[John:] E to the minus nought point three. Oh....
[speaker002:] Well it's E times itself nought point three times.
[John:] Minus nought point three times. Minus
[speaker002:] Yeah well the minus
[John:] How do you multiply some Thanks. How do you multiply something by itself a erm a minus number of times.
[speaker002:] Same as you do it a plus number of times except the other way round.
[John:] the other way round? Erm
[speaker002:] On the other side of the scale basically.
[John:] On the other side of the scale. I mean, once i negative numbers themselves are they don't exist.
[speaker002:] Mm.
[John:] I mean, it's just a a figment of someone's imagination. Anybody can use them to get
[speaker002:] Mm. It makes life easier. [LAUGHTER]
[John:] It's once you try and when you think, Well they must be real and I'm missing the point. Now what do they really mean. You know, there's sort of minus forty three bricks in my house. Mm. he's off his jump sort of thing.
[speaker002:] Mhm.
[John:] Erm multiply something together s Well say, ten to the six, that's ten multiplied by itself
[speaker002:] That's
[John:] a lot of times.
[speaker002:] Mhm.
[John:] How many tens are there altogether?
[speaker002:] Six
[John:] Six. Okay ten to the minus three? Well that's ten multiplied by itself a lot of times. How many tens are there altogether? Minus three. Oh yes? Ooh yeah. Well it doesn't doesn't really mean much does it. It's a load of rubbish really.
[speaker002:] Mm.
[John:] But it works.
[speaker002:] It does work.
[John:] And it gives us the right answers. So where did we get ten to the minus three from?
[speaker002:] Ten to the minus three?
[John:] Where did it How did we s we come up with this idea of ten to the minus three?
[speaker002:] Because it's giving us a value whether it be a value which
[John:] Why did you Why did you decide on minus three. Let's s let's say if we have erm Well what does ten to the minus three mean?
[speaker002:] It's ten times itself minus three [LAUGHTER] times []. There's no other way [LAUGHTER] really you can describe it [].
[John:] Okay let's say we have Now we won't use X we'll make it even simpler we'll use two.
[speaker002:] Mhm.
[John:] Two cubed times two to the fifth. What does that come to?
[speaker002:] Er s two to the eighth.
[John:] Right, so you added those, two to the three add five.
[speaker002:] Mhm.
[John:] And you do that because what does two cubed mean?
[speaker002:] Two times itself three times.
[John:] Right. Times two the fifth is
[speaker002:] Two itself five times yeah.
[John:] Right, we won't bother counting them we'll just
[speaker002:] Mhm.
[John:] say, there are five of them along there.
[speaker002:] Yeah.
[John:] And there are three of them along here so how many have we got altogether multiplied b well we've got three add five.
[speaker002:] Yeah.
[John:] Okay. Two to the three add five. That's okay. Now if we have let's say two to the five, divided by two to the three.
[speaker002:] two to the two.
[John:] What does it mean? Two times two times two times two times two. Mm one, two, three
[speaker002:] One, two, three, four, five
[John:] That's it.
[speaker002:] Mm.
[John:] Over two times two times two. Now why did we did we subtract this three from that five. Cos we're cancelling.
[speaker002:] Mhm.
[John:] We can cancel those three on the bottom.
[speaker002:] Yeah.
[John:] And how many we're left with?
[speaker002:] Two.
[John:] Five take away three, two.
[speaker002:] [cough]
[John:] Two to the five
[speaker002:] [cough]
[John:] minus three.
[speaker002:] Excuse me.
[John:] Now try this one. Two to the three, over two to the five. Mm.... Okay we're going to cancel the twos. But what
[speaker002:] Mhm.
[John:] we do what we did here, we cancelled all the twos on the bottom, with as many as it would cancel on the top. So we cancel all the twos on the bottom, right that's good, yeah that goes, yeah that's wh Oops, what do we do with these? Well
[speaker002:] Put in some imaginary twos yeah.
