Why do some people state that 'Zero is not a number'?

Every now and then I read about people who wonder whether zero is a number. It never occurred to me to question this, so I checked the Wikipedia page which, when talking about the Rules of Brahmagupta explains

In saying zero divided by zero is zero, Brahmagupta differs from the modern position. Mathematicians normally do not assign a value to this, whereas computers and calculators sometimes assign NaN, which means "not a number."

I did consider whether this difference in position may be the reason why some people state that "Zero is not a number".

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The question in the title differs from the question in the final paragraph. Which question are you asking? –  Bill Dubuque Nov 16 '12 at 16:56
@BillDubuque: Sorry for the imprecision; the question in the title is the one I meant to ask, the final paragraph was just me thinking out loud. I did adjust the question body accordingly. –  Frerich Raabe Nov 16 '12 at 17:12
Related: (In the sense that your question may be an answer to it, or vice-versa) math.stackexchange.com/questions/12323/… –  ShreevatsaR Nov 16 '12 at 17:15

A few distinctions:

In your first link, the discussion is regarding whether $0$ is a natural number, not that it is/isn't a number.

Your second link is predicated upon fairly weak, insubstantial first-year undergraduate arguments that can be dispelled with a more rigorous construction of numbers. I also question any blog that concludes a logical argument with "what's your opinion?" (I'd also question the veracity of a mathematical argument coming from a theological blog -- in the same way that I would question the veracity of a monetary policy argument coming from a sports blog).

Finally, the $0/0 = 0$ argument is considered non-standard. That most mathematicians define $0/0 = \text{NaN}$ is not the same as $0 = \text{NaN}$ because these are competing definitions.

So, in short, people often claim "zero is not a number" because they lack the background to understand the formal, rigorous definitions of the number system.

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Strictly speaking "NaN" is distinct from "is not defined". NaN comes from the specification for floating point values, which computers generally use as an approximation of the real numbers. NaN is a well defined value (actually several, as there are distinct flavors of NaN) with specific properties (such as being unequal to all floating point values, including itself). –  camccann Nov 16 '12 at 18:13
Yeah, I was a little loose on my terminology there, overloading NaN with both a computer science definition and an unrelated "not a defined number" definition. –  Arkamis Nov 16 '12 at 18:15
Yeah, that's part of the distinction I'm trying to make--the expression 0/0 has no defined value, i.e., the division operation is not defined on those arguments in the same way that division by "cucumber" is not defined. There is nothing denoted by 0/0 to be a number or not. –  camccann Nov 16 '12 at 18:21
Division by cucumber clearly leads to salad. –  Arkamis Nov 16 '12 at 18:22

Firstly, they're just definitions.

Secondly, its true that $0$ behaves differently to other numbers.

Here's a few examples.

1. Its never the case that $x+y$ equals $x$, unless $y$ is zero.

2. Its always the case that $xy = xy'$ implies $y=y'$, unless $x$ is zero.

3. As Michelle Lee said, certain procedures result in a sequence that will always converge to the golden ration $\phi$, unless the first term is $0$.

So, this is evidence that $0$ is special, which in turn suggests that it may be convenient to exclude $0$ from numberhood. Lets consider this possibility.

Well, suppose we did exclude it from being a number. Then we would notice some annoying things.

1. Its no longer true that the sum of two numbers is a number.

2. In particular, its no longer true that $x + -x$ is a number.

3. Its no longer clear how to even define $-x$.

etc.

Therefore, many mathematicians have thought about these kinds of things, and most have decided that "although $0$ is special in a variety of ways, its simply not convenient to define that its not a number."

Edit. It occurs to me that by "number" you might mean "natural number." There's certainly good algebraic reasons to exclude $0$ from being a natural number, however if you want every finite set to have a natural number of elements, then you had better include $0$. Thus, perhaps its best to have two different symbols available:

1. the usual $\mathbb{N} = \{0,1,2,\cdots\}$ which is useful in set theory, and generalizes naturally to the cardinal and ordinal arithmetics, and

2. the less typical $\mathbb{M} = \{1,2,3,\cdots\}$ which is algebraically a bit a nicer.

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