>> Hi. This is Julie Harland
and I am Your Math Gal.
Please visit my website
at yourmathgal.com
where you could search any
of my videos organized
by topic.
Okay. We're gonna multiply
and divide
with the number zero.
We're gonna start
with multiplication.
Any number multiplied
by zero is zero
and also zero times any number
is zero, so for instance,
if we've got 7 times zero,
that's a number multiplied
by zero, so it's just zero.
What multiplication really
means is--
what would happen
if you added seven zeros
together, so zero plus itself
seven times is gonna be zero.
So how about if we had zero
times 3, what would
that equal?
Well now you've got zero times
a number that's still gonna
be zero.
What if you had a fraction,
one-half times zero?
That's still gonna be zero.
And if you know
about negative numbers,
what if you had a negative
number like negative 9 times
zero, what would that equal?
That's also gonna be zero.
What if you had zero
times zero?
That's also going to be zero.
What if you had zero times,
let's say negative 1,
what would that equal?
That's also gonna be zero.
So are you getting the hang
of it?
If one of the numbers is zero
and you're multiplying the
answer is going to be zero.
Now this works
if you know a little bit
about algebra
and it even works
if you have a variable
like N times zero that's
also zero.
Or let's say,
have you ever heard
of square roots,
have you ever seen a
square root?
So if you have zero times the
square root of 2,
that's also zero.
Now, if you've never seen
this, don't worry about it.
You'll eventually get to it
when you're taking algebra,
but it doesn't matter what
you're multiplying zero times
or zero times something else,
something times zero,
you keep getting zero.
In fact if you have a bunch
of stuff and you're
multiplying by zero,
the answer is going
to be zero.
Lastly, you don't have
to use a dot
to show multiplication,
all of these could be written
a little bit differently.
You could for instance say 6
the little times sign
that looks like a little X,
that's also a zero.
You could also use parentheses
for multiplication,
you may have seen this
if you have like 7
and then parentheses zero
that means 7 times zero.
Or you might have zero
and then a parentheses
negative 8,
that also means zero.
So there're different ways
to show multiplication
but the important point here
is that if you have zero times
anything or if you have
anything times zero your
answer is gonna be zero.
Alright, we're gonna go
on to how to divide
with the zero
but first we're gonna just
talk about the algebraic
definition of division
and if you've never had
algebra this might be a little
bit tricky but just try
to catch on.
Division, what does it
really mean?
A divided by B,
so those are just numbers
like 6 divided by 2,
we say there's some answer
that that's going to be right.
And it means this other
equation would have
to be true, A would have
to be equal to B times what's
in the box.
But the number
in the box must be unique,
that means we want one answer,
okay?
So when you do a division
there can only be one answer.
So, let's look at an example,
something that we know
hopefully the answer to,
what's 6 divided by 2?
I bet most
of you know the answer
to that.
But we say well, it would have
to make this equation true.
So then the question is 2
times what will equal 6?
What would that be?
Three. So that's why 6 divided
by 2 is 3.
Alright, you know,
you could have fractions
and negative numbers,
all sorts of different things
but the main point is
that this is the algebraic way
of thinking about division.
It's saying you can think
of the answer
to a division problem
by thinking
about a different
multiplication problem.
So then, what
if we were trying to divide
by zero, so instead
of 6 divided by 2 what
if I tried to do 6 divided
by zero, okay?
So I'm trying
to find the answer
to 6 divided by zero
and that would mean 6 is equal
to zero times whatever would
go in the box here and hmm,
zero times whatever is
in the box, what could I put
in the box so that
when I multiply it times zero
I get 6?
Well, you know,
zero times anything is gonna
always be what number?
It's always gonna be zero.
So it doesn't matter what I
put in that box.
I'm gonna get the right-hand
side of this equation zero.
It's never ever going
to equal 6
so I have a big problem here.
There is no number I could put
in the box
to make this equation true.
So what we say
when that happens is
that 6 divided
by zero is undefined.
Okay, it's not a number
and this is the language
we use.
We say that it's undefined.
So what happened?
When I was dividing
by zero my answer was
undefined and you could
probably convince yourself
that it doesn't matter
if it's a 6 or a 3, or a 5
or a 2, or a negative 1 I'm
gonna have the same problem
over and over again.
The right side will be zero,
but the left number is some
other number.
So 6 divided
by zero is undefined.
Now, you could, of course,
write this 6 divided by zero
like this and you're gonna
have exactly the same problem,
okay, because you still would
have to try
to solve this equation
and there's still nothing you
could put in the box.
So no matter how you look
at it, 6 divided
by zero is undefined.
Let's try zero divided
by zero, okay?
