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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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    @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.2010-11-03

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

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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.

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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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    It is not mistaken: what N denotes is a convention. I prefer not to include zero in N.2018-02-09
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I write, e.g., \mathbb R_{>0}, $\mathbb R_{\geq0}$, \mathbb N_{>0}.

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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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    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.2010-08-23
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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.

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    Doesn't number theory need zero on the contrary?2018-11-22
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If I remember correctly the convention in economics is that $\mathbb{R}^+$ includes zero and $\mathbb{R}^{++}$ is "strictly positive" (does not contain zero).