Welcome to a lesson in which
we'll discover the rules for multiplying integers
by using opposites and the commutative property.
So let's start by reviewing the definition of opposites
and the commutative property of multiplication.
Opposites are two numbers that have a sum of zero.
Since four plus negative four equals zero,
four and negative four are opposites.
It's also true that if we plot opposites on the number line,
for example, four and negative four,
they're on the opposite side of zero,
but they are equidistant from zero.
The distance from positive four to zero is four units.
And the distance from negative four to zero
is also four units.
Now there is one more thing I do want to mention.
If we consider the integer negative six,
we do normally say negative six,
but there's nothing wrong with saying
the opposite of positive six or just the opposite of six.
Now let's review the commutative property of multiplication.
The commutative property of multiplication states
that a times b equals b times a.
So if we change the order of multiplication,
it does not affect the product.
So for example, we know that two times four equals eight,
and so does four times two.
But I do want to point out one slight difference
between two times four and four times two.
If we wanted to model two times four,
this means we have two copies of four or two sets of four.
So here would be one copy of four
and here would be another copy of four.
Notice how the result is eight.
But if we wanted to model four times two,
this means we actually have four copies of two.
So to model four times two, it would look more like this.
Here's one copy of two, two copies of two,
three copies of two, and four copies of two.
Of course, in both cases the product is eight,
but if we want to model them, there is a slight difference.
Now let's take a look at multiplying integers.
First let's consider six times three.
Well we know a positive times a positive is positive,
and therefore this product is positive 18.
And now for the next product we have negative six
times positive three.
But instead of thinking of this as negative six
times positive three, we could think of this as
the opposite of six times three,
which we could write as the opposite of
positive six times positive three.
But we know that positive six times positive three
is positive 18.
So this product must be the opposite of positive 18,
which would be negative 18.
Therefore, a negative times a positive is negative.
And then for the next product of six times negative three,
we'll start by using the commutative property
of multiplication, meaning we can write this as
negative three times positive six.
And now we can think of this just as the previous example.
Instead of thinking of this as negative three
times positive six, let's think of this as
the opposite of positive three times positive six.
Well positive three times positive six
we know is positive 18.
So we'd have the opposite of positive 18,
or just negative 18.
So a positive times a negative is also negative.
And then finally, for negative six times negative three,
let's think of this as the opposite of
positive six times negative three.
So we can write this as the opposite of
positive six times negative three.
Well we know positive six times negative three
is equal to negative 18 because you did it above.
So now we can think of this as the opposite of
negative 18,
but the opposite of negative 18 would be positive 18.
So now we've discovered that a negative times a negative
is positive.
Let's go ahead and summarize this outcome.
So we started with a positive times a positive is positive,
which we see here.
And then we discovered that a negative times a positive
is negative.
As well as a positive times a negative is negative.
So we discovered these two rules.
And then finally we had a negative times a negative,
which we discovered was positive 18,
our last rule here.
So what we've shown here is that
when multiplying two integers,
if the signs are different, then the product is negative.
And if the signs of the integers is the same,
both positive or both negative,
the product will be positive.
Let's take a look at this one more time.
Here we have seven times nine.
Well we know a positive seven times positive nine
is equal to 63.
And now to consider negative seven times positive nine,
let's think of this as the opposite of
positive seven times positive nine,
which we can write as the opposite of
seven times nine.
We know seven times nine equals 63,
so this would be the opposite of 63.
And therefore the opposite of positive 63
is negative 63.
Again, verifying that a negative times a positive
is negative.
Next, we have positive seven times negative nine,
which is the same as negative nine times positive seven,
or the opposite of positive nine times seven,
which we can write as the opposite of nine times seven.
And we know that positive nine times positive seven
is positive 63, so we have the opposite of positive 63
which is negative 63.
And then finally, negative seven times negative nine
can be viewed as the opposite of positive seven
times negative nine.
And we know that positive seven times negative nine
is negative 63, because we had it here.
So we can think of this as the opposite of negative 63.
And the opposite of negative 63 would be positive 63.
So while it wouldn't be common to show all this every time,
it is one other way that we can justify
the rules for multiplying integers.
So again, to summarize this one more time,
we knew that a positive times a positive was positive,
which we have here.
We also discovered that a negative times a negative
was positive, which we have here,
showing that when multiplying two integers,
if the signs are the same, the product is positive.
And then we also had a negative times a positive
was negative,
and a positive times a negative was also negative.
So in multiplying integers, if the signs are different,
as we see here, then the product will always be negative.
So again, this is just one of several ways to
try to understand where these rules for multiplying integers
come from and why they make sense.
I hope you found this explanation helpful.