# 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

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

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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
• $[0, \infty)$.
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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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All the mathematicians I ever met , ( a lot), understood that $R^+$ meant the positive reals. –  user254665 Aug 25 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 2 days ago

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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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
@hwong557, interesting. I'd think $(-\infty, a)$ would be almost as compact. –  cardinal Mar 19 '11 at 19:04
Interval notation does not per se fix the basic set. –  Raphael Mar 19 '11 at 21:11
@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 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 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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The following is also pretty common notation for the non-negative reals: $\mathbb{R}_{\geq 0}$ or $\mathbb{R}_{+}$.

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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'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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