[John:] We're two short on the top to do the complete cancel. We're two missing if you like.
[speaker002:] Mhm.
[John:] And we just call that, because it works,
[speaker002:] Yeah.
[John:] This thing of subtracting, we carry on doing it even when it goes negative. So we say, Well this is
[speaker002:] Two to the minus two.
[John:] Two to the That's it. Two to the three, minus a f a five.
[speaker002:] Mhm.
[John:] Two to the minus two. What does two to the minus two mean then?
[speaker002:] In this case it's when it's been cancelled down, there was two more needed than
[John:] Right.
[speaker002:] than there was in the first [LAUGHTER] place []. [LAUGHTER]
[John:] Right. I mean that's a good way of putting it. That's exactly
[speaker002:] Mhm.
[John:] what it does mean.
[speaker002:] Mm.
[John:] That when we cancelled it all down on the bottom, we were two short on the top.
[speaker002:] Yeah.
[John:] So it's just a a notation. And it's not a negative number at all, it's a positive number.
[speaker002:] Mhm.
[John:] But it's got this funny thing in, a negative index, so what value is, what's the value of two to the minus two?
[speaker002:] Erm [sigh] it's sort of nought minus... that figure. It's two to the minus Sorry yeah.
[John:] No it's not a negative number. We'll put it's sign in front of it so we can just make sure of that. We started off with a positive number here, and a positive number there, and we divided one positive number by another positive number. We got a positive answer, but we got this funny bar. I mean in in logs log tables they'd write it this way.... Cos two to the two isn't a brilliant example, let's let's think what would this mean? Ten to the minus two. What would be the value of that?...
[speaker002:] It's just ten times itself
[John:] Mm.
[speaker002:] Two times in a minus direction.
[John:] It's not minus not minus. Try it with your calculator, see what it comes to.
[speaker002:] Erm doing a log?
[John:] Just do row
[speaker002:] Mm.
[John:] te erm
[speaker002:] ten two.
[John:] X to the X to a power. What does it give you?
[speaker002:] Point nought one.
[John:] Point nought one. Okay. Try ten to the minus one. Have a guess first, what do you think ten to the minus one will come to?
[speaker002:] Point one.
[John:] Right. Okay. Erm so have a guess at what ten to the minus six will comes to.
[speaker002:] Er point... Well six zeros one. [LAUGHTER]
[John:] Ten to the
[speaker002:] Er five zeros one sorry.
[John:] Good. Ten to the minus one, had no zeros before the one. Ten to the minus two has got one. Ten to the minus six will have five.... It's a positive number, but it's a fraction. What would one over ten squared come to?
[speaker002:] Erm... [cough]...
[John:] Mm work it out in your head. Right you've done it there....
[speaker002:] One over ten squared's just a hundredth.
[John:] What's ten Right a hundredth.
[speaker002:] Mm.
[John:] Well that was interesting wasn't it? What would one over ten come to?
[speaker002:] Erm point one.
[John:] ...
[speaker002:] One
[John:] Ten to the one. ten squared. What would one over
[speaker002:] It's just like an inverse it's of that number
[John:] Right.
[speaker002:] Mm.
[John:] What would one over a million come to? A millionth.... So... what would one over a thousand come to?
[speaker002:] It'd be nought point nought nought one....
[John:] Okay. One over a thousand is a thousandth. And that would be ten to the...
[speaker002:] Power of three....
[John:] Okay. So, if someone gives you like something like this say, erm ten to the minus eight, and you want to know I mean you keep seeing this. Negative num that minus sign is the thing that when you've almost certainly got recollections of.
[speaker002:] Mhm.
[John:] eight take away five, yeah okay, I can do that. Five take away eight, I can't do that. And then one day, they've been telling you for years, You can't do that. One day they suddenly say, Well yes you can actually, we call them minus numbers, negative numbers, directed
[speaker002:] Mhm.
[John:] numbers. Temperature numbers. Oh hang on. [LAUGHTER]. And it sort of conditions you a bit, so later on whenever you see it, you sort of get hypnotized mesmerized
[speaker002:] Mhm.
