How is it true that zero is neither a positive number nor a negative number? 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!
 A: 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").
A: 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 vice-versa, 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.
A: 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.
A: 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. 
