Does negative zero exist? [closed]

In the set of real numbers, there is no negative zero. However, can you please verify if and why this is so? Is zero inherently "neutral"?

Solution 1:

There is a negative $0$, it just happens to be equal to the normal zero. For each real number $a$, we have a number $-a$ such that $a + (-a)=0$. So for $0$, we have $0+(-0)=0$. However, $0$ also has the property that $0+b=b$ for any $b$. So $-0=0$ be canceling the $0$ on the left hand side.

Solution 2:

My thought on the problem is that all numbers can be substituted for variables. -1 = -x. "-x" is negative one times "x". My thinking is that negative 1 is negative 1 times 1. So in conclusion, I pulled that negative zero (can be expressed by "-a") is negative 1 times 0, or just 0 (-a = -1 * a).

Solution 3:

A common definition of negative is "less than zero". In this sense, zero isn't negative (nor positive for a similar reason).

But the opposite of zero is well defined: it is zero. The unary $+$ or $-$ operators can very well be applied to $0$, with no effect.

One can admit that zero has no sign. The $\text{sign}$ function is usually defined to be $-1,0$ or $1$.