Does excluding or including zero from the definitions of "positive" and "negative" make any consequential difference in mathematics? I was absolutely certain that zero was both positive and negative. And zero was neither strictly positive nor strictly negative.
But today I made a few Google searches, and they all say the same thing: zero is neither positive nor negative.
I suppose that the definition of "positive" and "negative" depend on which country we're living in. In the U.S. "positive" and "negative" exclude zero. In France "positive" and "negative" include zero.
My question therefore is: does excluding or including zero from the definitions of "positive" and "negative" make any consequential difference in mathematics?

When multiplying value A by a positive value B, the sign of the result is identical to the sign of A:


*

*If A is positive, then the result is positive

*If A is negative, then the result is negative



When multiplying value A by a negative value B, the sign of the result is opposite to the sign of A:


*

*If A is positive, then the result is negative

*If A is negative, then the result is positive



Let's prove by contradiction that $0$ is not positive:


*

*Assume that $0$ is positive

*$-1$ is negative

*Therefore $(-1)\cdot0$ is negative

*But $(-1)\cdot0=0$, and $0$ is positive



Let's prove by contradiction that $0$ is not negative:


*

*Assume that $0$ is negative

*$-1$ is negative

*Therefore $(-1)\cdot0$ is positive

*But $(-1)\cdot0=0$, and $0$ is negative

