A few blogs have discussed an intuitive explanation of why multiplying two negatives produces a positive. That would be an explanation which makes sense to a non-mathematician.
The most intuitive argument from those blogs, to my mind, was from The Math Less Traveled and talks about a negative as a reflection about zero on the number line (image of this at The Number Warrior).
Here is how I’d explain it:
The first question to be asked is: What is a negative value? Like what does really mean?
The answer: A negative value, let’s say , is exactly a value where . That’s formally an additive inverse.
Now let’s look at . If we can believe that then we’re in good shape.
What does mean? Per above, it means the number that, when added to , equals zero. And that is obviously .
I have previously been underwhelmed by Wolfram Alpha. But today I finally saw a real-life query produce good results.
Admittedly, I am probably in the minority of people excited by this: I was wondering how many megawatt hours are produced by 8000 tons of oil. And what do you know, the query on Alpha worked.
Thanks, Wolfram researchers.