[John:] by that by Oh not gonna like this one. So get rid of it. As soon as someone gives you ten to the minus eight, you can say, Well if I want to convert that into real money,
[speaker002:] Gives you one over ten to the eight.
[John:] Okay. So that is not affecting the sign.
[speaker002:] It's still a positive number.
[John:] It's a positive number still. I mean you could have a negative number to the minus eight. You could have minus ten to the minus eight. If you wanted but that is not affecting the sign of the number. That's just telling you, when did the cancelling, we were eight short on the top.
[speaker002:] Mhm.
[John:] So in other words we got... eight tens left over on the bottom.
[speaker002:] Mhm.
[John:] So it's one over ten to the eight. Erm so going back to this one, E to the minus nought point three. What's that equal to?
[speaker002:] One over E to the point three.
[John:] Right, equals one over E to the plus nought point three. And what's E to the point three? Roughly? What's what's E as a number? Just approximately?
[speaker002:] Oh er two point seven something or other isn't it?
[John:] point three.
[speaker002:] Mm.
[John:] Okay. Call it three. And what does something to the power point three, mean?
[speaker002:] Roughly a third.
[John:] Does it.
[speaker002:] Erm
[John:] What does what does a hundred to the power half mean?
[speaker002:] It's [whispering] let me get this straight now []
[John:] Okay....
[speaker002:] It'll be a hundred to the pow erm a hundred times a half of itself so fifty.
[John:] No [cough] No.
[speaker002:] Oh.
[John:] It won't be. Now let's just have a look at what we're doing. You've looked at numbers all your life.
[speaker002:] Mhm.
[John:] Counting numbers, lovely. Know where you are with those. Then you get negative numbers, then you get fractions. And eventually you think you've just about got it sussed and they throw in indexes. You just about got these worked out. Oh not too bad these indexes, if you multiply then you just add them, if you're dividing
[speaker002:] Mhm.
[John:] you take them away. Raise them to a power, then you do multiply. So of one level up from what you think or one level down from what you think you should be doing.
[speaker002:] Mm.
[John:] And then they start throwing in fractional indices and negative indices. Well we've looked at negative indices. They're not negative at all. They're one overs. They're really the fractions.
[speaker002:] Mhm.
[John:] Okay so what do the fractional indices mean? Well we know the rules. The thi th See the thing about maths is it's very easy to learn the rules but but hard to understand when we're doing things that don't make sense.
[speaker002:] Mhm.
[John:] Erm so very easy to learn the rules for arithmetic, like say the rules for adding. Three add four is always the same as four add three.
[speaker002:] Four add three.
[John:] No problem. Erm seven take away three, that's not the same as three take way seven.
[speaker002:] Mhm.
[John:] You sort of learn things like that. Times, that's okay, three times four, the same as four times three. But not the same when you divide, it goes a bit odd.... mu multiplying using indices couldn't be simpler. Add the indexes, indices
[speaker002:] Mhm.
[John:] which do you prefer to call it?
[speaker002:] Well.
[John:] Indexes. Call it indexes it's simpler. Add the indexes. Right. So if I've got the number here, equals a hundred to the power half. I'm going to multiply it by another number which also happens to be a hundred to the power a half. And what's the answer?
[speaker002:] A hundred to the power of one.
[John:] Right. A hundred to the power of half, Add a half cos we're multiplying which is a hundred to the power one, which is just a hundred. So that number is the same as this one, they're both whatever they mean, they're both the same cos they're both a hundred to the power half. So what is this number, that you multiply it by itself and it makes a hundred?
[speaker002:] Ten.
[John:] Ten okay. Erm so a hundred to the power half, is actually equal to ten, let's say if we did erm sixteen to the power half, times sixteen to the power of a half. What would that come to?
[speaker002:] Sixteen to the power of one.
[John:] Okay, so what does sixteen to the power a half mean? What is sixtee or w what value has it got?
