Why a negative times a negative is a positive

Lets say you are an Ancient Philosopher who was building up mathematics who was building mathematics from the ground up And you already have a reasonable of what a negative number could or should represent and you know how to add and subtract negative numbers But now you are faced with a conundrum What happens when you multiply negative numbers? Either when you multiply a positive number times a negative number Or when you multiply two negative numbers So, for example You aren't quite sure what should happen if you were to multiply (and im just picking two numbers where one is positive and one is negative) What would happen if you were to multiply 5 times negative 3 You're not quite sure about this just yet You're also not quite sure what would happen if you multiply two negative numbers. So lets say negative two times negative 6 This is also unclear to you What you do know, because you are a mathematician, is however you define this or whatever this should be It should hopefully be consistant with all of the other properties of mathematics that you already know And preferably all of the other properties of multiplication That would make you feel comfortable that you are getting this right. and later we can think about other ways to get the intuition for what these might be allowed you to actually make sense but to make this make consistent with the rest of mathematics that you know, you go into a little bit of a thought experiment you say, well, what should five times three plus negative three equal well you already have a philosophy of adding negative numbers or adding positive numbers to negative numbers, you know negative three is the opposite of three, but you add three to negative three you're going to get zero, so this is going to be equal to five times zero based on how you already thought about adding a negative number to a positive, and anything times zero is going to be zero, so this expression right over here should be zero but you see, I want to multiply positive and negative numbers to be consistent with this distributive property so I should should be able to distribute this five and for math to be consistent, and math should be consistent, I should get the exact same answer, so let's distribute this five so we get five times three is going to write out as five times three let me write this multiplication sign, not this dot five times three, so i distributed there plus five times negative three i'll do that in yellow, five times negative three and this whole thing we just said should be equal to zero it should be equal to zero, well five times three those are two positive numbers, we should know what should should be, that is going to be fifteen now we get this thing, fifteen plus times whatever five times negative three is needs to be equal to zero in order to be consistent with all the other mathematics that we know, well what plus fifteen is going to be equal to zero, well the opposite of fifteen in order for this to be true, in order for this to be consistent with all the other mathematics we know this right over here needs to be equal to negative fifteen until you say five times negative three in order to be consistent with all other mathematics we know, needs to be equal to negative fifteen. That's also consistent with the intuition of adding negative three repeatedly five times, now look above above us slightly higher so you can see ideas of multiplying two negatives, but we can do the exact same product experiment. We want whatever this answer to be consistent with the rest of mathematics that we know so we can do the same product experiment. What would negative two times six plus negative six to be equal to. Well, six plus negative six is going to be zero. Negative two times zero, anything times zero, needs to be equal to zero, but then once again, we can distribute negative two times six so we get negative two times six, then plus negative two times negative six plus negative two times negative six, then once again all of this is going to be equal to zero, now based on the five experiment we just did, we said "well this needs to be equal to negative twelve" or we can view this as going to the six twice left direction on the number line which gets us to negative twelve or you could say repeatedly adding negative twos times six would also get you to negative twelve and now we also saw over here we want to multiply a positive and a negative we got the negative so this could be, you know, going to be equal to negative twelve so we have negative twelve plus whatever this business is going to have to be equal to zero (repeated) in order to be consistent with all the other mathematics that we know and so what plus negative twelve is going to equal to be zero Well, positive twelve plus negative twelve is going to equal to zero so this needs to be equal to positive twelve in order to be consistent with all the other mathematics we know so there we get the idea that this is going positive twelve. I'll leave you there and I'll see if I can make a few other videos that can also give you a conceptual understanding of why these are true