{+} \backslash \Bigl\{\left((\mathbb{R}^{+} \backslash \mathbb{R}_0^{+}) \cup (\mathbb{R}_0^{+} \backslash \mathbb{R}^{+})\right)\Bigr\} \cup \{1\}$ will be unambiguous :-) Jan 28, 2018 at 1:13

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 $a$.

• This definitely strikes me as nonstandard, at least in the U.S. I'd be curious to know where all this is used. (Not saying it's a bad notation, just never seen it in any texts of common mathematics publishers, for example.) Mar 19, 2011 at 18:55
• I learned this from my math prof who grew up in Canada. But yeah I've never seen it outside her notes, but it does make writing $\{ x \in R \mid x < a\}$ easier! Mar 19, 2011 at 19:00
• Interval notation does not per se fix the basic set. Mar 19, 2011 at 21:11
• @cardinal: I've seen it used many times in Europe (but rather as subscript: $\mathbb{R}_{\geq 0}$) and some people even write $\mathbb{Z}_{\geq 0}$ instead of $\mathbb{N}$ because the latter is ambiguous as to whether $0$ is in it or not. And of course all obvious variants such as $\mathbb{R}_{\lt t}$ and so on are also used. But certainly, interval notation is more common.
– t.b.
Mar 21, 2011 at 3:00
• @cardinal: I think I can confirm that to a certain extent. I'm pretty sure we exclusively used interval notation à la Bourbaki in elementary and high school in Switzerland (I had at least 6 math teachers at various places) and it is exclusively used in at least four elementary texts on (what we call) algebra in my bookshelf.
– t.b.
Mar 21, 2011 at 13:30

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

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

• And I learned in school that $\mathbb R^+ = (0,\infty)$, and $R_0^+ = [0,\infty)$. Well, except that we would have written those intervals as $]0;\infty[$ and $[0;\infty[$ … Jun 13, 2017 at 6:21

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

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: $$f: \mathbb R_{\mkern1mu+\mkern-1mu+} \rightarrow \mathbb R_{\mkern.5pt+\mkern-1mu+}$$.

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).