At first, the number zero looked like it was positive to me because positive numbers can be written with or without a plus sign to the left of them, but it's false. I was surprised when I heard that zero is neither positive nor negative, but it's still a number and it's still even. At least I know it's in between the positive and the negative numbers, so that must be why. Also, because zero is neither negative nor positive, it's also known as neutral. Am I on the right track? Also, is there such thing as ±0, since it's neutral? That's why I put the plus/minus sign there. Is this how zero is neither negative nor positive? I can see happy faces in your answers!

1Look: the law of trichotomy – ThePortakal Oct 29 '14 at 21:03

3Now, there are some ways in which $0$ is more like positive numbers than negative ways. For example, the square root of a positive number is another positive number. The square root of a negative number is an imaginary number (have you learned about those yet? it can be a very difficult concept to comprehend). But the square root of $0$ is $0$ itself, and the only other number that is its own square root is $+1$. – Mr. Brooks Oct 29 '14 at 22:29

Not to get carried away with closing for being "offtopic" or anything, but here's a very similar question: http://math.stackexchange.com/questions/26705/iszeropositiveornegative – Lisa Oct 30 '14 at 22:02

2If zero would be positive then $(1) \cdot 0$ would be negative... – N. S. Jun 07 '15 at 17:14
4 Answers
Would it make sense to adjust the definition of positive or negative so that one of them includes $0$? The following pretty theorems
 The product of a positive and a negative number is negative
 The product of two negative numbers is positive
 The product of two positive numbers is positive
would require ugly adjustments, for example:
 The product of a positive number and a negative number is negative, except when the positive number happens to be zero, in which case the result is zero, hence positive
The emergence of ugly theorems is often a hint that the definitions are bad (as in: not very useful  recall that definitions cannot be "wrong").
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2Worth noting: In French, 0 is both “positif” and “negatif.” To express English “positive” in French, say “strictment positif” or “positif et non nul.” https://fr.wikipedia.org/wiki/Nombre_positif – Steve Kass Oct 29 '14 at 21:30

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So much ground to cover, I'm going to try to address every part of your question, though not quite in the order you presented it.
How is it true that zero is neither a positive number nor a negative number?
Do you know about additive inverses? Define $f(x)$ to be the number such that $x + f(x) = 0$. Turns out that $f(x) = 1 \times x = x$. The additive inverse of a positive number is a negative number. For example, the additive inverse of 8 is $8$. The additive inverse of a negative number is a positive number. For example, the additive inverse of $\frac{3}{2}$ is $\frac{3}{2}$. We can say that $x \neq x$. Except if $x = 0$, in which case $1 \times 0 = 0$. This means that 0 is its own additive inverse. The point is that no positive number is its own inverse, nor does any negative number have this property either.
Hagen von Eitzen has already brought up the multiplicative perspective, but allow me to expand on his point. Consider the equation $ax = b$. Suppose I tell you exactly what $a$ and $b$ are. Can you determine what $x$ is?
 $a = 5$, $b = \frac{22}{7}$
 $a = \sqrt{43}$, $b = 3698$
Now let's make the game a little more difficult. I'll still tell you what $b$ is, but instead of telling you what $a$ is, I will give you hints.
 $a$ in an integer and $a < 0$, $b = 197$
 $a$ might be an integer and $a \geq 0$, $b = 0$
In the former, there are two possibilities for $x$. But in the latter, there are infinitely many possibilities. If $a \neq 0$, then $x$ must be 0. But if $a = 0$, then $x$ can be anything, including 0.
At first, the number zero looked like it was positive to me because positive numbers can be written with or without a plus sign to the left of them ... Also, is there such thing as $\pm0$
Think of 0 as the point of origin, and think of positive as meaning to the right and negative to the left (or viceversa, if you like). $4$ means you go 4 to the left, $+7$ means 7 to the right. So $0$ means you go 0 to the left, that is, you stay at the point of origin, and similarly for $+0$.
However, this reminds me about something I read somewhere about two's complement, which is used by almost all computers today. Without two's complement, 0 might have more than one representation as 0s and 1s inside a computer, including a bona fide $0$ that is different than $+0$. But I guess that's a digression into computer programming.
but it's still a number and it's still even.
Yes, it's still an integer, and it's still even. If $m$ and $n$ are both even, then $m  n$ is also even, correct? What if $m = n$? Then $m  n = 0$. For example, $12  10 = 2$ and $12  14 = 24$, so what is $12  12$?
Also, because zero is neither negative nor positive, it's also known as neutral.
Sure, you can call it neutral and people will understand what you mean, but this usage is hardly common.
I can see happy faces in your answers!
Not from me. I've had kind of a confused face on. I've been going back and forth beteween thinking this stuff should be obvious and thinking this stuff is taken for granted but shouldn't be.
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I've never heard it called "neutral," but if you must absolutely have an adjective for it, I suppose that's as good as any.
Think about your bank account. If the balance is negative, then that means you owe the bank money. If the balance is positive, then that means you have money with which to pay for goods or services. But if your bank account is exactly $\$0.00$, you don't owe the bank money but your don't have any money to spend. It's a good thing that you don't owe, but a zeroed out balance is neither positive nor negative.
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This is just a matter of definition.
If you define positive numbers as all the real numbers $x$ such that $x=x$, then $0$ is a positive number. If you define positive numbers as all the real numbers $x$ such that $x=x$ except $0$, then $0$ is not a positive number.
Added: Maybe I have to admit, sometimes it is not only a matter of definition, but also a matter personal preference.
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Well, absolute value _is_ zero or positive unless it has a negative sign in front of it to make it zero or negative. – Mathster Oct 29 '14 at 20:55

