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Problems with Zero - Numberphile

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Oct 25, 2012

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Problems with Zero - Numberphile
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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.
  • 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--

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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

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