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!
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 positve 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, e.g.


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")
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.