# 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. – Yuval Filmus Mar 19 '11 at 15:08
• Also, I wouldn't agree that $R_+$ usually includes $\infty$. The extended real line is used only in certain areas. – Yuval Filmus Mar 19 '11 at 15:09
• I removed the set theory tag since this isn't a set theory question. – Apostolos Mar 19 '11 at 15:09
• $[0,\infty)$ or if you want to work with the extended real line, $[0, +\infty]$. – cardinal Mar 19 '11 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. – ThR37 Jun 17 '14 at 9:53

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

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

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

• The last objection makes no sense since one could simply use $\mathbb R_+^3$. – Did Oct 25 '15 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. – celtschk Jun 13 '17 at 6:08
• Would that itwere so simple. In Probability with Martingales Williams tells me "Everyone is agreed that $\mathbb{R}^+$ is $[0,\infty)$. – Addem Oct 17 '17 at 19:38

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. – DanielWainfleet Aug 25 '15 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 "$>0$". – Hendrik Vogt Aug 28 '15 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 :-) – Kusavil Jan 28 '18 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.) – cardinal Mar 19 '11 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! – hwong557 Mar 19 '11 at 19:00
• Interval notation does not per se fix the basic set. – Raphael Mar 19 '11 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 '11 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 '11 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 $[0;\infty[$ … – celtschk Jun 13 '17 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 writing functions $f$ defines as $f:\mathbb R_{\geq 0}\rightarrow \mathbb R_{\geq 0}$).

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

I like $[0,\infty)$, but it can be awkward in certain settings such as $f:[0,\infty)\times (0,\infty)\rightarrow \mathbb [0,\infty)$ or

$$\left\{E\times[0,\infty)\times (0,\infty)\right\}$$

Instead I prefer $\mathbb{\bar R_+}$ for the nonnegative reals and $\mathbb R_+$ for the positive reals. This fits with the notion of closure in $\mathbb R$. (This might not suit those who regularly deal with the extended reals, but given that $\mathbb R$ is so standard, it seems natural to take the closure there.) The function $f: \mathbb{\bar R_+}\rightarrow \mathbb{\bar R_+}$ is then clear and reasonably compact. Moreover, $$\left\{E\times\mathbb{\bar R_+}\times \mathbb R_+\right\}$$ and $f: \mathbb{\bar R_+}\times \mathbb R_+\rightarrow \mathbb{\bar R_+}$ seem to be substantially easier to read than the interval versions above.

Consistency then dictates that $\mathbb Z_+$ denotes the positive integers and whilst $\mathbb {\bar Z_+}$ is arguably unsatisfactory notation for the nonnegative integers because the closure story no longer applies, I would adopt it in order to be consistent. You could use $\mathbb N=\mathbb Z_+\cup\{0\}$, but that seems worse.

I guess it depends on the problem at hand.

ps. I have also seen $\mathbb R_{++}$ for the positive reals and $\mathbb R_+$ for the nonnegative.

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