Why a Negative Times a Negative

Let's say that you are an ancient philosopher who is building up mathematics from the ground up and you already have a reasonable understanding of what a negative number could or should represent and you know how to add in 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 I'm just picking two numbers where one is positive and one is negative. What would happen if you were to multiply five times negative three? You're not quite sure about this just yet. You're also not quite sure about what would happen if you multiply two negative numbers. So let's say negative two times negative six. 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 consistent 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 and why it actually makes sense. But to make this consistent with the rest of the mathematics that you know you and you go into a little bit of a thought experiment you say well what should five times three plus negative three B. Well you already have a philosophy of adding negative numbers or adding positive to negative numbers. You know that negative three is the opposite of three. If you add three to negative three you're going to get zero. So this is going to be equal to this is going to be equal to five times zero. Five times zero based on how you already thought about adding a negative to a positive and anything times zero is going to be zero. So this expression right over here should be zero. But on the other hand you say well I want I want multiplying positive and negative numbers to be consistent with the distributive property. So I should be able to distribute this five. I 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 if we get five times three so five times three is going to be right it out. This is going to be five times three. I let me write it with a multiplication not the dot. I'll write the x sign for multiplying five times three. So I distributed there plus five times negative three and I'll do that in yellow. Five times negative three. Five times negative three and this whole thing we just said should be equal to zero. This should be equal to zero. Well five times three those are two positive numbers. We know what that should be. That is going to be 15. So now we get this thing 15 plus plus whatever five times negative three is plus whatever five times negative three is needs to be equal to zero in order to be consistent with all of the other mathematics that we know. Well what plus 15 is going to be equal to zero? Well the opposite of 15 in order for this to be true in order for this to be consistent with all of the other mathematics we know this right over here needs to be equal to negative 15. And so you say five times negative three in order to be consistent with all the other mathematics we know needs to be equal to negative 15. And that's also consistent with the intuition of adding negative three repeatedly five times. A slightly harder to conceive idea is multiplying two negatives. But we can do the exact same thought experiment we want whatever this answer to be to be consistent with the rest of mathematics that we know. So we can say so we can do the same thought experiment. What would negative two times six plus negative six 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. We can distribute negative two times six so we get negative two negative two times six plus plus negative two times negative six plus negative two times negative six and once again all of that's going to need to be equal to zero. Now, based on the thought experiment we just did, we said, well, this needs to be equal to negative 12. Or we could view this as going to the 6 twice in the left direction on the number line, which would get us to negative 12. Or you could say repeatedly adding negative 2, 6 times. That would also get you to negative 12. And now we also saw it over here that we multiplied a positive times in negative. We got to negative. So this could be, or we know that this is going to be equal to negative 12. And so we at negative 12 plus plus whatever this business is, whatever this business is, is going to have to be equal to 0. It's going to have to be equal to 0. In order to be consistent with all of the other mathematics that we know. And so what plus negative 12 is going to be equal to 0? Well, positive 12 plus negative 12 is going to be equal to 0. So this needs to be equal to positive 12. And ordered be consistent with all of the other mathematics we know. And so there we get the idea that this is going to be positive 12. 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.