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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$\begingroup$You will often find $ \mathbb R^+ $ for the positive reals, and $ \mathbb R^+_0 $ for the positive reals and the zero.
$\endgroup$ $\begingroup$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.
$\endgroup$ 2 $\begingroup$I write, e.g., $\mathbb R_{>0}$, $\mathbb R_{\geq0}$, $\mathbb N_{>0}$.
$\endgroup$ $\begingroup$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.
$\endgroup$ 5 $\begingroup$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.
$\endgroup$ 3 $\begingroup$If I remember correctly the convention in economics is that $\mathbb{R}^+$ includes zero and $\mathbb{R}^{++}$ is "strictly positive" (does not contain zero).
$\endgroup$ $\begingroup$R+ includes only positive real numbers.As 0 is neither positive nor negative,hence it is not included in R+
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