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