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.

[@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)

[@sail2byzantium]

Alright! James Tanton has a KZbin channel! This was a very useful video on the logic of why the product of two negatives equal a positive. Love your work and it methodological underpinnings and have definitely subscribed.

[@joelvigne9505]

Great explanation... discovery of why the rule exists because it is the consequence of the other rules and beliefs.

[@barthennin6088]

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

[@joelvigne9505]

Yes I would love that video

[@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!

[@Giannis_Krimitzas]

Thank you James

[@ReidarVik]

Absolutely fantastic.

[@iguana1677]

This is fantastic.

[@icePalm00]

Great explanation

[@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

[@jayciheppler4902]

He's literally breaking down those equations into the boxes so we have visuals

[@brandonzx100]

You didn't learn, you memorized a concept about counting that doesn't apply to this situation.

[@jayciheppler4902]

Thank you for making me realize yall dont actually know the reason it just fits

[@chocolateangel8743]

I think about in terms of + and - numbers being on the opposite sides of a number line, which I guess forces you to come to the same conclusion. If 2 X (-3) = -6 (negative side of number line), introducing the opposite of 2 (-2), forces the solution to be on the opposite side of the number line.

[@aletikarthik2523]

They are just verifying whether it is correct or not

[@robertleboeuf9598]

The *approach* of FOIL always works. (a+b+c+d)(e+f+g)=ae+af+ag+be+bf, etc.

[@TheButlerNZ]

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

[@CecilyP-c7m]

Sooooooo you’re talking about calculus?

[@J0hnnySlns]

James Munsey sent me.

[@CecilyP-c7m]

Cause this is still not true