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This seems to be (from what I've heard from various math people of various statures) a heated debate. One of my previous professors proclaimed very strongly that $0$ is not a natural number. Another recently said the same. But then there are people who say it should be. One very good logical reasoning given was that, if the natural numbers are supposed to be the "counting" numbers, then $0$ is useful in the sense that it denotes the lack of something.

I do realize that the natural numbers were defined prior to the discovery of the concept of "zero", but that shouldn't be a reason why we can't formally agree that it should be a natural number.

And for the cases where using $0$ breaks things, there's no reason why calculations can't explicitly omit $0$.

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marked as duplicate by Zev Chonoles, Ross Millikan, Andrés E. Caicedo, Asaf Karagila, Lord_Farin May 28 '13 at 22:37

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    $\begingroup$ It's up to you as long as you're clear what you intend. However, it makes sense that in reality you can't really count zero of something. Think about collecting apples -- you start with one, then two, etc. Zero is "artificial" in much the same way negative and complex numbers are. $\endgroup$ – oldrinb May 28 '13 at 22:11
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    $\begingroup$ Why can't you count zero of something? Zero. There: I just counted the blue unicorns in my office! $\endgroup$ – Mariano Suárez-Álvarez May 28 '13 at 22:12
  • $\begingroup$ @MarianoSuárez-Alvarez haha :-p there's no foolproof reason as to why it shouldn't be considered a natural number. It's all about convenience. I suppose it might have been considered "unnatural" to count something in absence. $\endgroup$ – oldrinb May 28 '13 at 22:13
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    $\begingroup$ I am surprised at "heated." In ancient Greek mathematics, $1$ was not considered to be a number either. I am more comfortable with indexing that starts at the natural number $0$. And in mathematical logic, $\mathbb{N}$ is generally considered to include $0$. $\endgroup$ – André Nicolas May 28 '13 at 22:15
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    $\begingroup$ @iostream007: Your teacher ought to know better. In mathematics natural number is a technical term, so arguing from what supposedly occurs in ‘nature’ is silly. Unfortunately, there are two competing definitions of the term, so one must always make clear which convention one is using. For me $0$ is a natural number; if I don’t want to allow $0$, I talk about positive integers. $\endgroup$ – Brian M. Scott May 28 '13 at 22:24