Problems with Zero - Numberphile

MATT PARKER: 0 is a perfectly good number.
And you ignore it at your peril.
The problem is it's a dangerous number.
And a lot of things can go horribly wrong with 0.
And because it is a slightly more unusual, nuanced number,
you have to be a little bit more careful with
how you handle it.
And so there are some things that you can't do with it.
So you can't divide something by 0.
And you can't have things like 0 to the power of 0.
And I get asked about these all the time.
People are constantly, why can't I divide by 0?
I want to divide by 0?
Isn't it just infinity?
Blah, blah, blah.
And so I thought I would do two things.
First of all, I'm going to show you why, no, you cannot
divide by 0.
It's not just infinity.
It's a bit more complicated than that.
And then, I'm also going to look at why you can't have 0
to the power of 0.
JAMES GRIME: OK, so this is something that we've been
asked a lot at Numberphile.
Well, you may know that something like multiplication
is just glorified adding, really.
You want to do 5 times 10?
You just add on 5 plus 5 plus 5 plus 5 10 times.
Division is just glorified subtraction.
So if I want to take a number like, oh, 20, and then divide
it by 4, I just keep subtracting 4.
So you take away 4, take away 4, take away 4.
You do that five times.
And that number, 5, that's your answer.
20 divided by 4 is equal to 5.
So it's just glorified subtraction.
That's really what it is.
Now, if I divide by 0, then that means I'm subtracting 0
over and over.
So 20 divided by 0 means I take away 0.
I've got 20.
And then, I take away 0 again.
I've still got 20.
I take away 0, and 0, and that would go one forever.
You would never get very far doing something like that,
keep taking away 0.
So 20 divided by 0?
That's infinity, isn't it?
Surely--
surely it's infinity.
And that's what I expect people to think.
Surely only a nerd would tell you differently.
That's when you cut to Matt telling them differently.
MATT PARKER: Because first of all, everyone goes, why can't
you just say that's something divided by 0?
So let's say I'm going to do a function.
I'm going to have the function of 1/x.
JAMES GRIME: We don't say something is equal to
infinity, OK?
So infinity is not a number, and it can't be
treated like number.
It's an idea.
So we can't say 1 divided by 0 equals infinity.
We can no more say that than we can say 1 divided by 0 is
equal to blue.
But if I am naughty and I do this, 1 divided by 0 is equal
to infinity, you would get just as equally 2 divided by 0
is equal to infinity.
And obviously you get the problem here.
That's one seems to be equal to 2.
Oh, and we see that's nonsense.
And that's why we don't-- so for a very good reason, we
don't say it's equal to infinity.
You're going to get nonsense like 1 equals 2.
MATT PARKER: But what if you take a limit?
What if you just take the limit as x gets
really close to 0?
Doesn't this equal infinity?
And so you would say that actually, dividing by 0, you
could therefore conclude that 1
divided by 0 equals infinity.
And I'm going show you why you can't do that.
So if you imagine your number line here.
This is the number line.
I'm going to put 0 right there.
So there's 0 in the middle.
And out here, this might be 1 and so on, all the way up.
And as you go along, this here, I'm going to draw on
this axis going up.
This here is 1/x.
I'm going to have 1/x on that.
And over here where it's 1, this would be about 1 there.
When you come back to, let's say, about 1/2, this is going
to be a bit bigger.
It's going to be twice as big.
By the time you get down to about 1/4, that's going to be
twice as being again.
And if you come-- as you get closer and closer, this does--
it does, I absolutely agree-- this gets bigger and bigger.
This goes racing off.
And it does tend to infinity.
This is absolutely correct.
But this only works if you're approaching 0 from the
positive numbers, if you're coming in from the right on
your number line.
If you come in from the left, it's completely different.
So if you start over here at negative 1, then your value is
actually down here at 1.
If you then go to negative 1/2, it's down here at
negative 2.
And as you get closer and closer, the value goes racing
off in this direction.
In fact, it goes racing down to negative infinity.
So yes, if you approach 0 from one
direction, you get infinity.
But if you come in a different way to exactly the
same place, you get--
well, you can't get much more different
than negative infinity.
And people will yell at me if I say it's infinitely
different from positive infinity, blah, blah.
Maybe this line goes all the way around and wraps around
the entire universe and then comes back up here.
But as far as I'm concerned, if you're coming from one
direction you get one answer.
If you're coming from the other direction, you get a
different answer.
You're going to the same place.
There is no one limit as you get closer and closer to
dividing by 0.
There's more than one limit with
completely different answers.
And that's why we say it's undefined.
Mathematically, what we would say is we say--
I want blue this time, sorry.
If you approach the limit as x approaches 0 from the positive
direction, equals positive infinity.
And then separately, down here, the limit as x
approaches 0 from the negative direction of 1/x equals
negative infinity.
And these are different.
They equal different things.
We simply cannot just assume that 1/0 equals infinity.
JAMES GRIME: If you go to 0 from this direction, it's
going off to plus infinity.
And if you go to 0 from this direction, it goes off to
minus infinity--
two different answers.
BRADY HARAN: When I type 1 divided by 0 into my
calculator or my computer, it can't do it.
It can't handle it.
What's it trying to do, though?
What can it not do?
What happens in those circuits?
What did it try and fail to do?
Or has the calculator been taught?
MATT PARKER: Oh, that's a very good question.