[speaker002:] Is it a square root?
[John:] That's it. I mean, you don't need to ask sort of why or what or we've n we've got this rule multiplying.
[speaker002:] Yeah.
[John:] We use it, and we find out that the square root of anything comes out as half of it. So sixteen to the half
[speaker002:] [cough] Oh excuse me.
[John:] square root of sixteen.
[speaker002:] Square root of sixteen.
[John:] We had E to the power of one third, times E to the power of one third, times E to the power of one third.
[speaker002:] Means E to the power of one.
[John:] That comes out to E. So what does E to the one third mean?
[speaker002:] Erm E well it's cube root E.
[John:] Right. the cube root, third root of E.... So what would erm... what would let's say... sixteen to the power of one quarter be?
[speaker002:] Erm er how you describe this one. Erm... the f I suppose the fourth
[John:] Go on.
[speaker002:] root.
[John:] The fourth root.
[speaker002:] Yeah.
[John:] Mm. That was the
[speaker002:] Of sixteen.
[John:] the second root, the twoth root. special name for it the square root.
[speaker002:] Mhm.
[John:] And that's the cube root, cos when we cube something it fourth root. so and what would that come to? What would the fourth root of sixteen be?
[speaker002:] Eesh erm Oh hell's bells. It's about two isn't or [LAUGHTER] something []?
[John:] It's very very close to two, yes. If we if you're not happy with fourth roots, you could do square roots.
[speaker002:] Mm. It'd be the square root of the square root.
[John:] If we had sixteen to the power of half, what's that? What does that mean?
[speaker002:] Erm well it's the square root of sixteen.
[John:] Square root of sixteen. That's some number, whatever it is,
[speaker002:] Mhm.
[John:] we might not be able to find the square root of sixteen, but whatever it is, it is a number.
[speaker002:] Mhm.
[John:] just treat it as a number. And we can raise that to the power of a half.
[speaker002:] Mm.
[John:] Means find the square root of this number. Well what's this number, this number is the square root of sixteen. Now what do we do with indexes when we're raising to a power?
[speaker002:] Now you times them.
[John:] Right, so we should finish up with
[speaker002:] Sixteen to the fourth well the one over four yeah.
[John:] One over four. Right? So what's this? This is the square root of the square root.
[speaker002:] Mm. Mm.
[John:] And that's what the fourth root means. And erm comes out to two. Two square it, you get the four, and then square four...
[speaker002:] Sixteen.
[John:] What would So we know what sixteen to the one quarter means, what about sixteen to the three quarters. Now it doesn't mean anything now in terms of multiply sixteen by itself three quarters times That's gone. That's what we started off with when we were doing the nice simple stuff with positive whole numbers for indexes. But then we developed it a bit further like we did with negative numbers.
[speaker002:] Mhm.
[John:] You know, three add four, four add three, yeah this is fine. And then we go beyond that, and develop it into this sort of slightly murky negative area. And then we develop this further. So what would that mean? How could you have finished up with an answer like sixteen it's a bit like integration this where we're working backwards.
[speaker002:] With a great deal of difficulty. [LAUGHTER]
[John:] What would you have been doing to what, to comes up with
[speaker002:] Erm... Mm
[John:] Have a look at what happened here, and what were we doing there.
[speaker002:] The only thing I could think of
[John:] Go on.
[speaker002:] would be
[John:] Just play about with it, see what you get....
[speaker002:] Three quarter. That's awkward. [LAUGHTER] That really is awkward. []
[John:] Well c I could make it erm nine sixteenth if you like if it's
[speaker002:] yeah.
[John:] that's any more help.
[speaker002:] Erm I'll make a guess.
[John:] Go on. Good.
[speaker002:] At that
[John:] Right,
[speaker002:] erm
[John:] So we started off with that. What would you do to it to turn it into this?
[speaker002:] Into that.... Times it by three.
[John:] Mm. Along the right lines. On the right lines. What were you timesing by three?...
