Problems with Zero - Numberphile

MATT PARKER: 0 is a very good number. You can ignore it, but at your own risk. He is a very dangerous figure. Many things can go terribly wrong with 0. Because it is a slightly unusual but subtle number, you need to be a little careful when working with it. And because of that, there are some things you can't do. For example, you can't divide by 0. And you can't talk about 0 to the power of 0. I often get questions about this. People keep asking me, why can't I divide by 0? I want to divide by 0! Doesn't that just become infinite? Bla bla bla.

So I'm going to do two things now. First, I'll show you why you can't divide by 0. It's not just infinite. It's a little more complicated. Secondly, I'll show you something about 0 to the power of 0. JAMES GRIME: Well, this is something we get a lot of questions about at Numberphile. You probably know that multiplication is actually repeated addition. Let's take 5 times 10 as an example? You do 5 plus 5 plus 5 plus 5 10 times. Division is actually repeated subtraction. So, for example, if I divide 20 by 4, I always subtract 4. So minus 4, minus 4, minus 4 and that's 5 times. There's your answer, 5. 20 divided by 4 is 5. So it's repeated subtraction.

That's all. So dividing by 0 means subtracting 0 and repeating it over and over. So 20 divided by

zero

; minus 0 I have 20 20. And then 0 again. Still 20. And then 0 off and 0 off. Keep it up. This makes no difference if you keep subtracting 0. So 20 divided by 0? That's infinite, right? Of course... of course it's infinite. That's what I hope people think. Only a 'nerd' will tell you otherwise. That's why we have Matt. MATT PARKER: Everyone always asks why can't you divide by 0? Let me make a graph. the graph of 1/x. JAMES GRIME: We never say anything equals infinity, okay? Infinity is not a number.

And it can't be treated like that. It's a concept. Therefore, it cannot be said that 1 divided by 0 equals infinity. You could also say that 1 divided by 0 is blue. If I say that 1 divided by 0 equals infinity, then it also applies that 2 divided by 0 equals infinity. And then you get into trouble. Because then 2 equals 2. And we know that's nonsense. And that's a good reason why we don't say it equals infinity. To avoid nonsense like 1 equals 2. MATT PARKER: But what if we look at it as a limit? What happens if you take the limit when x is very close to 0?

Don't you then reach infinity? So you can see that this leads to the conclusion that 1 divided by 0 equals infinity. Now I'll show you where that goes wrong. If this is your number line. This is your number line. Then I put 0 here. 0 is exactly in the middle. And this is 1 and so on. If you move forward and up from this axis. So 1/x is here. There is 1/x. And with a value of 1 it becomes 1. And back to 1/2, the value becomes a little bit larger. It will be twice as big. And when you get to 1/4, it becomes 2 times bigger.

And the closer you get to 0, the bigger it gets. He's almost going crazy. It seems to go to infinity. That is absolutely true. But it only works if I approach 0 from the positive numbers, if you approach 0 from the right side of your number line. If you come from the left things become very different. If you start at minus 1, your result is below the value 1, and then at -1/2 you get -2. And the closer you get to 0, the faster the value moves in this direction. Fly to negative infinity. So if you come from one side you get infinity.

But if you come from a different side, you arrive at the same place, there is no major difference, then you get minus infinity. So there are always people who say it's the same thing, blah, blah. Maybe this line has to do with the world or the universe and then ends here. But for me it's simple, if you come from one side you get an answer. If you come from the other side, you will get a different answer. You go to the same place. But there is no clear limit for dividing by 0. There are more than 1 answers that are completely different.

And that's why we say it's undefined. Mathematically we would say... I want a blue one now, sorry. The limit of 1/x as you approach 0 from the positive side is equal to plus infinity. And then here, as you approach 0 from the negative side, 1/x equals negative infinity. And that's different. The results are uneven. So you can't simply say that 1 divided by 0 is infinite. JAMES GRIME: If you go to 0 from this side you get more infinity. And if you go to 0 from this side, you get negative infinity - two different answers. BRADY HARAN: If I type 1 divided by 0 on my computer calculator, I can't do anything with it.

