Let's think back. Where do we - what was the first thing we began with, under indices, powers, exponents, What's the definition of an index? If I wrote, you know, A to the power M, what does that even mean? It means A times A times A, how many times? M times right? Okay. So let's actually write that. It's A times A times A - I don't know however many times M is, right, But it was one or five or 60, whatever it is, right, so I'll write that that is M times That's what a to the m and the power of m rather, is Okay, so then we took this idea And we said look, we can we can build on this. Right? if you have two numbers like this A to the m and a to some other base right, so this is what we wrote And if you multiply them by each other right you could rewrite it like this and count it up But that took forever. Instead, We wrote down. What we call an index law, right? It's equal to that base, and what's the power? What do you do? When you when you multiply you add the the indices right? So it's M plus N case it's ringing bells. Yes, okay? in exactly the same way if I go division like this Okay, you subtract the indices okay, you take the difference, so far so good and then one of the most important things We dealt was you know if these two numbers are the same m and n right A to the m divided by A to the n, You get a to the power of 0 right? But then we said well that must be 1 that must be one. that was the last "More", if you like, that we established, so there's 1, 2, 3 laws, then we raise this question, okay? We said for any number the power of 0 it's 1. The same way you know I can put this on the side 0 To the power of any number that's 0. right? 0 squared, 0 Cubed, 0 to the power of 100... 0 to the power of are you paying attention waiting what? (student:) I've been waiting for you.. I've been waiting for you. (teacher:) cool. Thank you.
(student:) your welcome So yeah you have asked us about zero to the power of zero? Ah good yes, so what happens when you put these together, right? any number to the power 0 is 1, any 0 the power any number is 0. so what happens when you put them together? ok? now! Yeah, therein lies a question now here's where we're going to do it, and you'll need your calculators for this ok in fact I'm going to get mine out We're going to need everyone's calculators as well, so I'm going to ask you to help me do some numbers, okay? so get your calculator up ok. so here's the thing right? if you're looking from this angle, if you're approaching along this way from this guy, you'd say it should be 0 If you come from this direction you'd expect 1 so is it either of those or is it something different entirely? ok here's what we're gonna do um I'm gonna draw up a table. You. Don't need to draw this table, but you're gonna help me fill it in, ok? mmm What I want us to think about is What these numbers are equal to if I'm starting to get close to.. You know.. really small numbers. so have really small numbers. so if you want, what we're about to do is something called limits I'll explain that shortly But you seem to know it, right? and here's what you need on your calculator. in the middle up the top There's a power button ok? it looks like an x with a little square sort of filled in at the corner, okay? so if you type that in for example, if you go, let me turn this on.. if you go Seven then you press that power button right and that little box will appear So this is what I typed, alright. 7, then I'll type that x with a box now if we type something like 3 so now three appears up in the power right so seven is the base and three is the pound you press equals and you get 343! okay, so seven cubed is 343 here's what we gonna do. I want us all to get a whole bunch of different numbers, okay? So hands up who's got a proper calculator like not another phone at proper one good, okay? So here's what I'm gonna do: I'll gonna write down a series of things right and then I want us to work out what their value is. okay? so value and and Mmm. I'm sure like all this I'll leave that plain. I'll think of a bad name in a second okay, so Really important. We do the, um, the calculations with your calculator So raise your hand again so I can assign some numbers, okay? Everyone raise your hand so I can see okay.. all right. jagr you get my hardest one Okay, you get one of the power of one okay? Keep your hand up kyle you get 0.9 to the power of 0.9. (kyle: it's just fine by me, yeah) Okay, Jack you can do 0.8 to the power of 0.8. You see I'm going right? Tom - .7. was it right good, okay? All right, let's start from the top and again remember three decimal places Okay, Jagr make sure you read it carefully don't miss out any details. What did you get? (jagr : 1)
one but any decimal places? Okay, so I'm going to go to zero zero zero is that okay? there you go. all right! Kyle you a next! .9!
(Kyle: 0.909)
0.909?! ok Good um will you next shot zero point eight three six Eight three six, okay good Tom Seven seven nine, okay. I want to pause for a second before we get to Lucas's number Just look up for a second and [let's] have a look at what's happening just have a look at what's happening, okay? I'm trying to establish a pattern can you see the numbers again [explorance]? But [I'm] trying to get to this right try it closer and close up at this point if you had to guess at this point What would you think would happen when we get to zero power zero? What does it look like it's going to it Looks like they going to zero right? They were dropping.. hmm.. well let's keep going! who's up to? Lucas? lucas: 0.736 0.736 okay, who is next was it - nelson - yeah? Nelson: zero point sev.. amm Nelson: seven zero seven..
