[ Algebra ] Double Negatives

In linguistics, double negatives are a no-no. Two wrongs do not make a right. Or do they? Well, if you ask a mathematics teacher, this might be the case. For example, -(-2) = 2. But why? Why? Why? Tell me why! Did it drop from the sky?

Let me try to explain this. Imagine Mr Sandman at the beach with his shovel. Or let us call a spade a spade. Anyway, he can build castles in the air, or at least sand-castles in the sand. With sand, he can literally make mountains out of molehills. For example, if he makes two sand mounds, that is “2”. He can also dig holes. Starting from flat ground “0”, if he takes away 2 mounds’ worth of sand, that means he is digging two holes. So the result is “-2”.

Now imagine there is this Mr Upside-Down guy living in his upside-down world underground. To him, our holes are his “mounds”. Our mounds are his “holes”. Mr Upside-Down calls “-2” two “mounds”, because everything to him is the opposite of what we call it. So what is - (-2)? That is what we get if we asked Mr Upside-Down to dig two of his “holes”.

From the diagram (turn it upside down if you cannot imagine it), when Mr Upside-Down digs two “holes”, he is actually creating 2 of our mounds. That is why - (-2) = 2. In general, -(-x) = x.

You see? Cool, eh?