Someone recently asked me why a negative $\times$ a negative is positive, and why a negative $\times$ a positive is negative, etc.
I went ahead and gave them a proof by contradiction like so:
Assume $(-x) \cdot (-y) = -xy$
Then divide both sides by $(-x)$ and you get $(-y) = y$
Since we have a contradiction, then our first assumption must be incorrect.
I'm guessing I did something wrong here. Since the conclusion of $(-x) \cdot (-y) = (xy)$ is hard to derive from what I wrote.
Is there a better way to explain this? Is my proof incorrect? Also, what would be an intuitive way to explain the negation concept, if there is one?