teacher: seven zero seven? Okay, alright now to Bill *Bill starts murmuring Okay, good now We'll pause there for a minute to the next number The people who were suspicious before should now have caused to be even more suspicious because look The numbers are decreasing right ah but not much Look at these guys right. They're getting quite close together. They are decreasing, but they're slowing down Yeah, that's right. Well actually we call that's exactly what it is. We call this exponential growth, but what we're looking at seems to be Exponential decay, but anyway, that's an assign Gets even more interesting okay, who had point three yeah? 696
696... what ?! What happened? Why am I slowing down is not even slowing down? It's gone the other way! turn around. Okay? Let's keep going! point two (0.2) Hold on... two four.. we're climbing again. We're climbing. okay. Chloe go ahead Cloe: 0.793 okay, now this is interesting I'm getting closer. I know I'm getting there, but I'm number one. I'm not there number two you see how from here to here We were slowing down, right? these gaps are getting very very small, but now that I turned around The gaps are getting bigger. See that's that gap like point three this gap like point Seven right, so what's going on? Okay? Now? We need more numbers, so we need your calculators again here the numbers. I'm going to suggest Let's go. Let's get half of this. what's half of point one? Think it's .05 isn't it so Jarrod, you got gypped out by doing one to the power of one. so why don't you do the 0.05 to the power of 0.05? okay? um Kyle. Do you want to do, Let's go smaller again. let's go on .02 to power 0.2. okay Jack join to 0.01 Okay, I'll tell you what the next one is in a second Jack can tell me what yours is Jarrod: 0.860 zero point eight six zero Hmm still in cooking increasing yeah Kyle: 0.924 0.924 still increasing yeah Jack: 0.954 zero Point nine five four Hmm Okay, now this time We sort of been going towards zero, but in smaller chunks okay? I want to get there faster. Okay, so this time Tom What I want you to put in is and other people can see if they get the answer faster feeling let's go ten times Smaller at zero point zero zero one okay? What do you get? Yes 993? okay, let's go again. Let's go ten times smaller again 0.0001 to the power of 0.0001. what's that equals to? you got it Lucas? yeah, yeah Ah now just because we've hit nines. What are the - out of curiosity - What are the other decimal places? Okay, all right ali ali can you do the next one let's do? 0.00001 that's four zeros, and to the power of zero point zero zero zero and one yep. four zeros one yeah four zeros, and then one what do we get? And I'm going to need more decimal places this time What happened to the break? ah You missed a 1 there, oh Go go back to the beginning Okay Math error what did you get zero point? Yeah? I'm (reading the number) okay cool [alright]. I'll go there. Yeah, okay Five zero is Five.. whoops, sorry - one two three four five yep? Okay, okay signing at the point, nice, now. This is interesting If I'm correct if you can go far enough with your calculator. I think you will convince your calculator Then it gets to one. I suspect. okay. I can't well you can sit down you can try and prove it But can you see we've done two things, okay? first, We saw it dropping down, right? But then it turned around [it] Started to increase like this But then it didn't increase forever. [oh], really Okay, did anyone get a one how many [zeros] I have to put on I use you can spam you Just kept on going to your rows until okay five minutes ago 20 20 zeros [I] can t Go, so what you've done by the way to get one all you've done is you your calculator has run out of nines That's all its own like if we had a longer Calculator or maybe some of your phones have more digits on those that don't you go? It doesn't have it doesn't have notation so that one and also The 9s don't repeat forever there are other numbers [here], so you can't use a repeater okay, so therefore therefore We never got to 0 [it's] not 1 0 but the well. It's not quite either is it but the the best Definition that you get like what it seems to be going toward is it seems to be going toward 1 okay? That's the best we can say but I don't bother Yes, well see it starts to break down if you go further cuz with negatives. It doesn't quite work Well see here's [the] thing This answer like. This is an answer We decide the numbers and the you know [we] start from our definition, right so it's what makes the most sense in this context okay Let me finish it off by coming back to this idea What you are all doing with your calculators is called taking a limit? There's this idea this thing which is approaching a certain number We can never get there, but you can get very close So if you're the curious type you can write this and you can wow your friends with it, okay? It's a bit of weird notation, but I'll explain what it means in a second. You should write it down then it looks like this okay, this thing what does it mean that that either it means see this thing right it means as x approaches 0 right you can see the error x is going there okay as x gets closer and closer and closer to 0 What happens .. to ... This guy here? this.. that to the power of itself And the answer is, it goes to that! That's what it does, okay? It never gets there, but that's where it's going if it could get there. That's what happened, okay? So I hope that gives you a conclusive answer um That's why if you go and look up on the internet, which I know some of you did. That's why you'll find this is the conclusion. Yup You mean why does it turn around here? Now that's a good question, and I don't have an answer to that at least I mean I don't have one right now probably because I haven't really thought about it all that much to be honest um I Wonder I mean there's always there's always reasons for everything. So maybe we could find out where they go
This is great
They never explained the part I wanted I wanted explained.. why do the numbers seem to go the other way.
I think his conclusion is pretty much correct, and this is a great lesson that is widely understandable. But it's not really rigorous.
The real reason 00=1 is because mathematicians said so. We've defined it that way because then exponentiation has useful properties, especially in the fields of combinatorics and algebra. For instance, the number of mappings from a set of size X to a set of size Y is XY. And there is no need to couch that statement in special cases for empty sets... the number of mappings from the empty set to the empty set is 1. Similarly, the binomial theorem holds for x=0 or y=0. (Both of these examples come from Knuth)
Analysis is the field that is most often a hold-out where 00 is undefined. That's because it's a field often concerned with the limits of functions, and ironically, the approach used in the video is on a bit shaky ground here. Specifically, why are we calculating the limit as x approaches 0 of xx? Why not the limit of x0 or 0x or even x100x ? We could actually choose any two functions f(x) and g(x) that both approach 0 as x goes to 0, and ask about the limit f(x)g(x). If you pick those functions randomly, probably most will have a limit of 1 together. But there are still non-trivial examples with limits other than 1, like f(x)=e(-1/x) and g(x)=x.
Note that we don't have this problem for other common operators like addition and multiplication, nor for exponentiation anywhere other than 00. So there's an argument to be made that it's more useful to consider it as undefined. It's certainly less defined than 0+0 or 0*0 or 23.
For everyone curious about the students question about the minimum point:
https://www.wolframalpha.com/input/?i=minimum+of+y%3Dx%5Ex