# What does $\mathbb Z_+$ mean?

I am not so sure whether the meaning of $$\mathbb Z_+$$ is very clear. How many different definitions are there? Does the definition that is used depend on whether the writer is English or German?

In French maths, this notation doesn't exist.

• I think that we use the notations : $\mathbb{Z}=\{\ldots,-2,-1,0,1,2,\ldots\}$, $\mathbb{Z}_+=\{0,1,2,\ldots\}$ and $\mathbb{N}=\{1,2,\ldots\}$ – Thibaut Dumont Jun 7 '13 at 13:55
• It depends on the context. A guess is that it means the additive group of $\Bbb Z$, i.e. $\Bbb Z$ when one considers the addition only. – Andrea Mori Jun 7 '13 at 13:56
• @DamienL, alas there is doubt. – vadim123 Jun 7 '13 at 14:00
• @DamienL Not true. I define $\mathbb{N} = \varnothing$, then $0\not\in\mathbb{N}$. My point is that you can't argue in general without stating your definition. BTW, there are people who define $\mathbb{N} = \{1,2,\cdots\}$ – mez Jun 7 '13 at 14:08
• @vadim123: In fact $0\in\Bbb N$ is a counterexample to the principle of the excluded middle: it is a proposition that contains no variables, yet it is neither true nor false ;-( – Marc van Leeuwen Jun 7 '13 at 14:38

Many people would interpret this to mean $\{1,2,3,\ldots\}$, although some might argue for $\{0,1,2,3,\ldots\}$. Absent any other context I don't think any other interpretations are likely.

Sadly, many authors use notations without defining them, because they are "standard" in their little corner of mathematics.

• That would more likely be $\Bbb Z_{>0}$ or $\Bbb Z_{\geq0}$, but everything is possible! – Andrea Mori Jun 7 '13 at 13:58
• Some people still uses : $\mathbb{R}_+=[0,\infty)$ for $\mathbb{R}_{\geq 0}$. – Thibaut Dumont Jun 7 '13 at 14:05
• Which causes my stupefaction about how people could "resolve" the ambiguity of the symbol $\Bbb N$ with another one that raises exactly the same problem! – Marc van Leeuwen Jun 7 '13 at 14:41

Symbol "$+$" means "positive", so $\mathbb{Z}_+$ shoud properly be understood as $\{1,2,3,\ldots\}$. It's less confusing that undefined $\mathbb{N}$ in some paper, where we don't know if it includes $0$.

• It's interesting because in french "positif" means non-negative. So it needs context probably. – Thibaut Dumont Jun 7 '13 at 14:07
• Actually $+$ means addition, not positive. If $n<0$ then $+n$ is (also) negative while $-n$ is positive. Therefore $\Bbb Z_+$ is an abomination that only makes the embassement of not being able to assign a uinque truth value to $0\in\Bbb N$ even worse; it should have been $\Bbb Z_{>0}$ (or even better $\Bbb N_{>0}$), which would never have raised the current question. – Marc van Leeuwen Jun 7 '13 at 14:36
• It depends on context. For example, when we are talking about electric fileds, positive electric charge is labaled by "$+$", and negative by "$-$" (not by $">0"$ nor $"<0"$). Thus I don't see anything wrong in labeling positive numbers by "$+$". This example could be used also as a respond to @Thibaut Dumont: I think that putting "$0$" as a positive number (like in french notation) is not too good ($0$ is not positive nor negative; it's neutral). – Bartek Pawlik Jun 7 '13 at 15:38

For the French notation $\mathbb N=\{0,1,2,3,\ldots\}$ and $\mathbb N^*=\{1,2,3,\ldots\}$ and and the notation $\mathbb Z_+$ is not used in general but it means $\mathbb N$.

• In the U.S. it is far from standard but common is $\mathbb{N}=\{1,2,3,\ldots\}$ and $\mathbb{N}_0=\{0,1,2,\ldots\}$. – vadim123 Jun 7 '13 at 14:15