[speaker002:] It's your actual
[John:] What were you multiplying by three?
[speaker002:] S erm... that.
[John:] Were you? That's a num That's a number. Were you multiplying that number by three? A quarter. Right.
[speaker002:] Oh, sixteen to the power of one over four, yeah.
[John:] Now were you multiplying t that's a number, sixteen It actually comes
[speaker002:] Mhm.
[John:] to two as you said.
[speaker002:] Yeah.
[John:] Were you multiplying two by three, or were you multiplying something else by three to finish up with sixteen to the three quarters?
[speaker002:] It's the quarter itself.
[John:] You were multiplying the index by three.
[speaker002:] Mhm.
[John:] So if you're multiplying an the index of something by three Where you had say I had X squared
[speaker002:] Yeah.
[John:] and I did something strange to is and it became X to the six, what would I have done to it.
[speaker002:] You'd have well basically timesed it by three.
[John:] No I wouldn't
[speaker002:] Timesed the index by three.
[John:] Yeah. I would have timesed the ind
[speaker002:] Mm.
[John:] And what what do you do when you times the index by three?
[speaker002:] Erm Oh, you're doing this again you're you're
[John:] Right.
[speaker002:] raising the power.
[John:] [whispering] Raise it up to the power. []
[speaker002:] Of three.
[John:] So X squared to the power of three
[speaker002:] Mhm.
[John:] is X to the two times three.
[speaker002:] Times three.
[John:] So sixteen to the fourth...
[speaker002:] Times three.
[John:] Right or cubed, raised to the power three, is equal to sixteen to the
[speaker002:] Sixteen to the thr
[John:] one quarter times three. Which is
[speaker002:] Three over four.
[John:] sixteen to the three quarters.
[speaker002:] Mhm.
[John:] Now sometimes they might give it you as sixteen to the three quarters and you can sort of read that off, What am I doing here? Well look the the one quarter part of it means, find the fourth root of sixteen and then the three means cube your answer.
[speaker002:] Yeah.
[John:] So sixteen to the three quarters is find the fourth root and cube it or if you like cube sixteen and then find the fourth root. It doesn't matter, it comes out to the same thing. Erm they're not always as helpful as that. They might say, sixteen to the nought point seven five. And you think, What on earth does that mean?
[speaker002:] It's still three quarters though.
[John:] It still means three
[speaker002:] Three over four.
[John:] So if you might get something like erm We'll go away from sixteen, we'll use two for a change. We'll use ten. Ten to the erm let's say, nought point nought three. Well, what we've really got is ten to the three hundredths.
[speaker002:] Mm.
[John:] Which what does that mean?
[speaker002:] Right, you've got ten to the power of one hundredth.
[John:] [cough] Okay and what does ten to the power of one hundredth mean?
[speaker002:] Right you've got ten to the power of one hundred.
[John:] Okay and what does ten to the power of one hundredth mean?
[speaker002:] Erm... basically Something not very pleasant. It'd be the hundredth root. [LAUGHTER] [LAUGHTER] Which I would not like to have to work out. []
[John:] The hundredth root of ten. Okay. And then well we haven't got one
[speaker002:] Times ten.
[John:] hundred, we've got three hundred.
[speaker002:] Mm. Times it by
[John:] So the index is times
[speaker002:] three.
[John:] The index is times three.
[speaker002:] .
[John:] So we must have raised that to the power three. So ten to the nought point nought three, is the hundredth root of Find the hundredth root of ten and then cube it. [LAUGHTER]
[speaker002:] Yeah.
[John:] Or if you like, cube ten then find the hundredth root of that. Erm and that's all this means. So just to make it a little bit more exciting [LAUGHTER] And then this is this is it finished off then. You you know it all. Ten to the minus nought point nought three. Well what on earth does that mean?
[speaker002:] Mm.
[John:] Well
[speaker002:] [LAUGHTER]
[John:] Let's let's slow it down a bit. Break it down a bit to something that's a bit easier to
[speaker002:] [cough] Oh dear.
