Why is a Negative times a Negative a Positive?

[Retro]

We all (hopefully) know this and use this everyday, but do we all understand exactly why it's like this? This video explains it entertainingly in various ways using lots of squiggly math.

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[Mars]

To me, the clearest explanation was the first one. All the rest are also entertaining and of course, convincing.
The concept of a Negative number seems so totally absurd. Yet, step by step, the 'logic' of the multiplication of two negatives yielding a +ve product, seems totally rational and easily understood.
This is a clever and simple demonstration of a 'weird' concept.