Is it attempting to do something, and then it's not
getting an answer?
Or has it just been rote taught to not divide by 0?
I honestly don't know.
I suspect it's just been taught that if someone hits
divide by 0, say error.
Or what it might do is actually try to get to that
answer by an iterative process, which it then finds
exploding in one or the other.
And so it's got some kind of built-in cap or some kind of
safety switch which goes off to say, this calculation is
getting out of control.
Call it off here.
Just say maths error.
But I imagine it might even vary from device to device.
But it'd be one of the two.

The other thing that people get very annoyed about is when
you've got 0 to the power of 0.
And the reason they get annoyed about this is when
you've got anything, anything at all, to the power of 0, you
always say it equals 1.
And when you've got 0 to the power of anything, you always
say it equals 0.
So what happens when these collide?
And people, to be honest, argue different ways depending
on what they need.
More often than not, people argue for 0 to the 0 equals 1
in my experience, although the video I did on 345 for
Numberphile, people in the comments argued that 0 to the
0 should be 0, which is, of course, equally insane.
And I'm going to show you why you can't have this.
And this is absolutely lovely because when you start with
your number line here--
this is a normal number line.
There's 0 in the middle.
This time, you can look at the limit as x approaches 0.
So this time, our function is x to the power of x, right?
And we're going to slide it in.
And in fact, we have to do it from both directions.
We have to come in from the positive direction.
And as we know, we have to come in, the limit as x
approaches 0, from a negative direction of x to the x.
And we'll see what we get.
And obviously, if they're different, then things are
going horribly wrong.
So if I draw in my y-axis here.
This is where I'm going to be graphing x to the x.
As you get closer in--
and to be honest, the path we follow is irrelevant.
But what happens is as you come in from one
side, you hit 1.
As you come in from the other side, you hit 1.
In fact, these have exactly the same answer.
They both give you one.
And so you say, well, if it doesn't matter which side
we're coming in from, if we can come in along the number
line this way into the middle, or we can come along the
number line this way into the middle, and both cases, the
function has the same limit, surely we can just call it 1.
But it's slightly more complicated because this is
only the real number line.
I'm not going to go into this.
But the real number line is very boring because it's one
dimensional.
You can go backwards and forwards on your numbers.
You've also got the complex numbers.
And for that, you need to put in the imaginary.
So I'm going to put in-- this is my imaginary axis.
And so now, you've got this entire surface of numbers.
And you've got the real in one direction,
imaginary in the other.
And any single point in there is part of the complex plane.
In fact, now there are loads of different ways to come in
towards the origin.
And you could approach it from anywhere on the complex plane.
And then, these approaches, you get different limits.
You don't get 1 anymore.
It starts to fall apart once you go to the complex plane.
And so this is why, even though on the surface of it it
might look like the limit should be 1, it doesn't work
once you go to complex numbers.
And that's why mathematicians still get very emotional when
you try to say that 0 to the 0 has a value.
In fact, it is still undefined because the limits vary.
JAMES GRIME: How about something like x divided by y?
So I'm going to draw--
here's x and here's y.
If I think about x divided by y--
BRADY HARAN: Slide that page around a bit?
JAMES GRIME: If I think about x divided by y, this is going
to be fine except here.
This is called the origin.
It's the point 0, 0.
x is equal to 0 and y is equal to 0.
So at this point, we have something that is
0 divided by 0.
That doesn't sound like good news at all.
What is that?
Is it 0?
Is it infinity?
What is it?
In fact, it can be any answer you want it to be depending on
the angle you come from.
I'll show you what I mean.
Now, this line is y equals x.
This line.
Now, if I travel along that line, then
this thing here, x/y--
why did I say this? y equals x?
This is actually x divided by x now, which is 1.
So this is 1.
Everything on this line is 1.
So it would be OK if I'm only traveling along that line.
I would be quite happy to say that that is a 1 as well.
Everything else is.
So I'm going to say, yeah, that is.
That's called a removable singularity.
That's it's proper name.
If I travel along this direction, this is the line y
equals minus x.
If I do that, y equals minus x.
In that case, you get x divided by y.
y is equal to minus x.
So this is minus 1.
Everything on this line is minus 1.
Now, let's try this.
I'm going to travel along the x-axis.
X-axis--
In other words, this is y equals 0.
That's what the x-axis is.
So y equals 0.
If I do that, then I get x divided by y.
I said y equals 0.
So here's x divided by 0.
Oh, dear.
Well, we know that this is a problem.
But it's going to be something like I'm going to be naughty.
It's going off to infinity--
plus infinity, minus infinity.
But it's something like that.
If I go along this direction, which is the y-axis, here x
equals 0 down here.
But you have the same thing, right? x is equal to 0.
So I'm going to say 0.
x is equal to 0.
Divide it by y.
That's 0 divided by 1.
Everything on this line is equal to 0.
So I would be justified to say, well, that point is the
only problem.
Take it out, and call it 0.
So it just depends on which angle you approach from.
In fact, I can make any number.
I've made minus 1, plus 1, infinity, and 0.
And depending on which angle you come at, you can make any
number you want out of that.
So 0 divided by 0 is this property called undefined.
Frankly, we could make it to be anything we want it to be
depending on the angle we come at it from.
[INTERPOSING VOICES]
MATT PARKER: It's all to do with the angle that the match
takes as being sort of--