[John:] Right, ten to the three. Right no problem.
[speaker002:] Got that one.
[John:] Right. Right ten to the minus three.
[speaker002:] Right [LAUGHTER]
[John:] What's that.
[speaker002:] Ten to the minus three, erm... is basically, when it's been broken down and we've got three left
[John:] On the bottom.
[speaker002:] on the bottom
[John:] And so on the bottom [cough] On the bottom we've got three left over.
[speaker002:] Yeah.
[John:] That's it.
[speaker002:] Mhm.
[John:] So write it was just
[speaker002:] One over that.
[John:] What about ten to the three quarters? What does that mean?
[speaker002:] Right, it's ten to the power of one over four.
[John:] So it's ten to the one over four.
[speaker002:] All raised to the power of three.
[John:] Yeah, so... what's ten to the minus three quarters?...
[speaker002:] Ah.
[John:] Ah.
[speaker002:] Mhm.
[John:] Good I like that.
[speaker002:] We're getting somewhere.
[John:] Ah. We're getting not... just somewhere a very long way away,
[speaker002:] It's ten to the power of three over four.
[John:] Mhm.
[speaker002:] .
[John:] That's it. That's it.
[speaker002:] Horrible way of putting it [LAUGHTER] but it's [].
[John:] as soon as you get a negative index anywhere, if someone says, Oh we've got erm X to the minus nought point six. Well let's get rid of that thing before we start thinking it's minus X or something. Well we'll just put that one over and that's got rid of that bar if you like as long as it
[speaker002:] Mhm.
[John:] has the bar there. One over X to the nought point six.
[speaker002:] X to the nought point six.
[John:] And then we can work out what on earth X to the nought point six means. Well I'll leave that to you. What does X to the nought point six mean then?...
[speaker002:] X to the nought point six is erm... X to the power of... six over ten.
[John:] Right. And what does that mean? What does that come to?
[speaker002:] Right
[John:] What what how did we finish up with X to the six over ten. What were we doing?
[speaker002:] X to the tenth.
[John:] Right we raised it.
[speaker002:] point six.
[John:] Okay. So we might have finished Is is X to the is X to the six over ten, the same as X to the three over five?
[speaker002:] Yes.
[John:] What does this mean?
[speaker002:] Hang on. Er oh. [LAUGHTER].
[John:] What does this mean? This means X to the one over five,
[speaker002:] Mhm. Three times.
[John:] Cubed. Not three times but cubed. So what this means, is that the same as that?...
[speaker002:] That's awkward.
[John:] [LAUGHTER]
[speaker002:] That really is awkward.
[John:] But that's it that's finished on indexes.
[speaker002:] Mhm.
[John:] You y erm I'm very very delighted actually you've really picked this up.
[speaker002:] Oh right.
[John:] It's wonderful, it is. Sometimes this can take several lessons. I mean it's not easy it's not there's nothing obvious about it, it's all what's this weird stuff we're doing now? But we use it in everything but particularly
[speaker002:] Yeah.
[John:] in engineering. Erm you need to be really on top of indices particularly negative indices and particularly negative fractional indices and usually expressed as a decimal because when you're looking at circuits and stuff like that, you'll get the erm you know, the answer will be given in this sort of form.
[speaker002:] They are the same.
[John:] Okay. So what how how did you what was your what was your first
[speaker002:] [LAUGHTER]
[John:] what was your first thought about it? Your first thoughts?
[speaker002:] Well the first thing that normally would run through me head would be, yes it is the same, cos I've seen that. And I know they're the same.
[John:] Right, it's
[speaker002:] I'm breaking it down to the cake again. It's the fractions of the cake. Right. Okay. It's got to be because three fifths is always the same as six tenths. Ah but because it doesn't mean this now.
[John:] Then you see that. [LAUGHTER]
[speaker002:] That's right. Yeah.
