Welcome to a lesson on
the multiplication and division of integers.
Let's go ahead and get started.
Let's first start by talking about some preliminary ideas.
First, the multiplication of natural numbers
can be viewed as repeated addition.
So for example, we can view three times five
to mean three groups of five
or five plus five plus five, which would equal 15.
The same idea, six times four means six groups
of four, which would give us 24.
The second idea is the commutative property.
When two numbers are multiplied together,
their product is the same regardless of the order.
So for example, four times two
is equal to two times four, which is equal to eight.
Let's start by talking about the product
of a negative number and a positive number.
Most textbooks state the rule as to multiply
a positive number and a negative number,
you multiply their absolute values
and the answer is negative.
So four times negative three would give us negative 12.
But to justify this, we could think of four times
negative three as we want four groups of negative three
or we want to add negative three to itself four times,
which, of course, would give us negative 12.
Again, we want four groups of negative
three and then we sum them.
On the second problem, negative two times positive five
is going to equal negative 10.
But again, we could rewrite this as five times negative two,
which of course is still negative 10,
but now we can apply the same idea.
We want five groups of negative two,
so we want to sum five groups of negative two,
which would give us the negative 10.
The idea of multiplication as repeat addition
breaks down a little bit when we start
talking about multiplying positive and negative fractions.
This video does focus mainly on integers,
but at this point when we introduce fractions,
we do just want to think a positive
times a negative will be negative.
Then we just multiply across the top.
That would be two times four, which would give us eight.
Three times five would be 15.
Really what we're doing here is we want
two-thirds of negative four-fifths.
So we could think of breaking this into
three parts and then taking two of them,
but that gets a little more complicated
when we deal with fractions.
So, again, by this point we would just follow the rule.
Next, the product of two negative numbers.
To multiply two negative numbers,
we multiply their absolute values
and the answer is positive.
Again, this rule is correct, but it doesn't really justify
why a negative times a negative is a positive.
So let's take a look at these two examples first.
Negative four times negative three would
be a positive 12 based upon this rule.
Negative two times negative five would be positive 10.
But let's take a moment and justify why this would be true.
Remember that when we write down the number negative four,
we could think of this as the opposite of positive four.
So when I multiply this by negative three,
I could view this as
the opposite of positive four times negative three.
Well, based upon our previous argument,
positive four times negative three is negative 12,
and the opposite of negative 12 would be positive 12.
The same thing for this second example.
We could think of this as the opposite
of two times negative five.
That would be the opposite of negative 10 or positive 10.
If that's still not convincing enough,
we could go back to the idea
of multiplication being repeated addition.
If I take a look at negative four times negative three,
going back to that idea of multiplication
being repeated addition,
well, here, if I want a negative number of negative three's,
this would translate into repeated subtraction.
So if I want negative four of negative three,
I'd have to subtract negative three four times.
Well, we know from adding and subtracting integers,
subtracting a negative is the same as adding a positive.
So this translates into three plus three plus three
plus three, which is equal to positive 12.
So you could think of multiplying a negative
times a negative as repeated subtraction,
but I think the idea of taking the opposite
of a positive times a negative is a little clearer.
Next, when three or more numbers are multiplied,
we can just multiply two at a time
and follow the rules that we've already discussed,
or the product of an even number of negative numbers
is positive, and the product of an odd number
of negative numbers is negative.
So for example, on part a here we have three negative
numbers being multiplied together.
Three is odd, therefore, this product must be negative.
Then we can just multiply the absolute
value of these together.
Two times three, which would be six,
times five would give us 30.
But we already know our answer's going to be negative.
On part b, we have four negative numbers
being multiplied together.
So we have an even number of negative numbers,
therefore, our product will be positive.
So now we can just multiply the opposite value of these.
So one times five is five, times two is ten,
times three is 30.
Again, our product is positive.
I have a couple other ideas I want to discuss.
The next one is the multiplication property of zero.
For any real number a, zero times a is equal
to a times zero which is equal to zero,
which means the product of zero
and any number is going to be zero.
So five times zero is zero.
Zero times five is equal to zero.
Now, let's talk about the general rules
for, not only multiplication, but also division.
The rules for multiplication and division
of sign numbers are the same.
If you're multiplying or dividing two
numbers where the signs are the same,
both positive or both negative, the answer is positive.
And if the two numbers have different signs,
meaning one's positive and one's negative,
the answer will be negative.
So to summarize here, if we have a positive
times a positive, we get a positive,
but also, a negative times a negative is positive.
So if the signs are the same, the answer will be positive.
If the signs are different, the answer will be negative.
The same is true for division.
So if we use those rules here,
a positive divided by a negative will be negative.
14 divided by two is seven, so we have negative seven.
A negative divided by a negative will be positive.
24 divided by six equals four.
On number three, we'll just take it two at a time
since division and multiplication have the same priority.
We're performing order of operations.
A negative divided by a negative will be positive.
So here we have positive four
times negative two.
And now a positive times a negative is negative,
so we have negative eight.
OK, let's take a look at division involving zero.
For any real number k, k divided by zero us undefined.
So the two examples.
First, we have 12 divided by zero.
The second example we have is zero divided by five.
The way I remember this is I take a look at this
and I think some number k over zero.
So if you think K over O or KO,
if you know anything about boxing or mixed martial arts,
I think undefined when I see the word KO.
On the second example, zero over five,
I think zero over K or OK,
and OK means we can perform this division.
It is defined and it equals zero.
We'll end by thinking about a poem
that we can use to remember the rules
for multiplying and diving integers.
Here's how it works.
The word Hate represents a negative number,
and the word love represents a positive number.
So if you love to love, or you have a positive
and a positive, then you love.
So the product or the quotient will be positive.
If you love to hate, then you hate,
so that means a positive and a negative
results in a negative when you're multiplying or dividing.
Next, if you hate to love, or you have a negative
and a positive, then the result will
be negative when you multiply or divide.
Lastly, if you hate to hate, you have a negative
and a negative, then you love.
So if you multiply or divide a negative
times a negative, you get a positive.
So one more time, if you love to love, then you love.
If you love to hate, then you hate.
If you hate to love, then you hate.
If you hate to hate, then you love.
Thank you and have a good day.