Why is (-1)*(-1)=1?

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Even though this question seems trivial to most people with some mathematical expertise, it actually cuts quite deep into an intuitive understanding of what multiplication truly is. When one writes an expression like "5*2=10" what they really mean is that if I take two piles of 5 apples (or 5 piles of 2 apples) I get a total of 10 apples, but sadly this intuitive approach doesn't really generalize well to negative numbers. What does it mean to take (-1) of some object? To intuitively understand what multiplication is a different approach is required. Imagine I take a meter-stick with length 5 and scale it by a factor of 2. I clearly get a new meter-stick that has a length of 10. I could also scale it by something like a factor of pi and get another meter-stick with some length, so clearly this approach doesn't rely on the fact that the number is an intiger. In this case multiplication by a negative number would simply reverse the direction of the meter stick. If it was initially pointing to the right (the positive number direction), after multiplying it by -1 it would simply point in the opposite direction. Multiplying by -2 would just be a combination of two operations: first of all we would scale the meter stick by a factor of 2 and follow the scaling by an inversion. So, clearly, if we start with the number -1, which is simply a meter-stick of length one in the negative direction, and multiply it by -1, we invert it and get a meter-stick in the positive direction, which also has length 1. Hence (-1)*(-1)=1

Rudolfs T. IB Physics tutor, IB Maths tutor

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