Multiplication with Negative Numbers


Uploaded by MuchoMath on 10.08.2008

Transcript:
>> Professor Perez: Hey!
This is Professor Perez from Saddleback College.
Today, we're going to do multiplication with negative numbers.
Of course, we can't get started without our student of the semester and that's Charlie.
He better be ready to go!
Hey, Charlie, you ready to go?
>> Charlie: Yeah!
>> Professor Perez: Yeah you better be.
Okay, today we're doing multiplication with negative numbers.
That's right.
Now before you watch this video, you should watch the video on absolute value because in
that video we talk about the concept opposites and we're going
to be using that concept in this video.
Anyway, let's get started with a review of 1 times 3.
There we go, Charlie.
All right, now.
Pay attention Charlie, remember, what did 1 times 3 actually mean?
>> Charlie: 1 plus 1 plus 1.
>> Professor Perez: Yeah, 1 plus 1 plus 1.
Remember, multiplication was presented that way in this class.
Multiplication is really addition.
So, how do you figure out 1 plus 1 plus 1, Charlie?
>> Charlie: 1 plus 1 plus 1.
>> Professor Perez: It's 3, okay.
Now, why we're showing it this way is so that we can understand
that a positive times a positive will always be positive
because if you're multiplying two positive numbers together, you're always going
to be adding positive numbers, and therefore, your answer will always be positive.
That is why a positive times a positive is positive.
All the time!
Okay, Charlie, now, let's do one with a negative one.
Oooh! Don't get scared.
Here we go, Charlie.
Negative 1 times 3.
Now, what does that mean, Charlie?
Negative 1 times 3.
>> Charlie: Negative 1 plus negative 1 plus negative 1?
>> Professor Perez: That's right.
Negative 1 plus negative 1 plus negative 1.
Now, you should know how to add negative.
Remember, that was the video you should have watched before you're watching this one.
Anyway, some people like to see negative 1 plus negative 1 plus negative 1,
or some change the addition to subtraction and write negative 1 subtract 1 subtract 1.
It doesn't matter.
Somebody prefers the top one, some of you prefer the bottom one.
Anyway, how do you add a negative 1 and a negative 1 and a negative 1 on a number line?
Here it is.
Negative 1 plus a negative 1 plus a negative 1 is negative 3.
So notice here, hey, whenever you're taking negative times a positive like this one,
negative times a positive, you're always going to be adding negative numbers together, right?
So, negative times a positive means you're going to add negative
and the result will always be negative.
So that means, that a negative times a positive will always be negative.
There you go.
And now, by the commutative property, Charlie, which says a times b is the same as b times a,
and it must be true, hey, if a negative times a positive is negative,
then a positive times a negative will also be negative.
That's right!
Now, one thing to notice here, Charlie, here we go.
We're going to talk about the opposite again.
Notice if you multiply negative times 3, you get negative 3.
Now, Charlie, what's the opposite of 3?
>> Charlie: Negative 3.
>> Professor Perez: It is this negative 3.
And so what we're seeing here is that multiplying a number by negative 1 is the same
as taking the opposite of the number.
Okay? Opposite of 3 is negative 3.
Negative 1 times 3 is negative 3.
It's the same thing.
So, the opposite of 3 is negative 3.
Now, using mathematical symbols, remember, the opposite of 3 is...negative 3.
There you go.
And so we just found out multiplying a number by negative 1, right?
is the same as taking the opposite.
So negative 1 times 3 must be negative 3.
So there you go.
All right, Charlie, let's do this with another number now.
Let's do negative 1 times 5.
Negative 1 times 5, Charlie, means what?
>> Charlie: Negative 1 plus negative 1 plus negative 1 plus negative 1 plus negative 1.
>> Professor Perez: Okay, that's right, and remember it's the same as this,
negative 1 subtract 1 subtract 1 subtract 1 subtract 1, however you want to look at it.
Okay, what's our result though Charlie?
>> Charlie: Negative 5.
>> Professor Perez: Negative 5, that's right,
because negative 1 plus negative 1 plus negative 1 plus negative 1 plus negative 1 is negative 5.
There you go.
And so once again, notice Charlie, what's the opposite of 5?
>> Charlie: Negative 5.
>> Professor Perez: Negative 5.
