What makes negative numbers different from positive numbers other than their being (almost) opposite? To quote from Wikipedia's article on negative numbers 

Negative numbers represent opposites. If positive represents movement to the right, negative represents movement to the left. If positive represents above sea level, then negative represents below level. If positive represents a deposit, negative represents a withdrawal.

This leads one to think that anything one does to a negative number will result in opposite (or just opposite signed) results to that of its positive counterpart. 
This is obviously not true though because, for example $(-3)^2=9$ and $3^2=9$.

With that in mind, I ask, are negative numbers really the opposite of positive numbers? What makes them fundamentally different from positives in a way that leads to this discrepancy?


This leads one to think that anything one does to a negative number will result in opposite (or just opposite signed) results to that of its positive counterpart.

Well don't think that, because this is overgeneralizing.
It is "good" to pour a cup of water into the mouth of a thirsty man.  It is "evil" to punch him in the nose.  Good and evil are opposites.  However if you change both activities in the same way, they need not remain opposites. If you punch him in the nose 1000 times, that is still evil.  If you pour 1000 cups of water into his mouth, that is evil as well, as he will drown.
