Why is $\mathbb{Z}^{+}$ sometimes used to denote the natural numbers? This does not really make sense to me for several reasons: 


*

*The integers are usually constructed using a given construction of the natural numbers

*Historically natural numbers were conceived of before the integers

*The notation $\mathbb{Z}^+$, is more cumbersome to use than $\mathbb{N}$.


However, I see some good reasons to use this notation, for instance if the basic object of study is the integers, then (for some reason) one might want so signify that the natural numbers is a subset of the integers and hence use $\mathbb{Z}^+$.
What is the history here (if any)? Why do some mathematicians insist on using the notation? 
 A: I really think it is just a notation thing. For sure humans considered the natural numbers before we considered the negatives and even zero. 
Why I think it is a handy notation. There is no universal agreement on what we call the natural numbers, some people consider zero to be a natural number, while I have never had a book that did not start the enumeration of the naturals with the number one. So the notation $\mathbb{Z}^+$ rids us of this confusion.
A: There has always been some ambiguity as to whether $\Bbb N$, the natural numbers, includes zero or not due to inconsistent usage by past authors.
Where as using $\Bbb Z^+$, the positive integers, or $\Bbb Z^\ast$, the nonnegative integers, is unambiguous about the matter.
A: $\mathbb{Z}^+$ means "positive integers."
$\mathbb{N}$ means "natural numbers," which includes zero, no, wait, it doesn't, yeah, it does... Look at this page: http://oeis.org/wiki/Latin_alphabet On the rightmost column of the table, it says:

$\mathbb{N}$ The set of natural numbers, including 0[31], though sometimes also used to mean the same thing but excluding 0.[15]

The "[31]" is a citation for Princeton University Press book, the "[15]" is for a Springer-Verlag book. Both books were published less than a decade ago. Both books have tables of notation, because readers are almost certain to be confused as to whether the author means for $\mathbb{N}$ to include 0 or not.
I can't remember when was the first time I saw $\mathbb{Z}^+$. But if it wasn't in a table of notation, I'm sure I instinctively understood that it means "positive integers."
