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--
That guy draws fantastic x's
I disagree that the reason you can't divide by zero is that the limit of 1/x as x->0 doesn't exist. It ultimately still rests on the fact that 1/0 doesn't exist. It's a topological statement about something inherently algebraic.
The real reason you can't divide by zero is because it violates algebraic axioms. In particular, in a unital ring we have
a+0 = a for all a
for all a not 0for all invertible a, aa-1 = 1So we have for any a, 0a = (0+0)a = 0a + 0a => 0a = 0. So then 0-1 cannot exist unless 0=1, in which case you have a degenerate ring.
Does anyone know the answer to calculators and division by '0'?
Interestingly, ARM designed processors are equipped to handle division by zero and computations with infinity. The only operations that cannot be done are indeterminate forms, for good reason.
http://infocenter.arm.com/help/index.jsp?topic=/com.arm.doc.dui0378d/BABFBEJJ.html
So his iPhone could divide by zero if the software people didn't catch it.
I think 00 is very much defined, and is 1. Consider:
Just because the limit of xx as x goes to 0 in the complex plane doesn't exist doesn't mean that 00 is undefined.
Sorry if its a bit of a dull question but it's been plaguing me for months. Can you take the limit of the |x/0| and if so is infinity a legitimate answer past the whole x=blue thing.