How do you explain -1*-1=+1 (without using velocity,
direction and time)
Multiplication by -1 is reflection through the origin. A number
becomes its counterpart on the other side of zero. Clearly, if you
reflect a number twice through the origin (multiply twice by -1), then
that is the same as doing nothing at all (multiplying once by 1).
My guess is that this is a concept, a convention, an assumption, or an
agreement, based on which other relationships can be expressed and derived.
I recommend looking into a reference book on the history of mathematics to
learn about its invention and justifications.
Dr. Ali Khounsary
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Update: June 2012