# $\mathbb{Z}^{+}$ includes zero or not?

Does $\mathbb{Z}^{+}$ includes zero or not? I think that $0$ is not involved in the set of positive integers, but my book included zero in the set of positive integers in an answer.

• You can count it as the book's authours' mistake, zero is not a positive integer. Jun 27 '14 at 9:02
• Oh, so basically when we define integers, we can categorise the set of integers into 3 sub-categories : \begin{align} & \bullet \mathbb{Z}^{+} \implies \text{Positive Integers only.} \\ & \bullet \mathbb{Z}^{-} \implies \text{Negative Integers only.} \\ & \bullet 0 \implies \text{Zero only.} \end{align} is this right? Jun 27 '14 at 9:03
• If I was decorating $\mathbf{Z}$ to describe the nonnegative integers, I would write $\mathbf{Z}^{\geq}$ or $\mathbf{Z}^{\geq 0}$. It is a quirk of English language (or at least, English usage) that the word "positive" is not well-defined on whehter or not it includes $0$.
– user14972
Jun 27 '14 at 9:04
• While I consider non-negative and positive are actually different terms. Jun 27 '14 at 9:06
• @Kusha: I agree that should be true, but one must be aware of how the word "positive" gets actually used, because one will encounter people who use the word positive when they mean nonnegative.
– user14972
Jun 27 '14 at 9:07

A number $x$ is defined to be positive if $x > 0$. Is $0 > 0$? No, so it is non-positive (and it is also non-negative).
$\mathbb Z^+$ is a notation, so it is difficult to argue about it, because some authors do use non-standard notation and it's alright as long as they're consistent. But $\mathbb Z^+_0$ is a better notation for the set of non-negative integers.
• @KushashwaRaviShrimali Adding to the confusion, some textbooks include $0$ in the set of natural numbers $\mathbb N$, whereas others don't, and use $\mathbb N_0$ for $\mathbb N \cup \{0\}$. Personally, I don't include $0$ in $\mathbb N$, and this also seems to be the convention followed by the majority. Jun 27 '14 at 9:20
• Yeah, well, Natural Numbers are basically defined as $$\{ 1,2,3, \dots\}$$ Jun 27 '14 at 9:30
• @Kushahwa: In many places the natural numbers do include $0$. This is why they are usually denoted by $\Bbb N$ and not by $\Bbb Z^+$. Jun 27 '14 at 9:39
• @M.Vinay: It is easy to judge the "majority" if you haven't seen it. I could claim the same, since in my field $0$ is a natural number. So in all the books, and all the papers and all the classes... $0$ was always a natural number. Jun 27 '14 at 9:40
As some of the answers here show, in some languages the term "positive" may include $0$. In particular, it might be expected that authors whose native tongue is such a language may include $0$ in $\Bbb Z^+$.
If the book is consistent with this definition, then there's no real issue here. If the book suddenly becomes inconsistent with this definition (e.g. the author writes $\frac1n$ for $n\in\Bbb Z^+$) then it is likely a typo.