Welcome to another Mathologer video. Today i'll talk about these crazy expressions here. Now in school they tell you that if you don't stay away from these eternal damnation awaits. And that's actually a pretty good rule to pass on to the masses to prevent them from suiciding but if you actually stop there a lot of modern mathematics is not possible. So let me explain this a little bit. Alright, so why do they tell tell you that you can't divide 3 by 0 or that 0 / 0 is undefined? Well let's have a look at something that nobody has a problem with --- 3/8. So in maths 3/8 actually just stands for the one and only solution of this simple equation there 8 times x is equal to 3. Ok, so let's change 8 to a 0 and see what happens. Well we are immediately in trouble here because no number satisfies this equation. The equation will always be 0 is equal to 3 which is wrong. So it sort of makes sense to stay away from something like this, right? And so what about 0/0? Well a different problem comes up here. Every single number actually satisfies this equation so there is also trouble. Let's stay away from it. So for most people that's all they really need to know but now if we had stopped there in mathematics there would actually be no calculus and nobody would know Isaac Newton. That would be really sad right? Ok, so what's calculus about? Calculus is all about derivatives and integrals. Here I have drawn a nice function and we want to find the derivative of this function at a certain value. Now you all know that this stuff is incredibly important but it has a really simple geometrical interpretation. The interpretation is the derivative here is just the slope of this touching line. Now just reading off from the function it's not clear what that slope should be. Wow what's easy to calculate is actually the slope of a cutting line like this. And then the idea is that if I move these two points of intersection together then the more they come together the closer the cutting line will be to the tangent line and the closer the slopes of the cutting lines will be to the slope that I'm really interested in. Ok now how do I actually calculate the slope of the cutting lines? Well you can just immediately read off what the height here is and what the width is and then we get the slope, just height divided by width, pretty obvious. But now see what happens when i move the two points together. Both the height and width approach zero, so overall the slope approaches this forbidden 0/0. But now nothing really terrible happens, right, you would expect something terrible to happen but nothing terrible happens.We are just getting closer and closer to this slope of the touching line. That seems a bit strange but just remember that equation that corresponds to 0/0 has all numbers as solutions so what really happens here is that we've established a context, a very narrow context in which 0/0 kind of makes sense as one particular number and you can see that as the context changes we get different numbers. So let's have a look at a specific example. Let's calculate the derivative of x squared at 1/2. Alright we have to see here what's the width here and what's the height. Width call it w so that gives us a second value. Now we evaluate the function at these two values. That gives us 1/2 squared and 1/2 plus w squared. Now the height is of course the difference between the two so that one here. Expand the top, notice that these two things cancel out and you've got the height and with the height we've got the slope and you can see the slope really what does it do at the top it goes to 0 at the bottom it goes to 0 but as long as we staying off 0 we can actually cancel the w and what this gives us this nice expression here and just by looking at that it's completely obvious what the limit of this expression is as w approaches 0. That's the slope we are after. This means that the derivative of x squared at 1/2 is equal to 1, isn't this neat. Now 1/2 was actually not very special here. We could have used any initial value here and if we did that we would find that the derivative of x squared is 2x. So once we've got that we've got this whole thing under control as far as calculus is concerned. Alright so we write this like that in calculus books and actually if you have a really close look you also find that the 2 was nothing special. We could have done the same thing for 3 and you should really try this, or 4 or for 5 or just in general for any positive integer, not a big deal. And now we just run with the scheme and make up a huge table of derivatives for all sorts of functions that we are interested in. And so you have it in a nutshell calculus courtesy of 0/0, pretty neat isn't it. So really that apple that hit Newton there at some point in time and made him invented calculus was definitely a 0/0. Before we go on and maybe don't believe anything I said so far let's just ask our smartphone what 0/0 is. So Siri ... That's fun but now I also promised you all of these guys. It is very important to make sense of these and actually it's done in a calculus book in the chapter on "indeterminate forms" and if you look there you find that these slope quotients that we had a look at so far are actually just a special incarnation of this sort of setup. So you have a quotient of two functions g and h depending on a variable, in this case t. And as the variable approaches a critical value both g and approach 0. Whenever you get something like this happening then you say that the quotient takes on the indeterminate form 0/0 at the critical value and actually just here I better say the critical value can also be infinity so t can also go off to infinity. What makes this whole thing indeterminate, well, by just looking at the information we've got so far it's not a all clear what the quotient does as we approach the critical value. It