@Mathster yeah, but first of all, before using terms as "positive" or "negative", we should give them definitions, right? – Petite Etincelle Oct 29 '14 at 20:58

It does depend on how “positive” is defined, but I think it’s worth pointing out that no one (in any serious mathematical setting) actually defines “positive numbers” the first way you mention. In mathematics, there is only one meaning for “positive” as a property of real numbers, and zero is not positive. – Steve Kass Oct 29 '14 at 21:16

@SteveKass what is the only one meaning you are referring to? – Petite Etincelle Oct 29 '14 at 21:18

The meaning of “x is a positive number” for a real number $x$ is $x>0$. I should clarify (especially in light of Petite Etincelle’s username) that I’m answering this question in terms of Englishlanguage mathematics. In French, for example, “positif” does not mean what English “positive” means. https://fr.wikipedia.org/wiki/Nombre_positif – Steve Kass Oct 29 '14 at 21:21

@SteveKass Yeah I agree here using the order of real number is a good definition. – Petite Etincelle Oct 29 '14 at 21:26

2And for things other than numbers, "positive" may or may not include $0$. For example, a "positive operator" or a "positive linear functional" or a "positive measure" could be $0$. – Robert Israel Oct 29 '14 at 22:24

In math you can define anything to be anything, but at some point you need to step up and explain the worth of your arbitrary definitions. $0$ has some properties in common with the positive numbers, but also many not in common. – Lisa Oct 30 '14 at 22:04

@Lisa My answer is not profound but remains reasonable. But going from disliking to saying "define anything to be anything", isn't it too far away? – Petite Etincelle Oct 30 '14 at 22:21

@Petite Perhaps. It would be much crazier to define negative numbers as being all $x < 7$. But still, I doubt that we gain anything by considering $0$ to be a positive number. – Lisa Oct 30 '14 at 22:24

@Lisa Are you supposing that I am for considering $0$ to be a positive number? Btw, [probably you don't like to come to France...](http://fr.wikipedia.org/wiki/Nombre_positif) – Petite Etincelle Oct 30 '14 at 22:29

I don't like to come to France because the airports are awful and the restaurant waiters rude. Awkward math terminology by itself would not dissuade me. And it is awkward: why say "strictement positif" when you can just say "$x \geq 0$" and have that be easier to understand for the nonFrench speakers who might read your paper? – Lisa Oct 30 '14 at 22:34

@Lisa I'm not a French guy but with my limited knowledge in French, I have to correct you that "strictement positif" actually means $x>0$. France has had and is having and will have many great mathematicians, that's why it sticks to its own languages and others have to learn about it – Petite Etincelle Oct 30 '14 at 22:40

@Petite I admit I was wrong about that. In part because it helps me make my point. – Lisa Oct 30 '14 at 22:41

@Lisa Actually I am wondering where your confusion comes from? Is there a language for which "strictly positive"(if its corresponding term exists) means $x\geq 0$? Maybe you can enlighten me. – Petite Etincelle Oct 30 '14 at 22:44

Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/18300/discussionbetweenlisaandpetiteetincelle). – Lisa Oct 30 '14 at 22:48

1You both got offtrack. I was hoping to see someone explain: What do we gain from defining 0 as a positive number? – Oct 30 '14 at 23:03

1There *are* some things to be gained by defining 0 to be positive, but they are rather minor. For example, you can then say that the norm of a number in an imaginary quadratic field is always positive. But it would much easier to say that that norm is never negative. – Robert Soupe Oct 31 '14 at 02:14