For questions on teaching students about negative numbers. Both formally and intuitively.
Negative numbers are usually only defined within the context of the real numbers $\mathbb R$ or within a subset of real numbers. In any case defining negative numbers requires a totally ordered set with a neutral element.
We usually assume that we are dealing with numbers in $\mathbb R$. In that case we say:
A negative number is any number smaller than $0$. The opposite of a negative number is a positive number i.e. a number that is greater than $0$. The number $0$ is generally defined to be neither negative nor positive. Some people define $0$ to be both negative and positive. The former definition is however considered more usual.