Negative times Negative: A Deep Dive into Understanding It

It's an age-old question: Why is negative times negative positive? And school mathematics is not prepared to give a proper justification as the proclivity there, for fine reasons perhaps, is to seek real world models that "explain" math. (Where in the real world do you actually multiply two negative numbers together? "If I am traveling at a negative velocity and go backwards in time, then I've moved a positive distance forwards." Got that?)
Here's the math that leads us to say that negative times negative is positive. It all just hinges on one belief about how numbers work.

[@CylasConway]

James, I am a GED Instructor and your explanations are so helpful to get to the principals behind mathematics and explain those to my students! Thank you!

[@harrosheehan]

I highly recommend Professor Tanton's courses available on "The Great Courses". His Geometry one is amazing. Also, his guidebook on combinatorics made me stop hating the subject. Would love to know if there are any plans for releasing material for higher mathematics (college level)

[@IRVINMILLER]

Dear Jim, When I learned about negative numbers from a geometric point, I thought of the negative rectangles, as take aways from the original big rectangle. However, there was an overlap region which was the product of negative numbers which equaled the overlap region. If the area was positive it we compensate for the region being taken away twice. Your doing the problems two different ways plus the one I just described is what gives us confidence in what seems to be absurd in mathematics. Still another way of showing two negatives make a positive is to used patterns like: -2x2=-4 -2x=0 -2x-1=2 -2x1=-2 -2x-2=4. By the way you have one an assumption in your algebraic proof which is assumed rather than proved. Regards, Irvin

[@barthennin6088]

Excellent! Would be interesting to see what happens if we do NOT assume the distributive rule.

[@Giannis_Krimitzas]

Thank you James

[@iguana1677]

This is fantastic.

[@TheButlerNZ]

I was always taught that 2 positives always make a positive... Yeah Right!

[@J0hnnySlns]

James Munsey sent me.