# Problems with Zero - Numberphile

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Oct 25, 2012

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• 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.
• 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.
• 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
• 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
• 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
• 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--
• 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
• 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.
• 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.
• 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--

### Description

Dividing by zero, zero divided by zero and zero to the power of zero - all pose problems!
More links & stuff in full description below ↓↓↓

This video features Matt Parker and James Grime - https://twitter.com/standupmaths and https://twitter.com/jamesgrime

NUMBERPHILE
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