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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\} \; ?$

  • 0
    I highly doubt that $\mathbb R^+$ would include infinity, given that infinity isn't even in $\mathbb R$.2015-08-06

8 Answers 8

0

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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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.


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

  • 1
    Would that itwere so simple. In *Probability with Martingales* Williams tells me "Everyone is agreed that $\mathbb{R}^+$ is $[0,\infty)$.2017-10-17
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Not that I knew of. There are many, e.g.

  • $\mathbb{R^+_0}$,
  • $\mathbb{R^+}$ and
  • $[0, \infty)$.
9

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.

  • 0
    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 :-)2018-01-28
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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 $a$.

  • 0
    @Raphael: One might fix that by using the base set as index where not obvious, like $(-\infty,a)_{\mathbb R}$ (no, I never have seen that notation).2017-06-13
4

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

2

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

  • 0
    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[$2017-06-13
0

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