What are the pros/cons of the Natural numbers including 0? This question is posed to those primarily in Pre-Calculus / Secondary  education, but if you have anything interesting relating to your area of mathematics that'd be awesome to hear. Note, I'm a mathematics teacher, and my Masters in Mathematics was a few years ago... 
Since University I have always defined $ \mathbb{N} =\{ 0 ,1,2,3...\} $ but across curriculum I have taught they insist $ \mathbb{N} =\{1,2,3...\} $. I have a few questions (I know they are very mixed across expertise), but answers to any would be amazing): 


*

*Is there a preference at research level on how you define it or this dependent on what it is you are doing with the natural numbers?

*Is there a preference to how you define during University teaching?

*Is there a benefit to how we define it during secondary education? 


I can see benefits to discounting 0 for summations and sequences during secondary education. But also, I often see the set $ \mathbb{Z}^+ $ introduced, which at this level of study is treated the same as $ \mathbb{N} \setminus  \{0\} $ at this level. 
What are your explanations, advantages and disadvantages?
Sorry this question is a little vague. 
 A: In my experience: the matter of $0 \in \mathbb N$ depends on what you want to do with the set $\mathbb N$. If you want to use its algebraic properties, especially number theory, then usually $0 \notin \mathbb N$ is nicer. There is no particular reason to include it, and it will often create the need for special cases in theorems. Example: the fundamental theorem of arithmetic. Example: the equivalence between irreducibility and primality.
If you want to use the set $\mathbb N$ for counting things, either formally (the set of finite cardinalities) or more broadly (using it as an index set) then usually $0 \in \mathbb N$ is nicer. Example: if $0 \notin \mathbb N$, then the precise statement about finite dimensional vector spaces becomes "the dimension of a finite-dimensional vector space is a natural number or $0$". Example: writing $f, f', f'', f''', \ldots$ gets tedious quickly, so we write... $f, f^{(1)}, f^{(2)}, f^{(3)}$? Or do we write $f^{(0)}, f^{(1)}, f^{(2)}, f^{(3)}$?
A: As already mentioned, it depends on what you want to do with $\mathbb N$. For example:


*

*In the context of abstract algebra, where we are usually interested in studing operations on sets, it is good to include zero because then $\mathbb N$ will have an identity element for addition.

*In the context of analysis, where we often use sequences, it is good not to include zero because then the first term of $(x_n)_{n\in\mathbb N}$ will by $x_1$ (instead of $x_0)$, the second term will be $x_2$ (inastead of $x_1$), and so on.
What should be made clear to the students is that the choice has nothing to do with whether or not zero is considered "natural" as opposite of "unnatural/artificial". It is just a matter of mathematical convenience. In mathematics, the name of the things does not follow the same rules of the common sense: usually, the name does not tell what the thing is. For example, the "imaginary numbers" are as real as the "real numbers" and the "simple groups" are not simple at all (some of its examples are called "monsters").
Remark. This argument is due to the Brazilian mathematician Elon Lages Lima, published in 1982 in his text "Zero é um número natural?" [Is zero a natural number?] in the first edition of the "Revista do Professor de Matemática" [Mathematics Teacher Magazine].
