Problems with Zero: Division by Zero

Zero 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 zero Because it is a slightly more unusual New-want number you have to be a little 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 zero and you can't have things like zero to the power of zero I mean I get asked about these all the time people are constantly why can't I divide by zero? I line divide by zero isn't it just infinity blah blah blah blah And so I thought I would I would do two things I'm first of all I'm gonna show you why no you cannot divide by zero It's not just infinity It's a bit more complicated than that and then I'm also gonna look at why you can't have zero to the power of zero Okay, so this is something that we've been asked a lot at number file. Well, you may know that something like Multiplication is just glorified adding really you want to do five times ten You just add on five plus five plus five plus five ten times Division is just glorified subtraction so if I want to take a number like oh 20 and then divide it by four I just keep subtracting four So you take away four take away four take away four you do that five times and that number five That's your answer 20 divide by four is equal to five So it's just glorified subtraction. That's really what it is now if I divide by zero Then that means I'm subtracting zero over never so 20 divided by zero means I take away zero I've got 20 and then I take away zero again. I still got 20. I take away zero and zero That would go on forever you would never get very far doing something like that keep taking away zero so 20 divide by zero That's infinity isn't it surely surely is infinity that an us what I expect people to think right Surely only a nerd would tell you differently that's when you cut to Matt 10 and different First of all everyone goes while why can't you just say that something divided by zero? So let's say I'm gonna do a function. I'm gonna have the function of one over X We don't say something is equal to infinity Okay, so infinity is not a number and it can't be treated like a number. Okay, it's an idea So we can't say One divided by zero equals infinity we can know more say that than we can say one divided by zero is equal to blue All right, but if I am naughty right and I do this one divided by zero is equal to infinity You would get just as equally too divided by zero equal to infinity and obviously you get the problem here. That's one seems to be equal to two obviously that's nonsense and that's why we don't so for very good reason We don't say equal to infinity you're gonna get nonsense like one equal to but I say what if you take a limit What if you just take the limit as X gets really close to zero? Doesn't this equal? Infinity and And so you would say that actually dividing by zero you could therefore conclude that one divided by zero equals Infinity and I'm gonna show you why you can't do that. So if you imagine your number line. Yes. This is the number line I'm gonna put zero right there's zero in the middle and out here This might be one and then so on all the way up and as you go along This here I'm gonna draw on this axis going up This here is my one over X. I'm gonna have one over X on that and over here when it's one This would be about one there when you come back to let's say about a half This is gonna be a bit bigger. It's gonna be twice as big by the time you get down to about a quarter That's gonna be twice as big again if you come in as you get closer and closer This does it does 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 zero 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 one then your value is actually down here at one If you then go to negative a half it's down here at negative two and then 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 zero 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 rather maybe maybe this line goes all the way around the reps around the entire universe Then comes back up here but as far as I'm concerned If you come in from one direction you get one answer if you come in from the other direction you get a different answer you're going to the same place There is no one limit of as you get closer and closer dividing by zero 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 pressure zero from the positive direction equals positive infinity and then separately down here The limits as x approaches zero from the negative direction of one over x equals negative infinity and these are different The equal different things we simply cannot Just assume that one over zero equals infinity if you go to zero from this direction It's going off to plus infinity and if you go to zero from this direction It goes off to minus infinity two different answers when I type one divided by zero into my calculator or my computer It can't do it it can't handle it What's it trying to do though? What cannot not do what happens in those circuits? What did it try and fail to do or has the calculator been taught? That's a very good question. Is it attempting to do something and then it's not getting an answer is it just been wrote taught do a divide by zero? I honestly don't know. I suspect it's just been taught that if someone hits divide by zero say error Or what it might do is actually try to get to that answer by iterative process Which then finds exploding in one direction 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 calculations getting out of control Call it off here just a math Sarah. I imagine it might even vary from divis to the device, but it'd be one of the two The other thing that people get very annoyed about is when you've got zero to the power of zero and the reason they get annoyed about this is when you've got anything Anything at all to the power of zero you always say it equals one and When you've got zero to the power of anything you always say it equals zero So what happens when these collide and people to be honest Are you different ways depending on what they need more often than not people argue for zero to the zero equals one in my experience? Although the video I did on 345 and number file people in the comments argued that zero to the zero should be zero Which is of course equally insane and I'm gonna show you why you can't have this and this this is absolutely lovely Because when you start with your number line here This is this is the normal number line there zero in the middle this time you're gonna look at the limit As x approaches zero, but this time our function is x to the power of x right And we're gonna slide it in and in fact we're gonna 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 zero from the 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 wrong. I'm gonna be good on I think 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 one As you come in from the other side you hit one 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 if we can come in along the number line This way into the middle but 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 one But it's not anymore complicated because this is only the real number line I'm not gonna go into this but the real number line is very boring because it's one dimension Or 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 gonna put in this is my imaginary axis And so now you've got like this entire surface of numbers and you've got the real and one direction Imagine you're in the other and any single point on there is part of the complex plane in fact now There are loads of different ways to come in towards the origin and you can approach it from anywhere on the complex plane And then these Approaches you get different limits you don't get one 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 one. 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 zero to the zero has a value in fact It is still undefined because the limit's very How about Something like x divided by y So I'm gonna draw his x and his y If I think about x divided by y I think about x divided by y Those this is gonna be fine Except here this is called the origin is the point zero zero x is equal to zero and y is equal to zero So at this point we have something that is zero divided by zero That doesn't sound like good news at all. What is that? That zero is 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 right this line If I travel along that line then this thing here x over y Why I say it's y equals x this is actually x divided by x now It is one So this is one everything on this line is one So it would be okay if I'm only traveling along that line I would be quite happy to say that that is a one as well everything else is So I'm gonna say yeah, that is that's called a removable singularity. That's its proper name If I travel along this direction, this is the line y equals minus x If I do that so y equals minus x in that case you get x divided by y y y is equal to minus x So this is minus one Everything on this line is minus one now Let's try this. I'm gonna travel along the x axis x axis in other words. This is y equals zero. That's what the x axis is so y equals zero If I do that then I get x divided by y I've said y equals zero So this is x divided by zero O dear, well we know that this is a problem But it's going to be something like and I'm gonna be naughty You know, it's it's going off to infinity plus infinity minus infinity, but it's It's something like that If I go along this direction Which is the y axis here x equals zero down here I do the same thing Right, x equals zero. So I'm gonna say zero x is equal to zero Divide by y That's zero divided by y. Everything on this line is equal to zero So I would be justified to say well That point is the only problem take it out and call it zero So and it does depend on which angle you approach from in fact I can make any number I've made minus one plus one infinity and zero and You depending on which angle you come out You can you can make any number you want out of that so that Zero divided by zero 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 It's also the angle that the matrix is being sort of