Indeterminate: the hidden power of 0 divided by 0

by birtanpublished on August 29, 2020

welcome to another mythology video today I'll talk about these crazy expressions here now in school I 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 all right so why don't it tell you that you can't divide three by zero but it zero

Divided by zero is undefined well let's have a look at something that nobody is a problem with 3/8 so in mats 3/8 actually just stands for the one and only solution of this simple equation there 8 times X is equal to three okay

So let's change eight to zero and see what happens well we immediately in trouble here because no number satisfies this equation the equation will always be 0 is equal to 3 so it's 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's 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 that would actually be no calculus and nobody would know Isaac Newton that would be really sad right okay so what's calculus about calculus is all about derivatives and integrals

So here I've drawn a nice function and we want to find the derivative of this function at a certain value we all notice this stuff is incredibly important but it has a really simple geometrical interpretation

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 its slope should be and what's easy to calculate is actually the slope of a cutting line like this and

Then you know the idea is 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 okay 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 do width is and

Then we get the slope just height divided by weights pretty obvious but now see what happens when I move to two points together both the height and the width approach zero so over the slope approaches this forbidden zero divided

By zero but now nothing really terrible happens and you would expect something terrible happens but nothing terrible happens we're just getting closer and closer to this slope of the touching line so that seems a bit strange but

Just remember that equation that corresponds to zero divided by zero 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 as the context changes we get different numbers out you so lets us have a look at a specific example let us calculate the derivative of x squared at half right so we have to see what's the width here and

What's the height which we call it W so it gives us the second value now we evaluate the function at least two values so it gives us 1/2 squared and 1/2 plus W squared now hiders 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 width the height we've got the slope and you can see the slope really what does it do at the top it goes to zero at the bottom it goes to

Zero but as long as it's staying off zero we can actually cancel the W and what this gives us is this nice expression here and just by looking at that it's completely obvious what the limit of this expression is as W

Approaches zero that's the slope we're after that means the derivative of x squared 1/2 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 all right so we write this like that in calculus books and actually if you have

A really really close look you also find that the tool was nothing special so we could have done the same thing for 3 and you should really try this or for 4 or 5 or doesn't 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 were 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 day at

Some point in time and made him invent calculus was definitely a zero by zero Apple right before we go on and if maybe you don't believe anything I said so far let's just ask our smartphone what 0/0 is so Siri person good see it doesn't

Make sense there are no cookies you have no friends 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 calculus book in the chapter on indeterminate forms

And if you have looked there you find that these slope quotients that we had a look at so far actually are just a special incarnation of this sort of setup so you have a quotient of two functions G and H and their dependent

Variable in this case T and as the variable approaches a critical value both G and H approach 0 whenever you get something like this happening then you say that the quotient takes on the indeterminate form zero divided by zero

At the critical value and actually just here but let's say the critical value can also be infinity he can also go off to infinity what makes this whole thing indeterminate well by just looking at the information

Of course so far it's not clear or 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 zero to conclude that the product will go to zero and even Siri knows about indeterminate so if you ask about any of those strange expressions

That I showed you they will tell you they are all indeterminate so let's have a look at an example one to the power of infinity so infinity on that they 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 hate you're talking about the g ^ hich could approach any sort of finite value or it could go to

Infinity or it could go anywhere you want okay so here's a really famous example of something like this 1 plus 1 over T to the power of T so here the critical value is actually infinity so T gets bigger and bigger and as T gets

Bigger and bigger this 1 over T will approach zero 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 1 limit and then the

Other one it's really the speed at which one expressions goes to 1 and the other expression goes to infinity that determines at the behavior of the whole expression okay in this case it approaches this value here very famous

It's the base of the natural logarithm city and there's a whole video that I've done about this indeterminate form maybe what again after you finished with this video and so you have a look at this and you

Think yet well to make sense of all of these strange expression you have invented a lot of different sorts of mathematics but actually turns out that all of these expressions and that comes really as a surprise to many many people

Are just 0/0 in disguise so all of these expressions can be reduced to less the consideration of 0/0 and just want to show you how that works for one to the power of infinity so again that just corresponds to this now this whole

Thing's equal to e to the power of log that whole expression and now we're dealing with the logarithms that we can pull the exponent in front of the logarithm like that G goes to one our log of one is zero means that this whole

Thing he goes to zero now H goes to infinity but we can also write the whole exponent here in this form and then the one over H actually goes to zero so what we've done now is basically reduce one to the power of infinity to zero divided

By zero 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 expression so how do you actually do this in practice what I've

Tried to push here is that the Apple that hits Newton was really a zero by zero Apple a zero by zero op o makes calculus but you can also go the other way around once you've got calculus you can actually hit the zero divided by

Zero with it how does that work well that's a bit of magic and its called l'hopital's rule so here is a indeterminate form 0/0 at one so if we let this thing go to one but what's the top and the bottom go to zero and now

How would you actually figure out where the whole expression goes to we could try and you know evaluate this at T's that are very close to one that'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 a derivative so the derivative of the top is one over T and the derivative of the bottom is just 1 and now you see what happens

To this expression as T approaches the critical value and of course nothing terrible happens here at all this just comes one right and now if the function that we're dealing with here nice differentiable check out the details in

The calculus book then we can actually conclude at this stage that what we're really interested in also goes to one really really nice trick so overall what we've seen is that to make sense of 0/0 in calculus just means sort of setting a

Special kind of context and sneaking up on the zeroes so the same sort of thing if you want to make sense of 3/0 you also do this by sneaking up on zero and of course you know that things explore magnitude wise and you know it can make

Sense of this in this way but it's not the end of it at all so in higher levels of calculus it actually makes sense to treat infinity like a number and to actually write equations like three divided by zero 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 three as a number divided by zero as a number is equal to infinity sort of as a number but other branches of

Mathematics you sometimes find that it actually does make sense to set zero divided by zero equal to one that's just topic for another video so let's just leave it at that


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