Someone recently asked me why a negative × a negative is positive, and why a negative × a positive is negative, etc.

I went ahead and gave them a proof by contradiction like so:

Assume (−x)⋅(−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)⋅(−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?

**Answer**

This is pretty soft, but I saw an analogy online to explain this once.

If you film a man running forwards (+) and then play the film forward (+) he is still running forward (+). If you play the film backward (−) he appears to be running backwards (−) so the result of multiplying a positive and a negative is negative. Same goes for if you film a man running backwards (−) and play it normally (+) he appears to be still running backwards (−). Now, if you film a man running backwards (−) and play it backwards (−) he appears to be running forward (+). The level to which you speed up the rewind doesn’t matter (−3x or −4x) these results hold true.

backward×backward=forward

negative×negative=positive

It’s not perfect, but it introduces the notion of the number line having directions at least.

**Attribution***Source : Link , Question Author : Sev , Answer Author : Community*