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So you, as the ancient philosopher in mathematics

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have concluded in order for the multiplication of positive and negative numbers to be

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consistent with everything you've been constructing so far

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with all the other properties of multiplication that you know so far

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that you need a negative number times a positive

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number or a positive times a negative to give you a negative number

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and a negative times a negative

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to give you a positive number and so you accept

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it's all consistent so far.. this deal does not make complete concrete

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sense to you, you want to have a slightly deeper institution than just having to accept its

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consistent with the distributive property and whatever else and so you try another

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thought experiment, you say "well what is just a basic multiplication way of doing it?"

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So if I say, two times

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three, one way to

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to conceptualize is basic multiplication is really repeating

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addition, so you could view this as two threes

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so let me write three plus three

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and notice there are two of them, there are two of these

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or you could view this as three twos, and so this is the same thing as

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two plus two plus two and there are

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three of them, and either way you can conceptualize

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as you get the same exact answer. This is going to be equal

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to six, fair enough!

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Now, you knew this before you even tried to tackle negative numbers.

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Now let's try to make one of these negatives and see what

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happens. Let's do two

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times negative

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three, I want to make the negative into a different color. Two times

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negative three.

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Well, one way you could view this is the same analogy

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here, it's negative three twice so it would be

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negative.. I'll try to color code it

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negative three and then another negative

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three or you could say negative three minus three

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or, and this is the interesting thing, instead of

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over here there's a two times positive three

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you added two, three times.

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But since here is two times negative three

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you could also imagine you are going to subtract two, three times

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So instead of up here, I could

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written two plus two plus two because this is a positive

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two right over here, but since we're doing this over negative three

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we could imagine subtracting two, three times, so this would be

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subtracting two (repeated)

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subtract another two right over here, subtract another two

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and then you subtract another two

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notice you did it, once again, you did it

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three times, so this is a negative three, so essentially you are subtracting

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two, three times. And either way, you can conceptualize

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right over here, you are going to get negative six

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negative six is the answer.

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Now, so you are already starting to feel better about this part right over here

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negative times a positive, or a positive times a negative

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is going to give you a negative. Now lets take to the really un-intuitive one

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and measure negative times a negative, and all of a sudden negatives kind of cancel

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to give you a positive. Now why is that the case? Well we can just build from

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this example right over here. Let's say we had

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a negative two, lets say we had

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negative two, let me do it a different color,

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let's say we had a negative two, I already used this color

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negative two times

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negative three.

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So now, we can d- actually I'll do this one first.

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Let's do multiplying something by negative three so we'll

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repeatedly subtract that thing three times whatever that thing is

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so now the thing isn't a positive two so the thing over

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here is a positive two but the thing we're going to subtract is a negative two

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So let me make it clear, this says we are going to subtract something

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three times, so we subtract something three times, so

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subtracting something (repeatedly) three times

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That's what this part right over here tells us

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and we'll do this, exactly three times

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Over here, it was a positive two we subtracted three times, now we're going to

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do a negative two, now we're going to do a negative two

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and we know from subtracting negative numbers, we already

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built this intuition that subtracting a negative is the same thing

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it's the same thing as adding a positive, and so this

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this is going to be the same thing as two plus two plus two and

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we're told once again, gives you a positive

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six, you can same use the same logic over here, now

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instead of adding negative three twice, really I could have written this as

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negative three as this example

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negative three

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negative three, and we added it

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we added it, now let me put a plus here to make it clear

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over here we added it twice, we added negative three

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two times, or here since we have a negative two, we're going to subtract

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to negative three twice, so we're going to subtract something

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and we're going to subtract something again, and that something is going to be

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our negative three, it's going to be our negative three, so

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negative, negative and put our three right over here

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and once again, subtracting negative three is like taking away

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someone's debt, which is essentially giving them money,

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this is the same thing as adding three plus three which is once again six. So now

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you, the ancient philosopher, feel pretty good. Not only this

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all consistent with all the mathematics you know

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the distributive property is also the property of multiplying something

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times something all these things you already know, and now

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this actually makes conceptual sense to you, this is actually very consistent with

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with your notations, your original notations, or one of the positive notations

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of multiplication which is as repeated addition