[John:] So one of the best ways to to sort of get it fixed in your mind, is try it with something. Well let's try erm
[speaker002:] That's what I ended up [LAUGHTER] doing []
[John:] Well that's what everyone does. Erm sort of how high level they're at at maths. When they
[speaker002:] Mhm.
[John:] get to a bit where they get stuck. They sort of come down
[speaker002:] Pull it down a bit.
[John:] Pull it down to something a bit more real that you can get hold of. I mean if you're doing things like er you're working with X or something, erm and you want to know does Well so say you're working with X times and Y and you want to know
[speaker002:] [cough]
[John:] You don't know if it does i is X is X times Y always the same as Y times X? And you think, Well they're only numbers. So a three times two
[speaker002:] throw in a three and a two [LAUGHTER]
[John:] Would that always be the same as a two times a three?
[speaker002:] Yes of course it would.
[John:] Well these are only numbers. And they're following the same laws so yeah that should be. But
[speaker002:] So it works. No?
[John:] if someone just says sort of, straight off and asks you especially if you you think, Ooh he's asking me here maybe they're not, maybe they are. Is X times Y the same as Y times X. Oh. What what's this X and Y? [whispering] They're numbers. [] Put some numbers in, try it. So what numbers did you put in here to try it?
[speaker002:] It was two [LAUGHTER] actually [].
[John:] Good, it's a good one to use, two. So what did you try?
[speaker002:] Right.
[John:] Two to the six tenths.
[speaker002:] I did it to the fir first one.
[John:] Right.
[speaker002:] Erm, put it in there, and thought, well a fifth of two.
[John:] Fifth root of.
[speaker002:] Mm. Of two.
[John:] Fifth root of two. So something that you had to multiply by itself and get five of them all multiplied together and it equals two.
[speaker002:] Erm and it was as What was it I got? Bububububub [vocalized pause]
[John:] You worked that out in your head? Fifth root of two.
[speaker002:] I do I think so. It was something like that I just remember thinking hang on this works. I was
[John:] you could get a you could get a job on the stage doing that. [LAUGHTER]
[speaker002:] No I was pulling it through this, that's what I was doing. I was working it out through this.
[John:] Right. Mm.
[speaker002:] And I thought, well I'll break it down so That was what it was, I'd pulled all the fractions together, put them into a common denominator.
[John:] Mm.
[speaker002:] I thought well, there's two of them to every one of them.
[John:] Yeah.
[speaker002:] So double that's six. Six tenths so it's the same.
[John:] Right. Oh right, I mean I'm I'm I'm not disputing that. If if if you're talking of pieces of cake, then three fifths, six tenths, there's no difference.
[speaker002:] Is the same thing. Yeah.
[John:] Right? That's that's one thing, but is finding the tenth root of something and then raising to the power six, or raising something to the power six and then finding the sa the tenth root, is that the same as finding the fifth root and cubing it or cubing it and finding the fifth root. That's not as obvious. That's what we're looking at, these I mean normally, normally they're the same. And we're not bothered.
[speaker002:] Yeah.
[John:] But are they the same here? I mean would it be the same if we had ten and we raised it to the three quarters. What do we do with that? That means we w w what does ten to the three quarters mean?
[speaker002:] It's ten to the power of a quarter.
[John:] Right. The fourth root of ten and then
[speaker002:] Is
[John:] cube it. Or we can do it the other way round, it doesn't matter. We could have started off with ten cubed and found the fourth root of that.
[speaker002:] Mhm.
[John:] It still gives us the same answer cos we multiply to get the indices. But would that be the same as ten to the nought point seven five. Which is ten to the power seventy five and then find the hundredth root of it?
[speaker002:] Oh God. [LAUGHTER]
[John:] Or ten sorry find the hundredth root of ten and raise that to the power seventy five. I mean these two are equivalent cos our rules say so for multiplying the indices.
