“Negative Times Negative Is Positive”, But Why?

A basic rule we all learn very early in our mathematical education is that
“a negative number times/multiplied by a negative number is a positive number”. We are instructed to repeat this so often it almost becomes a mantra for these early years of mathematical development. What we don’t learn, and what we are not taught until probably college level, is why.

I’m sure we all have inherent reasons that come to mind to justify this. I still recall a teacher telling the class that it was because “multiplying a negative by a negative is like flipping from positive to negative and back to positive on the number line”.

Moving forward, just to save repetition, we will refer to the statement:
“A negative number times/multiplied by a negative number is a positive number” as the claim.

Now, what if I told you that it did not have to be the case that the claim was true? That, in fact, we could instead have it that a negative number times a negative number was negative. This feels quite strange to say, as it goes against something so ingrained in us that it feels almost unnatural. We will justify exactly why this choice would have been unnatural below, however, it is worth noting that this is exactly what it was, a choice.