Why a Negative Times a Negative Makes Intuitive Sense

Using the repeated addition model of multiplication to think about multiplying negative numbers

So you, as the ancient philosopher in mathematics

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

consistent with everything you've been constructing so far

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

that you need a negative number times a positive

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

and a negative times a negative

to give you a positive number and so you accept

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

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

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

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

So if I say, two times

three, one way to

to conceptualize is basic multiplication is really repeating

addition, so you could view this as two threes

so let me write three plus three

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

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

two plus two plus two and there are

three of them, and either way you can conceptualize

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

to six, fair enough!

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

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

happens. Let's do two

times negative

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

negative three.

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

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

negative.. I'll try to color code it

negative three and then another negative

three or you could say negative three minus three

or, and this is the interesting thing, instead of

over here there's a two times positive three

you added two, three times.

But since here is two times negative three

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

So instead of up here, I could

written two plus two plus two because this is a positive

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

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

subtracting two (repeated)

subtract another two right over here, subtract another two

and then you subtract another two

notice you did it, once again, you did it

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

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

right over here, you are going to get negative six

negative six is the answer.

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

negative times a positive, or a positive times a negative

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

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

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

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

a negative two, lets say we had

negative two, let me do it a different color,

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

negative two times

negative three.

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

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

repeatedly subtract that thing three times whatever that thing is

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

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

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

three times, so we subtract something three times, so

subtracting something (repeatedly) three times

That's what this part right over here tells us

and we'll do this, exactly three times

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

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

and we know from subtracting negative numbers, we already

built this intuition that subtracting a negative is the same thing

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

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

we're told once again, gives you a positive

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

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

negative three as this example

negative three

negative three, and we added it

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

over here we added it twice, we added negative three

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

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

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

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

negative, negative and put our three right over here

and once again, subtracting negative three is like taking away

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

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

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

all consistent with all the mathematics you know

the distributive property is also the property of multiplying something

times something all these things you already know, and now

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

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

of multiplication which is as repeated addition