He does not understand. So what is he trying? What can't he do? What happens in the circuits? What did he try in vain? Or did someone tell you that? MATT PARKER: Oh, that's a good question. Do you try something but get no response? Or did they just tell you? I really do not know. I think they told you that if someone writes divided by 0, say error. Or maybe he tries to get the answer through an iterative process that then explodes or something. And do you have some kind of limit or safeguard that says this calculation goes nowhere?

So stop here. It simply indicates that there is a bug in the operating system. It may differ depending on the device. But it has to be one of the two. The other thing that people ask a lot is about 0 to the power of 0. The reason this bothers people is because if you raise any number to the power of 0 you always get 1. But 0 to the power of 0 you always get 0. So, What about this? People always reason according to what they need. Most people say that 0 to the power of 0 is 1, although in my video I made on 345, most commenters claimed that 0 to the power of 0 should be 0, which is equally ridiculous.

Let me show you why this is so. This is really cool because if you start your number line here, your number line goes here... this is a normal number line. with 0 in the middle. Now we look at the limit as x approaches 0. This time the function is x to the power of x, right? We are getting closer to 0. We have to do it from both sides. We'll start on the positive side. We approach the limit as x approaches 0, from the negative side of x to the power of x. Let's see what we get. And if they differ, things go very badly.

So I'll draw my y axis here. And now I raise x to the power of x. And the closer you get to 0... To be honest, it doesn't matter how accurate it is. But what happens on one side is that you get closer to 1. And from the other side you get closer to 1. So they both give the same answer. They both give you 1. So if it doesn't matter which side you come from, whether you come from this side or this side, in both cases the limit is 1. Then we can say that it is 1. But this is only true for real numbers.

So the real numbers. I'm not going to go too deep into it. But the actual number line is boring. It's just one-dimensional. You can only come and go. But there are also complex numbers. To do this you have to make imaginary calculations. Then I add the imaginary axis. And now you have this whole surface of numbers. The real in one sense, the imaginary in the other. Each point is located in the complex plane. And now there are many more ways. to go to 0. You can start anywhere on the complex plane. And then you get different limits.

Not necessarily anymore 1. Things are going wrong at the complex level. And so, even though it seems like you're going to 1, things go wrong when you start using complex numbers. That's why mathematicians get excited when they say that 0 to the power of 0 has a value. It is not defined because you can get different values. JAMES GRIME: And what about x divided by y? I'll draw it... here's x and here's y. I'm going to use x divided by y... BRADY HARAN: Are you turning the paper a little bit? JAMES GRIME: x divided by y works fine, except here.

This is the origin. The point (0,0). x is 0 and y is 0. So at that point we have 0 divided by 0. And that's not good news. What's that? Is it 0? It's infinite? What is it? To be honest, you can get any answer. Depending on the angle you reach. I'll show you. This line is x equals y. This line. If I pass this one here x/y... Why did I say that? x is equal to y. Basically x divided by x. So that's 1. Everything on this line is 1. So I'm fine if I follow this line. So I would like to say that it will be 1.

Because everything is. So I think that's good. We call this removable singularity. That's the name. Now I'm going to follow the line y is equal to negative x. If I do that. and you divide x by y. y is equal to negative x. so you get minus 1. Everything on that line is minus 1. Let's try it. I'm going along the x-axis... Or y is equal to 0. That's the x-axis. So y is 0. If you divide x by y. I said y is equal to 0. Here x is divided by y. Oh darling. We know that's a problem. But let me be naughty.

Reach infinity... more infinite, less infinite. It's going to be something like this. If I go along the y-axis, where x is 0, you get this. It's going to be 0. x is equal to 0. divide it by y. That's 0 divided by 1. Everything on that line is 0. That applies everywhere except this point. Take it away and call it

zero

. So the answer depends on the angle. You can do any number like this. I did plus 1, minus 1, 0 and infinity. Depending on the angle you can do anything. Therefore, 0 divided by 0 is undefined. We can do anything depending on the angle we have.

MATT PARKER: It has to do with the angle of the match that determines whether...