could go to a specific value, it could go to infinity or it could do nothing reasonable. On the other hand, you know if you were to consider the product you don't have to know anything about g and h except that they both go to 0 to conclude that the product will go to 0 and even Siri knows about indeterminate. So if you ask about any of those strange expressions that i showed you Siri will tell you they're all indeterminate. So let's have a look at an example, 1 to the power of infinity. Infinity and I really have to say this, in elementary calculus never stands for anything like a number. It always stands for some sort of function or process that goes to infinity. The same sort of thing here. Both 1 and infinity actually stand for functions, the first one approaches 1, the second one approaches infinity as t goes to a critical value. And this is also indeterminate meaning that if you don't know exactly what you're talking, about which g and which h you're talking about, the g to the power of h could approach any sort of finite value or it could go to infinity or it could go anywhere you want. Ok so here's a really famous example of something like this: (1+1/t)^t So here the critical value is actually infinity. So t gets bigger and bigger and as t gets bigger and bigger this 1/t will approach 0 and so the whole orange bit will approach 1 and of course the exponent will approach infinity and it's really really important to say that you can't do first one limit and then the other one. Tt's really the speeds at which one expression goes to 1 and the other expression goes to infinity that determine the behavior of the whole expression. In this case it approaches this value here, very famous, it's the base of the natural logarithm, e, and there is a whole video that I've done about this indeterminate form. Maybe watch it again after you finish with this video. And so we have a look at this and you think that well to make sense of all of these strange expression you have to invent a lot of different sorts of mathematics. But it actually turns out that all of these expressions and that comes really comes as a surprise to many many people are just 0/0 in disguise. So all of these expressions can be reduced to just the consideration of 0/0. I just want to show you how that works for 1 to the power of infinity. So again that just corresponds to this. Now this whole thing is equal to e ^ ..... And now we're dealing with a logarithm so we can pull the exponent in front of the logarithm, like that. g goes to 1. Now ln(1) is 0. This means that this whole thing here goes to 0. h goes to infinity but we can also write the whole exponent here in this form and then the 1/h actually goes to 0. So what we've done now is basically reduce 1 to the power of infinity to 0/0 :) Pretty neat, right, and you can do this for all the other ones. So far, apart from this very, very simple x squared example we haven't actually figured out any other expressions. How do you actually do this in practice. What I've tried to push here is that the apple that hit Newton was really a 0/0 apple. The 0/0 apple makes calculus but you can also go the other way around. Once you've got calculus you can actually hit 0/0 with it. How does that work? Well that's a bit of magic and its called l'Hopital's Rule. So here is an indeterminate form 0/0 at 1. So, if we let t go to 1, both the top and the bottom go to 0 and now how would you actually figure out where the whole expression goes to. You could try and evaluate this at ts that are very close to 1 t,hat's a perfectly fine strategy but there's actually a really nice shortcut. So what you do is you take the top and find the derivative and you take the bottom and find the derivative. So the derivative of the top is 1/t and the derivative of the bottom is just 1 and we will see what happens to this expression as t approaches the critical value. Of course nothing terrible happens here at all, this just becomes 1, right? And now if the functions that we're dealing with here are nice, differentiable, check out the details in a calculus textbook, then we can actually conclude at this stage that what we're really interested in also goes to 1. A really really nice trick. So, overall, what we've seen is that to make sense of 0/0 in calculus just means setting a special kind of context and sneaking up on the zeros. The same sort of thing if you want to make sense of 3/0. You also do this by sneaking up on 0 and of course you know that things explode magnitudewise and, you know, you can make sense of this in this way. But that's not the end of it all So in higher levels of calculus it actually makes sense to treat infinity like a number and to actually write equations like 3/0 is equal to infinity and you really mean it. So these things don't stand for any kind of functions they really stand for, you know, 3 as a number divided by 0 as a number is equal to infinity (sort of) as a number. In other branches of mathematics you sometimes find that it actually does make sense to set 0/0 equal to 1. But that's a topic for another video, so let's just leave it at that.
"You've all been indoctrinated into accepting that you cannot divide by zero."
Uhhhhh. When you find derivatives, you're not dividing by zero. I haven't studied L'Hopital's rule, but I doubt anything there depends on having meaning for these expressions. As far as I'm aware, these expressions are there for convenience. What mathematical result cannot be made in their absence?
I really don't understand the point of this video. It gives me the impression that indeterminate forms are somehow an essential part of the theory. I fail to see that.
Bored math person I be. Here is a list of the ways that 0 ^ 0 and all other indeterminate forms given in the video can be made into "0" / "0".
A Hastily Written Public Google Doc On Indeterminate Forms