Now, of course,
I could write zero divided
by zero this way as well
but I would still have
to solve the same problem.
What could I put in the box?
Alright, so we look
at these two problems.
They're both the same problem.
I'm just writing the division
a little bit differently.
What could go on the box?
Zero times what will give
me zero?
Well, zero times zero
that could work, right?
I could put zero
in there, right?
But the question is is
that the only number
that works?
Well, let's see.
Can you put in like a 5
and make it work?
Yeah, 'cause zero was also
zero times 5.
So, you know what?
Anything I put
in that box will work.
Any number I put
in the box works.
[ Pause ]
>> Alright.
So we have the problem again.
We say that zero divided
by zero is also undefined,
okay?
And of course,
I could write it like that.
That's undefined.
So what did we find out?
We found out that-- sorry.
Zero divided
by zero is undefined
and we also found out that
if we had some other number,
I'm gonna just put a little
number symbol.
We tried like 6 divided
by zero.
When we tried other numbers
in the numerator it was
also undefined.
So no matter how we looked
at it, if I was dividing
by zero in all of these cases,
dividing by zero,
you could see you've got the
zero in the denominator.
It doesn't matter.
What was in the numerator
or what was in front
of the division sign
when I'm dividing by zero,
the answer is undefined.
So we found out,
any number times zero--
when I times zero times a
number it becomes a zero.
Any number divided
by zero is undefined, alright?
So we have to get
that straight.
So we could sort of write
that like this,
A times zero is zero
or zero times A is zero.
So in other words
if we're multiplying that's
how it works.
But if we have A divided
by zero, it's undefined
or if we have it written
like this.
Oh, actually,
I don't wanna write the word
equals because it doesn't
equal anything.
We say it is the undefined.
It means we can't even define
it to be any number at all.
Okay.
>> Alright,
we have one more thing
to think about here.
What if I'm doing zero divided
by a number
so I could write zero divided
by A or zero over A
where A is not zero?
Because we already decided
that zero divided
by zero is undefined.
So in other words,
what's zero divided by 5?
Again, we just go back
to this definition.
Sorry. Zero divided
by zero is--
the box means
that zero is 5 times what goes
in the box and what could go
in the box?
Five times what will give
you zero?
Zero again,
because we know 5 times zero,
right, will give you zero.
So zero divided
by 5 is zero and, of course,
we could write something
like this, zero divided
by negative 7.
If you don't know negative
numbers, don't worry you will
get to learn about them soon,
equals negative 7 times zero.
And what will go in the box
to make that true?
Negative 7 times what will
give you a zero?
Zero, 'cause we know
when you multiply any number--
when you're multiplying 2
numbers, if one of them zero,
the answer is zero,
so this is true.
So zero divided
by a number is zero.
So zero divided
by A equals zero
if A is not equal to zero.
We already know what zero
divided by zero is,
that's undefined.
Okay. There's one more way
to remember this
and it's called the
lollipop method.
Okay. Now,
the lollipop method,
here's how you think about it,
instead of using any number,
just use the number 1.
So if I have zero divided
by 1, we just determined
that that was zero,
and we had the 1 divided
by zero.
Remember when you divide
by zero, it's undefined,
so here's the way
to remember it.
This is a good lollipop
because it's right side up
and still good to eat,
so if it's right side up,
the lollipop is right side up,
the answer is zero.
Now, look at this other one,
when over zero.
This one is upside
down sticking to the floor.
It's yucky.
It's on the floor
so that's the bad lollipop.
[Laughter] The bad lollipop
is undefined.
That's the way some people
think of it and, of course,
it doesn't matter
if it's really a 1,
but it looks really
like a lollipop
if you put the ones there.
So let's do our conclusions.
So here are our conclusions.
Any number times zero is zero
so I'm using letters
because that could represent
any number you want,
so I'm writing that as--
I'm using A, B, C, D,
et cetera in these examples.
So A times zero, zero times B,
zero times zero,
those all equal zero.
Any number divided
by zero is undefined so all
of these are undefined
because I'm dividing by zero.
So if the zero is
after the division sign
or if it's in the denominator,
it doesn't matter what the
other number is it's
always undefined.
But zero divided
by any non-zero number
is zero.
So E and F are not zero.
So note, E
and F are not zero, okay?
So it's a non-zero number.
So if the zero is
in the numerator
or the zero is before the
division sign,
and the other number is not
zero, then the answer
to that is also zero.
And that's pretty much
everything you need to know
about multiplying or dividing
with the number zero.
[ Pause ]
>> Please visit my website
at yourmathgal.com
where you can view all
of my videos
which are organized by topic.