One thing that I do not get clearly is why 2 negatives cancel each other out to then become a positive, let alone why 2 different signs become negative. What is the reason/logic behind that?
Here are a three different ways to think about why a negative times a negative equals a positive. The final maths example involved some algebra and is a little beyond what’s been covered in the course so far, but hopefully it’s an easy enough example to follow.
Let’s first compare it to how double negatives work in English.
I often use the example “If you don’t know nothing, then you must know something”.
Here we’re basically saying that the “don’t” and “nothing” are cancelling each other out, leaving us with a positive statement.
A great analogy I once heard (I forget where) was to imagine filming a moving object.
If it’s moving forwards (+a) and you play the film forwards (+b) it will appear to move forward (+a x +b = +ab) .
If you play the same film (+a) in reverse (-b) it will appear to move backwards (+a x -b = -ab).
Now, if the object is moving backwards (-a) and you play the film forwards (+b) it will appear to move backwards (-a x +b = -ab).
So if you take that film (-a) and play it in reverse (-b) the object will once again appear to be moving forwards (-a x -b = +ab).
Finally, let’s look at an actual maths example. It’s not a rigorous proof but if you were to look one up it would probably be near identical but with letters instead of numbers.
So let’s use the example of (-2) x (-4) = ?
We know that: (-2) x 0 = 0
and we also know that: 4 + (-4) = 0
So we could rewrite our original equation as:
(-2) x (4 + (-4)) = 0
Here we’re taking our first logical statement and substituting in the second.
Then, distributing what’s inside the parentheses we get:
(-2) x 4 + (-2) x (-4) = 0
In this expanded form, we notice that part of our original equation is back.
So now we really just need to tidy things up.
We know that:
(-2) x 4 = -8
Therefore:
(-8) + (-2) x (-4) = 0
Finally, we do some rearranging and move the -8 to the other side.
Note that subracting from one side means you need to add to the other, so:
(-2) x (-4) = 8
We thereby prove that in this example a negative times a negative equals a positive.
Like I said originally, this isn’t a rigorous proof. However, try following this process with any two negative numbers and you’ll always end up with a positive result.
I strongly encourage you to try this process for yourself with some other negative numbers and prove that it always holds.
Anyway, I hope this helped. If you want me to clarify anything, feel free to ask. 
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Thank you very much for the insight. I found the second explanation about filming a moving object helped me understand it better.
The third example took me a few more minutes to understand, so I kept researching on the Internet and found a good video which tackled this: https://www.youtube.com/watch?v=nJqyr8h22nY .
The second part of the video of the same series actually had an example like the third example with letters as you mentioned.
Many Thanks!
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Glad I could help.
Algebra hasn’t been covered in the course yet and the third example does require some basic knowledge of how to manipulate equations.
However, now that you understand it better, is there anything you think would make that example easier to follow?
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