>>Math is cool and you can do it.
>>Okay we're going to go over how to multiply and divide.
We're working with positive and negative numbers.
Now if you're working with all positive numbers it's really nothing new.
But if you're multiplying or dividing and there are any negative numbers use the following rules
to decide if the sign, that means whether it's positive or negative
of the answer, is positive or negative.
The answer will be positive if there are an even number of negative numbers
and the answer will be negative if there are an odd number of negative numbers.
So we'll simply count the negative signs.
For every two of them you're going to get a positive for instance okay and then you're going
to multiply or divide as usual to get the numeric part of the answer.
Now keep in mind it will be different if there's a zero involved so I'm saying
if we're only working with positive and negative numbers.
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So just keep in mind zero is neither positive nor negative.
So any number times zero is still going to be zero and also remember
that any number divided by zero is undefined.
So zero times a negative number is simply zero or zero times a positive number is zero.
You don't say whether that's positive or negative and if you divide
by zero it's undefined, remember you can write it like this,
divide by zero that would also be undefined.
So those are the special cases if zero is involved.
All right let's go work with some positive and negative numbers
with multiplication and division.
All right, let's go through and just decide the sign of each of these numbers first.
Keep in mind it is not at all like addition or subtraction where you have to see
if there are more positives or negative,
which one outweighs the other, has nothing to do with it.
All right so in the first one, we have one negative sign so the answer is negative.
Here's a division, we have two negative signs so the answer is positive.
That means you could just leave it blank, you don't have to put a plus sign.
The third one, two negative signs, positive, next one,
one negative sign so the answer is negative, the next one,
one negative sign so the answer is negative.
Going up to the next column, one negative sign so the answer is negative,
two negative signs so the answer is positive.
All right, now these next three are dealing with zero.
Zero divided by anything is simply zero.
Zero times anything is zero and anytime you divide by zero it's not even equal
to anything, we'll say it is undefined.
So this one is actually just undefined, actually have to write out that in words.
All right, now let's go through, we've decided the signs now we do the regular old division
and multiplication, 16 divided by two, so we just ignore the signs,
we've already decided whether the answer is positive or negative,
16 divided by two is eight, 25 divided by five is five, three times two is six,
three times two is six, three times two is six, 16 divided by two is eight,
16 divided by two is eight and there you go.
So that's all there is to it.
You do regular multiplication and division.
The only thing new is deciding whether the sign is positive or negative.
So for every two negatives, watch, you could cancel them
out if you want to make them a positive.
Here we've got two negatives so we can cancel them out, we're not really cancelling out,
I'm not using perfect language here and negative times a negative, that's going to be positive.
So you notice on all the rest of them where the answer's negative it's
because there's just one negative sign left.
Anytime you've got one left, the answer is going to be a negative.
Okay let's go on to some more problems.
All right, try these.
Decide the sign, whether it's positive or negative and then go ahead
and just do the multiplication or division.
All right for the first one, watch my little trick here, for every two of them,
it doesn't matter which two, I'm left with one lone negative, this is going to be a negative
and that's because originally there were five negative signs which is odd.
And then I just multiply one times two times three times one times five that's going
to give you 30, so the answer is negative 30.
All right let's do the next one.
We've got two negative signs so if you want you can change for every two you can do that,
only with multiplication or division.
And if you want you can cancel them now or you can multiply and then divide by two
so you could either do 32 divided by two to get 16 or you could have canceled first
like the two goes into the four, eight times two is 16 so different ways of getting the answer
and that is all you need to know about multiplying or dividing with negative numbers.
Now why it works is much more complicated.
So I'm going to show you with counters how you might be able to see
that a negative times a positive, a negative times a positive why do you end
up getting a negative number, why does that make sense and then this is the trickiest of all,
why does a negative times a negative end up being a positive answer.
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Let's start with a physical model for three times two.
One way of thinking of three times two is it means three sets of two.
So two remember can be thought of as two positive signs, so if you want three sets
of those, it means you have two, two more, two more and that gives you a positive six.
So it makes sense that three times two is six.
Now let's do three times negative two.
Similarly this would mean put in three sets of hum, what does that mean if negative two?
I'm going to think about it as three sets of two negatives all right.
So we're going to think of those negative counters,
so I have one set of negative counters, two sets of negative counters,
three sets of negative counters, hum, that's negative six,
so three times negative two is negative six.
All right, so we saw that a positive times a positive is a positive
and a positive times a negative is a negative.
So hopefully that will help you see why a positive times a negative ends
up giving you a negative.
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Okay here's the tricky one.
What if they were both negative?
Negative three times negative two.
Well since the first number is negative it doesn't make sense to put in three
so we're going to think of since the first number is negative we're going
to think of that as taking away.
So we could think of negative three times negative two meaning take away three sets
of two negatives, well from what?
And here's the trick, from something, so you have to take it away from zero.
So remember how we represented zero, we could represent zero
as the same number of positives and negatives.
So let's say there's a zero and now since I'm starting off with zero I'm going
to take away three sets of two negatives.
Okay so let's do it.
Let's take one set of two negatives away, two sets, three sets
and what you have left are six positives.
So that's one little model of seeing why maybe a negative times a negative could possibly make
sense to be a positive.
It's still a little bit strange but that's just one model.
Here's another model, it just has to do with patterns.
So if you believe that any number times zero is zero,
this one in the middle negative three times zero should make sense.
And if you believe a negative times a positive is negative then these first two will
make sense.
So what we know is the, the first is always negative three
and the next number I'm decreasing, I'm starting multiplying by two then I'm multiplying by one,
then I'm multiplying by zero and I'm multiplying by negative one
and then by, sorry, negative two.
So these are going down and look what's happening to the answer.
If I start with negative six, hum, it went up three and that went up three.
So if you're just going to follow the pattern, it makes sense the next number would go
up by three and the next number would also go up by three.
So that's another way of possibly justifying why it makes sense negative times a negative is
a positive.
All right, so now you know how to multiply with positive and negative numbers.
>>I hope you learned something, remember math is cool.