Most of us have been taught (or indoctrinated, to be honest) to accept that a negative number multiplied by a negative number yields a positive number, but would be hard-pressed to explain why. Even adults and teachers. Traditionally, educationists have appealed to extrapolation from a pattern e.g.
3 × -2 = 6,
2 × -2 = 4,
1 × -2 = 2,
0 × -2 =0,
-1 × -2 =-2, (extrapolating)
-2 × -2 =-4, (extrapolating)
-3 × -2 =-6, (extrapolating)
This is fine and dandy, but it does leave a slightly bitter after-taste. As a good pupil, I understood it whan I was learning this stuff, but I did not quite like it. Most of the other pupils would be struggling with the concept of negative numbers. They give up trying to understand it, and teachers give up explaining it. Pupils just accept the idea, because the teacher says so.
But ... I was wondering: Is there a more visual way to do the indoctrination ... i mean ... er ... education? Yes! There is!
Imagine an array with 3 columns of 2 circles spinning clockwise (indicated by arrows) on a piece of green paper (refer to the green rectangle on top right side of the diagram above). This represents 3 × 2 = 6.
What if you want -3 × 2? Upon reflection (no pun intended), you can flip that green rectangle along the vertical axis and get 3 columns of 2 circles. Let us imagine that the obverse side of the paper is coloured pink, and that the ink from the marker that drew the spinning circles could seep through to the other side of the paper. As you use the marker to go over the spinning circles, do you notice something? The circles are now spinning anti-clockwise (or counter-clockwise for those of you speaking Americanese)! Anti-clockwise is bad! It is negative. So this is negative six (-6).
If you flipped the original green rectangle along the horizontal axis, you would also get a pink rectangle with the circles spinning anti-clockwise. So 3 × -2 = -6.
Now if you flip vertically (upside-down) from the pink -3 × 2 rectangle, or flip horizontally (left-right inverted) from the 3 × -2 pink rectangle, you will get the -3 × -2 rectangle, which is green and the circles are spinning in a clockwise manner. That means it is positive! Therefore -3 × -2 = 6!
Fantastic! This is my own spin (literally) on why Negative times Negative is positive. In case you are wondering whether this is quackery, let me tell you: it is not. This is based on insights from transformation geometry. Any sort of reflection has a negative determinant which changes the clockwise/anti-clockwise rotation sense of an object. I am just applying this insight to provide my visual explanation.
I hope you have enjoyed this presentation, and learned something useful and interesting!
P.S.: The colorful diagram in this article my original artwork. The theory of transformation geometry is "public knowledge", but as far as I know, no one else presents multiplication of negative numbers in the visual manner as I have done above.