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

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.

• 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 6 '10 at 9:05
• 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
• 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
• Let @zar post it as an answer and we are done. :) – Pratik Deoghare Aug 6 '10 at 13:38
• 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

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.

• Why is it mistaken to include zero in $\mathbb{N}$? It's the additive identity and the cardinality of the empty set. – user76284 Feb 9 '18 at 0:35
• It is not mistaken: what N denotes is a convention. I prefer not to include zero in N. – Mariano Suárez-Álvarez Feb 9 '18 at 1:07

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

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

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.

• Mh, here in Italy when we say "positive" we mean "$>0$", not "$\ge 0$". – zar Aug 6 '10 at 17:10
• 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
• 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 6 '10 at 22:35
• 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

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.

• Shouldn't the second line read $\mathbb{R}^+ \cup \{0\}$ or is it an abuse of set notation? – Reinstate Monica Jul 13 '15 at 15:18
• @Solomonoff'sSecret - fixed. Apparently you need double \ to escape { and }. – Maciej Piechotka Jul 13 '15 at 15:51
• Doesn't number theory need zero on the contrary? – Pavel Šimerda Nov 22 '18 at 23:42

If I remember correctly the convention in economics is that $\mathbb{R}^+$ includes zero and $\mathbb{R}^{++}$ is "strictly positive" (does not contain zero).

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