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I've been trying to find answer to this question for some time but in every document I've found so far it's taken for granted that reader know what $\mathbf ℝ^+$ is.

Carl Mummert
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AndrejaKo
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    It depends on the choice of the person using the notation: sometimes it does, sometimes it doesn't. It is just a variant of the situation with $\mathbb N$, which half the world (the mistaken half!) considers to include zero. – Mariano Suárez-Álvarez Aug 06 '10 at 09:05
  • @Mariano Suárez-Alvarez I was afraid of that... – AndrejaKo Aug 06 '10 at 09:15
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    It is just as Mariano says. In this case, the "correct" convention is that it should not include zero (after all, zero is not positive), but you certainly can't count on this: about half the time, the author means to include $0$. – Pete L. Clark Aug 06 '10 at 09:17
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    You will often find $\mathbf R^+$ for the positive reals, and $\mathbf R^+_0$ for the positive reals and the zero. – zar Aug 06 '10 at 10:20
  • Well, I just found out that in my particular case comment made by @zar is the answer. Anyway, how should we close this question? Comment made by @Mariano Suárez-Alvarez seems to be the answer in general case... – AndrejaKo Aug 06 '10 at 13:33
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    Let @zar post it as an answer and we are done. :) – Pratik Deoghare Aug 06 '10 at 13:38
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    @Mariano: Why not post your comment as an answer. That way it can be accepted. =) – Jens Aug 06 '10 at 13:47
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    I tend to use $\mathbb{R}{> 0}$ or $\mathbb{R}{\geq 0}$ and avoid that notation altogether. – Andrea Ferretti Aug 06 '10 at 14:22
  • @Andrea Ferretti That's the most unambiguous way, but my question came from a problem on an exam. It was something like "what kind of algebraic structure is ($\mathbf R^+$,), where is blah, blah blah". The answer depended on zero belonging or not in $\mathbf R^+$. – AndrejaKo Aug 06 '10 at 14:44
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    Just use (0, +∞). Problem solved. (Oh ]0, +∞[?) – kennytm Aug 06 '10 at 17:26
  • Other notations:

    $\mathbf{R}_{+}$ for ${x\in\mathbf{R}:x\geq 0}$

    $\mathbf{R}_{+}^{\times }$ for ${x\in\mathbf{R}:x>0}$

    (An Introduction to the theory of the Rieman Zeta-Function by S. J. Patterson)

     – Américo Tavares Aug 23 '10 at 21:48
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    To add to the confusion, I've seen it used to indicate the group given by $\mathbb{R}$ under addition. – Paul VanKoughnett Nov 03 '10 at 00:52
  • @Paul, I have seen that as well, but more commonly see $\mathbb{R}_{+}$ for group under addition (subscript vs superscript). But I have also seen that represent the set of positive reals, so in the end it doesn't help much. – bzc Nov 03 '10 at 01:05

6 Answers6

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You will often find $ \mathbb R^+ $ for the positive reals, and $ \mathbb R^+_0 $ for the positive reals and the zero.

Harsh Kumar
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zar
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It depends on the choice of the person using the notation: sometimes it does, sometimes it doesn't. It is just a variant of the situation with $\mathbb N$, which half the world (the mistaken half!) considers to include zero.

Mariano Suárez-Álvarez
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I write, e.g., $\mathbb R_{>0}$, $\mathbb R_{\geq0}$, $\mathbb N_{>0}$.

Christian Blatter
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As a rule of thumb most mathematicians of the anglo saxon school consider that positive numbers (be it $\mathbb{N}$ or $\mathbb{R}^{+}$) do not include while the latin (French, Italian) and russian schools make a difference between positive and strictly positive and between negative and strictly negative. This means by the way that $0$ is the intersection of positive and negative numbers. One needs to know upfront the convention.

marwalix
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    Mh, here in Italy when we say "positive" we mean "$>0$", not "$\ge 0$". – zar Aug 06 '10 at 17:10
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    Hmm. As above, for me $\mathbb{R}^+ = (0,\infty)$, but $\mathbb{N}$ includes zero. Hence I wouldn't call the latter the set of positive integers: for that I use $\mathbb{Z}^+$. Seems logical... – Pete L. Clark Aug 06 '10 at 17:35
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    As well in France the anglo saxon school is taking over, but if you read Italian mathematicians of the beginning of the 20 th century you will realise that like Bourbaki slightly later in France positive included zero – marwalix Aug 06 '10 at 19:14
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    Well, since we're sounding off countries, here in Serbia we have ℕ and ℕ_0 and use terms such as nonnegative and nonpositive to show if set includes zero or not. Also we don't have word integer so we use "whole number" instead. – AndrejaKo Aug 06 '10 at 22:35
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    Early editions of Bourbaki indeed defined zero to be both positive and negative, but by the 1930s even Bourbaki changed their mind.

    As a general rule, $\mathbb{N}$ excludes zero if and only if you are a number theorist.

     – JeffE Aug 23 '10 at 19:54
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I met (in IBDP programme, UK and Poland) the following notation:

\[\mathbb{R}^{+} = \{ x | x \in \mathbb{R} \land x > 0 \} \]

\[\mathbb{R}^{+} \cup \{0\} = \{ x | x \in \mathbb{R} \land x \geq 0 \} \]

With the explanation that $\mathbb{R}^{+}$ denotes the set of positive reals and $0$ is neither positive nor negative.

$\mathbb{N}$ is possibly a slightly different case and it usually differs from branch of mathematics to branch of mathematics. I believe that is usually includes $0$ but I believe theory of numbers is easier without it. It can be easilly extended in such was to have $\mathbb{N}^+ = \mathbb{Z}^+$ denoting positive integers/naturals.

Of course, as noted before, it is mainly a question of notation.

Maja Piechotka
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R+ includes only positive real numbers.As 0 is neither positive nor negative,hence it is not included in R+

Swatantra Singh Rathore
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