Welcome to a lesson on multiplication and division of integers.
Before we talk about the rules for multiplying and dividing integers,
I do want to start by talking about two preliminary topics.
First I want to talk about opposites.
Five and negative five are called opposites. They're called opposites
because their sum is equal to zero and they're also the same distance from zero.
But you can also think of negative five as the opposite of positive five.
Here's negative five,
and here's the opposite of positive five, and they mean the same thing.
And I also want to mention the commutative property.
When multiplying two numbers, the product is the same regardless of the order.
Three times five, or five times three is equal to fifteen.
So now that we've covered these two topics, let's talk about multiplying
and dividing integers.
To multiply or divide sign numbers, the formal rules say that when
multiplying, or dividing two numbers, we want to use the absolute values
and then multiply, or divide. And then if the signs are the same,
the answer is positive.
If the signs are different, the answer is negative.
But I think when you're first learning how to multiply and divide,
I think these tables here are very helpful.
Here's the one for multiplication and here's the one for division.
And the good news is they're exactly the same except for the operation.
What I mean by that is, notice that these first two rows
we're multiplying, or dividing two numbers with different signs,
the product and quotient will always be negative.
And if we're multiplying, or dividing two numbers with the same sign,
both negative, or both positive, the product, or quotient will always be positive.
So these two tables can be very helpful.
Let's take a look at some examples and then we'll also try to justify
why these rules are true.
Four times negative three, we have a positive times a negative,
so this will be negative twelve.
Here we have negative two times five, we have a negative times a positive,
that product is also negative, so this will be negative ten.
And let's talk about why these products are negative.
If we consider negative two times positive five, remember we can think of negative two
as the opposite of positive two times positive five.
Well we know that positive two times positive five is positive ten,
so this would be the opposite of positive ten,
which is equal to negative ten.
And we could perform the same argument for four times negative three,
remember we could rewrite this as negative three times four if we wanted to.
Now let's take a look at these last two examples.
Here we have negative six times negative seven.
Well a negative times a negative is positive, so this is positive forty-two.
In the last example, eight times one, that's a positive times a positive,
which is also positive.
And notice in these four examples we've also seen four ways to write multiplication.
Now for a quick justification of why a negative times a negative is positive.
If we look at negative six times negative seven,
remember you can think of this as the opposite of
positive six times negative seven and we just learned that
a positive times a negative is negative,
so this would be the opposite of negative forty-two
and the opposite of a negative forty-two would be positive forty-two.
Now let's take a look at some division examples.
Again notice the rules are the same, except now we'll be dividing instead of multiplying.
So positive thirty-six divided by negative two,
that'll be negative eighteen.
Negative forty-eight divided by positive six, and negative divided by a positive is still negative,
so that'll be negative eight.
And now we have a negative twenty-four divided by negative three.
A negative divided by a negative is positive, so this will be positive eight.
And then a positive divided by a positive is positive will also be a positive.
Thirty-five divided by five is equal to seven.
So I think when you're first learning how to multiply and divide
sign numbers, these two tables here are very helpful,
and I would highly recommend writing these down into your notes.
I hope you found this helpful.