Does set $\mathbb{R}^+$ include zero?

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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

Answer 4

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

Answer 5

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.