Hi! Welcome to Math Antics.
In our last video we learned how to add and subtract integers.
In this video, we’re gonna learn how to multiply and divide integers.
The good news is that multiplying and dividing integers is easier than adding and subtracting them
because even though it involves both positive and negative numbers,
the multiplication and division still works basically the same way.
As we learned in our video about negative numbers,
the negative numbers are like a mirror image of the positive numbers on the number line.
Each number on the positive side has a negative counterpart on the negative side.
There’s a 2 on the positive side, and there’s a -2 on the negative side.
There’s a 5 on the positive side and there’s a -5 on the negative side, and so on…
Realizing that can help us understand that
negative numbers are just like positive numbers, but they have a negative factor built into them.
For example, think about the number 3 for a minute.
Do you remember when we learned about factors, we learned that 1 is always a factor of any number.
And that’s because multiplying by 1 doesn’t change a number.
3 and 1 × 3 are the same value.
So if you have the number 3, you can factor out a 1 and you have 1 × 3.
On the other hand, what if you have negative 3 instead?
We could still factor out a 1 from it and nothing would change; you’d have 1 times -3.
But because the number is negative, we could also factor out a -1.
Doing that would give us -1 × 3.
So one way to think about negative numbers is to imagine that they are just like positive numbers,
but they always have a factor of -1 built into them.
That means, if you want to change a positive number into a negative number,
all you have to do is multiply it by a factor of -1.
5 times -1 is -5,
7 times -1 is -7,
10 times -1 is -10, and so on…
And one way to visualize this is to see
that multiplying by a factor of -1 just switches a number from the positive side of the number line to the negative side.
Ah… but that raises an interesting question.
What happens if you multiply a number that’s already on the negative side by a factor of -1 ?
…like -1 times -3 Is that going to make it EXTRA negative?
Nope! In fact it’s going to do just the opposite!
Multiplying a negative number by another negative factor is actually going to switch the answer back to the positive side of the number line.
Multiplying by -1 acts like a switch no matter which side of the number line you start on.
If you start with a positive, then multiplying by -1 switches it to negative.
But if you start with a negative, multiplying by -1 switches it back to positive.
And you can keep switching back and forth between the positive and negative side of the number line
by multiplying by another -1 as many times as you want…
Positive!
Negative!
Positive!
Negative!
Positive!
Negative!
So, multiplying by 1 negative gives us a negative
multiplying by 2 negatives gives us a positive
multiplying by 3 negative gives us a negative
multiplying by 4 negatives gives us a positive
multiplying by 5 negative gives us a negative
multiplying by 6 negatives gives us a positive, and so on…
Do you see the pattern?
In a multiplication problem, if you have an EVEN number of negative factors,
they’ll form pairs that will balance each other out and will give you a positive answer.
That’s because the pair (-1 × -1) just equals 1 which has no effect on the answer.
But, if you have an ODD number of negative factors that are being multiplied together,
after you balance out all the pairs,
there will always be one negative factor left over that will give you a negative answer.
In a minute, we’ll see some examples of how knowing this will help us when multiplying and dividing lots of integers,
but first, let’s look at the simple problem 3 × 5.
We know that the answer is 15. But now, thanks to negative numbers,
we know that there’s three more variation of this problem that we need to learn how to do.
This might seem a little complicated, but it’s really not.
That’s because we’re gonna get the same number for the answer in all four problems,
it’s just that the sign of that answer (whether it’s positive or negative)
will be different depending on how many negative factors we’re multiplying.
Basically, when doing integer multiplication and division,
you can just pretend that the negatives aren’t there while you multiply or divide.
And then, you count up how many negative factors you have to figure out the sign of the answer.
If there’s an even number of negative factors, the answer will be positive,
but if there’s an odd number of negative factors, then the answer will be negative.
So, in the first problem,
we don’t have any negative factors, so our answer is just gonna be positive 15, (which we already knew).
In the second problem,
we have only one negative factor being multiplied, which means our answer will be negative 15.
In the third problem,
we also have just one negative factor which means that our answer will also be negative 15.
And in the fourth problem,
we have two negative factors being multiplied. That’s an even number of negative factors,
so they’ll balance each other out (they’ll switch and switch back) and they’ll give us an answer of positive 15.
That’s pretty easy, right?
And the really great news is that because multiplication and division are so closely related (they’re inverse operations)
the rules about negative factors are exactly the same for division problems.
For example, if you have the problem 8 divided by 2 (shown here in fraction form) the answer is positive 4.
But if you have 8 divided by negative 2, then there’s one negative, so the answer is negative 4.
Likewise, if you have -8 divided by 2, then there’s still just one negative,
so the answer will also be negative 4.
But, if BOTH of the numbers you’re dividing are negative (-8 divided by -2)
then there’s two negatives in the division problem, so the answer will be positive 4.
And one way to see that a negative divided by a negative gives you a positive
is to realize that if you factor out the -1 on the top, and you factor out the -1 on the bottom,
then you have a pair of common factors that you can cancel, just like you would if you were simplifying a fraction.
Okay, are you ready for some more complicated examples?
In these problems, we’re gonna be combining both multiplication and division.
And the good news is that it doesn’t matter whether a negative factor is being multiplied or divided.
You still get to count it when figuring out if you have an even or an odd number of factors.
How about this one: (-3 times 8) over -2.
Well, on the top, -3 times 8 is gonna give us -24
(since 3 times 8 is 24 and there’s only one negative factor).
And then, -24 divided by -2 is gonna be positive 12
(since 24 divided by 2 is 12 and we have two negatives, which means our answer will be positive.)
Here’s another good example: -1 times -8 times -6 over the quantity -3 times -4.
All five of the numbers are negative and since that’s an odd number of negative factors,
we know that the answer is going to be negative. Watch and see…
On the top, -1 times -8 is positive 8.
Positive 8 times -6 is -48.
And on the bottom, we have -3 times -4 which is positive 12.
And then as the final step, -48 divided by 12 is -4.
So it worked! Our final answer is negative because we had an odd number of negative factors.
But what we if made one slight change to that same problem?
What if we made one of the numbers on top positive instead of negative?
Then we’d only have four negative numbers (which is even) so the answer should be positive 4.
Let’s see if it is…
On the top, -1 times positive 8 is -8.
-8 times -6 is positive 48.
And on the bottom, we have -3 times -4 which is positive 12.
And then as the final step, 48 divided by 12 is positive 4.
Just as I suspected!
So do you see what I mean? All the multiplication and division works the same.
It’s just that you have to figure our whether the answer is gonna be positive or negative.
And to do that, all you have to do is
figure out if you have an even or an odd number of negative factors in the multiplication or division.
If there’s an odd number of negative factors, then the answer will be negative.
And if there's an even number of negative factors, the answer will be positive.
And remember, this process ONLY works for integer multiplication and division.
If you have a problem that also contains integer addition and subtraction,
you need to do those operations using the rules we learned in the last video.
As always, the way to get good at math is to practice,
so be sure to work some integer multiplication and division problems to make sure you’ve got it.
Thanks for watching Math Antics and I’ll see ya next time.
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