Why is negative times negative = positive?

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?


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

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

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