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.