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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.

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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-Alvarez Aug 6 '10 at 9:05
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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 6 '10 at 9:17
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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 6 '10 at 10:20
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Let @zar post it as an answer and we are done. :) –  Pratik Deoghare Aug 6 '10 at 13:38
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I tend to use $\mathbb{R}_{> 0}$ or $\mathbb{R}_{\geq 0}$ and avoid that notation altogether. –  Andrea Ferretti Aug 6 '10 at 14:22
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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.

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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.

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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.

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Mh, here in Italy when we say "positive" we mean "$>0$", not "$\ge 0$". –  zar Aug 6 '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 6 '10 at 17:35
    
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 6 '10 at 19:14
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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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In my first semester of graduate school (in the U.S.), my algebra professor told us on the first day that $\mathbb{N}$ includes zero and $\mathbb{Z}_+$ does not. That afternoon, my analysis professor told us the exact opposite. How tedious! I prefer to say "positive" or "nonnegative" rather than use these symbols, but have also come to appreciate $\mathbb{Z}_{\geq 0}$ and $\mathbb{Z}_{>0}$ where brevity is desired. –  Jonas Meyer Aug 27 '10 at 23:06
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I write, e.g., $\mathbb R_{>0}$, $\mathbb R_{\geq0}$, $\mathbb N_{>0}$.

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I met (in IBDP programme, UK and Poland) the followin 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 set of positive reals and $0$ is neither positive nor negative.

$\mathbb{N}$ is possibly a sligthly different case and it usually differ from branch of mathematics to branch of mathematics. I belive that is usually includes $0$ but I belive 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.

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