I'm going to put on, to finish my definition,
I'm going to put on two more words.
That's it.
Just two more words.
Okay.
And I'm just going to use,
as best as I can,
the language we've already developed, right?
Because we just established
what adding and subtracting is, right?
Yeah?
Ok, this is coming on to what Caval was mentioning, ok?
Multiplication, multiplying I think
the simplest way to understand it is,
it's repeated adding.
That's all it is.
Isn't it?
C'mon, think about it, right?
Example.
Ok,
Ah, 3 times 5.
What is 3 times five?
Of course you could start to talk about factors,
You could start to talk about an area, or that kind of thing...
But really what it means is, well
it's 3, and then you add 3 again,
and then you do it again, and then you do it five times.
Right?
So you are putting together
you're adding on 5 sets that are
all the same size: 3, namely.
Yes?
(unintelligible question)
Yeah, that's right.
then it would.
If I'd written it like this,
then it would, right?
Right?
I know we tend to think of these as the same,
but they represent different ideas, right?
Like,
you know,
the fact that multiplication is commutative,
is not immediately apparent
because not all operations
are, right?
So,
anyway,
ok,
So therefore you can probably see where I'm going to go...
for division, it's...
it's repeated subtraction.
But it's a little different.
It's a little different.
Ah, sorry.
It's an ATM machine.
Ok, so,
it works a little bit different to this,
ok?
For instance, if I said
15 divided by 5
What does that actually mean?
Why, in what way, is it repeated subtraction?
What y..., what it means is:
how many times can I subtract 5
before I have nothing left, ok?
And the answer is
well, I shouldn't say equals
I can do it three times.
I can do it once
twice
three times, right?
And that's when I have nothing left in my set.
Ok?
So that's why
3 times, that's the answer.
Ok, now...
now you can see
why I had to establish this, because
that's going to be our gateway to think about what this means, right?
Division
by
zero.
Ok, so,
Example:
Ok?
Let's consider
What is 1 divided by zero?
Allright?
So if I take this basic, basic, basic definition, right?
It's, well,
how many times can I take away zero?
And you're like, "Well,
well I can take away one zero.
I can take another."
I still haven't,
I still haven't got to my set
which had nothing in it yet, right?
So, you notice
Aha!
There's three here.
How many here?
Answer:
answer:
infinite, right?
Infinity.
Ok?
So you might think:
Okay,
so,
does it seem fair to say
this?
Ok?
Now, I'm going to push further on this a little bit, right?
Because one of the ways mathematicians solve problems like this
is we say, "Well, ok, we've got a problem,
here's one way to approach it."
Let's approach it another way, ok?
Because, you know,
you guys actually know a lot more about division.
You know that it's more than just repeated subtraction, ok?
So, for instance, let's consider
let's consider these
let's consider these numbers, ok?
1 over 1.
1 over 0.1.
1 over 0.01.
Right?
And I, I can keep going, ok.
So,
There we go.
No prizes, what's it equal to?
(one)
1 divided by .1?
It's 10.
1 divided by .01?
Ok, allright, so you can see where this is going.
Right?
So now this is a different kind of argument
but this, if it kept going,
it seems reasonable to say: "Welllll...
Yeah?
Now this is interesting.
I've approached this same problem
from two completely different angles, and
I've got the same answer.
Ok?
So this seems promising.
Seems promising.
But there's a problem.
There's a big problem.
Ok?
And the problem is, you know,
this game, right, this game that I played
here with 1 divided by something and then
divided by other things, and so on.
I can play with other numbers too, can't I?
For instance:
I could do it with 2.
Right?
2 over .1.
2 over .01.
And so on.
Right?
Now all of your numbers, right?
They're going to be twice the size.
You're going to get 2,
and then 20,
and then 200.
But they're tending toward the same thing, right?
They're tending toward infinity.
So,
That means
this is where this is going is
2 over 0.
Right?
So apparently
1 over 0
is equal to infinity.
And 2 over zero is equal to infinity.
Right?
It's like well if you have
if you have these two things divided by the same
they should be
they should be equal
Right?
Hmmm.
Now, you see the reason why it was
so important to go establish this, right,
is that you can see this is not just playing a game.
We've broken, not just rules we decided or invented, right?
But something which we knew to be fundamentally wrong.
Here's one!
I should put it in this hand, sorry.
Here's one!
And here's two and they are not the same.
Right?
They are two sets and they are different sizes.
So this is, this is wrong, this is wrong.
It can't be.
Ok?
Which means that this can't be.
Which means that none of that is true.
Ok?
(therefore it's undefined?)
Now, ok, so...
Then comes the question:
Well then, ok,
we call it, we call it, right?
Undefined.
But why do we call it that?
Is it just because
It's like "Well I, I don't know."
And I'll show you why, I'll show you why.
Let's come back to this, ok,
let's come back to this.
Ok.
Now, I approached it from this angle
and I was heading from this bigger number, 1
to .1, .01, and I was getting towards 0.
That's the way I did it, right?
I approached it.
Now, there's other ways to approach zero you know.
You don't have to just come down from 1 towards 0.
You could go, for instance,
from -1
and go up towards 0.
I can do that, can't I?
1 over this
1 over this.
I'm still approaching 0.
And I'm still going to get there.
Minus .001
and so on.
I should still get to 1 over 0.
Ok, but what happens
Minus 1.
Minus 10.
Minus 100.
Minus 1000?
Where's this going?
This is not going to infinity.
It's going to negative infinity.
Ok, now,
This is where we get to the crux of the issue, ok?
Undefined.
Why do we call it undefined?
I used to think, y'know,
you get your calculator out
and you punch it in
and it's like "math error" cause y'know
cause it's "undefined".
You look it up on Wikipedia.
You're like "Are mathematicians just lazy?"
They're like "Ah, oh, I don't know what it is. We'll just not define it."
No.
In fact
Undefined. It's undefined because
it's undefinable.
Right?
There is no number you can pick
that makes sense, right?
Because you know you put this
and some other guy will say "But I can make it equal this
in exactly the same way you did."
Right?
So there's equal sort of, um,
bidders for the crown
of what 1 over zero is equal to. Yes?
(it looks like if you graphed it as a hyperbola)
Umhm.
(then it would never, cause it never touches zero)
Right.
Ok, so, so,
So this stuff here, right,
you can do it visually, right?
You've got this, right,
this is what we started with.
We started at 1 over 1
which is like here.
Right?
And then we decreased the value
of the denominator
1 over x, right?
So we were going this way
and off we go up to infinity.
But you guys know what the other side of the graph looks like, right?
That's the other side of 1 over x.
And off it goes to negative infinity.
Right?
So it's like, "Where am I going?"
Well the answer is, I can't be going anywhere.
Right?
(because it doesn't exist)
That's right, that's right.
The technical way of saying this is that the limit does not exist.
So, it's undefined because you can't define it.
It's not just because we haven't worked out what it's equal to yet.
It's because we tried to work it out
and you just get completely random things, ok?
So that's why 1 over 0 is undefined
it can't be defined.