And we're seeing, hey, negative 1 times 5 is giving us negative 5.
But again, we also see that a negative times a positive is a negative.
And by the commutative property, it must be true,
positive times a negative is a negative and there you go.
So, going back to this opposite concept, multiplying a number by negative 1 is the same
as taking the opposite of the number because negative times 5 gives you negative 5.
There you go.
Okay? Opposite of 5 is negative 5, using the symbols, the opposite...of 5...is...negative 5.
And most of us like to look at it as, hey, this is the same as negative 1 times 5, right?
That's what it is.
Negative 1 times 5 is negative 5, so there you go.
All right, now, we're going to attempt to explain why is it
that a negative times a negative is a positive.
That's a tough one.
But here's my approach.
Okay, here we go, Charlie!
Let's start with this.
Charlie, from our last video, what is the opposite of negative 5?
>> Charlie: 5.
>> Professor Perez: It was 5, right?
The opposite of this negative 5 is that 5 over there, right?
Okay, using symbols how do we write this?
The opposite of negative 5...okay?
Is 5. There we go.
Now, we just learned that multiplying a number
by negative 1 is the same as taking the opposite, right?
Okay? And so, negative 1 times negative 5 must be what, Charlie?
>> Charlie: Positive 5.
>> Professor Perez: 5.
Now, sometimes we look at this as the double negative rule because the opposite
of a negative number, the opposite of any negative number will always be positive.
We call that the double negative rule.
Or we can simply look at it, hey, if you take a negative and multiply it by a negative number
like here, negative 1 times negative 5, you're always going to end up with a positive number.
Therefore, a negative times a negative is a positive.
That's my approach, so, anyway, if you don't like this, well, you can talk to your parents,
your children, your facilitator, your teacher, your tutor, and ask them to explain
to you why a negative times a negative is a positive and if you hear something good,
e mail it to me, I'd like to hear it!
Anyway, let's move on here.
Our result, negative times negative is positive, and that's because the opposite
of a negative number will opposite will always be positive.
So there you go.
So let's just do some problems here.
Charlie, negative 1 times negative 8.
Is the answer positive or negative?
>> Charlie: Positive.
>> Professor Perez: It is positive, right?
And the answer is positive 8.
Very nice!
How about 3 times a negative 8.
This answer will be?
>> Charlie: Negative.
>> Professor Perez: Negative, and it's a what?
>> Charlie: 24.
>> Professor Perez: Negative 24, very nice.
Now, here we have 3 times a negative 5 times a negative 2.
Remember, multiplication, everything is being multiplied, we can multiply in any order,
so we can Kung-Fu this by first multiplying what, Charlie?
>> Charlie: Negative 5 and negative 2.
>> Professor Perez: Negative 5 times negative 2 is equal to what?
>> Charlie: 10.
>> Professor Perez: Positive 10 because a negative times a negative is a positive, right?
Bring down your 3.
And 3 times 10 is what, Charlie?
>> Charlie: 30.
>> Professor Perez: 30, very nice there, Charlie!
Okay, let's do one more.
Here's a negative 4 times a negative 5 times a negative 2.
Now don't get scared!
Remember, Charlie, negative times a negative is a what?
>> Charlie: Positive.
>> Professor Perez: Positive, so let's try some Kung-Fu on this.
Which two numbers are we going to multiply first?
>> Charlie: Negative 5 times negative 2.
>> Professor Perez: Negative 5 times negative 2.
Now, which is a positive 10, and bring down your negative.
Now notice here, we have three negatives, right?
To start with, and what's negative 4 times 10, Charlie?
>> Charlie: Negative 40.
>> Professor Perez: Negative 40, so your answer was negative.
When you have three negatives multiplied together, your answer is always going
to be negative because two of them are going to pair up and give a positive result,
and the one that's left over, negative times a positive, gives you this negative result.
In this 3 times negative 5 times negative 2, when you multiplied negative 5 and negative 2,
that gave you a positive result, multiplied by the left
over positive 3, which gave you a positive 30.
So, some people like to say, hey, if you have an odd number
of negatives being multiplied together, the answer will be negative.
If you have even numbers of negative, the answer will be positive.
So anyway, that was a tough lecture, phew!
So, next, we're going to go on to another tough lecture,
which is division with negative numbers, right?
All right, anyway, we'll see you all again soon!