[speaker002:] Yeah. It's
[John:] But is is that you know, is this three quarters, is that the same a is ten to the power three, all raised to the power and then find the fourth root, is that the same as ten to the seventy five and then find one hundred. Your calculator probably won't go up to ten to the seventy five.
[speaker002:] Erm.
[John:] But you could try it on these. On the six tenths and the three fifths. With ten, couldn't you.
[speaker002:] Mm.
[John:] Start off with ten and find... ten to the six, ten to the power six. and then find the tenth root of that.
[speaker002:] Er does my calculator like going backwards? Don't think it will.
[John:] Has it got a tenth root? Well you've got some number in there, so what do you want, you want that number to a power, to the power of one tenth.
[speaker002:] Mm.
[John:] So.
[speaker002:] X Y er one tenth. Er point one.
[John:] That's it.
[speaker002:] . There we are.
[John:] What does that come to?
[speaker002:] Three point nine eight something or other.
[John:] Three point nine eight something or other. okay.
[speaker002:] Put that in the memory.
[John:] Right, so that was
[speaker002:] That's that one.
[John:] ten to the six over ten. Right, now now try ten to the three over five. [whispering] Ten to the three fifths. []
[speaker002:] Ten X Y
[John:] So ten start off with ten and cube it first.
[speaker002:] Yeah.
[John:] So we've got ten cubed. And then find the fifth root of that.
[speaker002:] X Y erm... Yeah. Three point nine eight something or other [LAUGHTER] so it's the same [].
[John:] So it's the same. So it is the same. I mean it it's just as well. It's just as well
[speaker002:] Just check, do a memory recall. Yeah. It's the same.
[John:] because otherwise we would be in an awful lot of trouble with these.
[speaker002:] Mhm.
[John:] If someone says it's sort of, Oh it's six tenths, and Oh well I could cancel that down can't I. I can make that ten to the three fifth. And somebody else'll say, Oh we'll put that as erm ten to the point six. And if these weren't all equivalent, we would be in a lot of trouble, but they are.
[speaker002:] Mm.
[John:] So that's an interesting thing. If you multiply these out, and even when they're fractional. So you raise it to a power, multiply the indices, when it comes out as a fraction, it doesn't matter how you
[speaker002:] [cough]
[John:] I mean if you wanted to be really obscure
[speaker002:] Yeah.
[John:] you could put write something as ten to the minus erm... three percent. [LAUGHTER]. What's three percent? Well three percent's just three over a hundred.
[speaker002:] Three over a hundred yeah.
[John:] So ten to the minus three o No one would dream of putting of using percent. It is a fraction.
[speaker002:] [LAUGHTER]
[John:] I suppose you could use it if you wanted to. So back to the back to E.
[speaker002:] Right.
[John:] Erm I lost your page. Here it is.... What does E to the minus nought point three mean? I'd like you without using your calculator, just to get some idea of how big that is. We'll take E as three. We'll forget about the E and we'll put a number in instead. And we'll say, Three
[speaker002:] Mhm.
[John:] to the minus nought point
[speaker002:] Point three.
[John:] three recurring. Okay. make it any more. I won't say any more easy, but any less awkward.
[speaker002:] [LAUGHTER] Right,... Okay so that's three... jujujujuja [vocalized pause]... Ee, where's the start? Erm
[John:] Well what's the most awkward thing about this?
[speaker002:] Well the first thing is to minus. over.
[John:] Right, get rid of that. Good. So it's one over three to the
[speaker002:] Mm nought point three. Right.
[John:] And it's nought point three recurring.
[speaker002:] Right. Erm
[John:] give you a clue. We'll get rid of the decimal and
[speaker002:] Sign
[John:] change it into
[speaker002:] .... the power of a third.
[John:] Good. Excellent. And then what does that mean in in real money. Without all these indexes?...
[speaker002:] Erm dudududududududu [vocalized pause] That's where you're putting it now. Erm
[John:] If it was three to the half, what would you change it to?...
[speaker002:] I'm trying to think back now erm
[John:] Well when we were doing that Okay? A hundred to the half times a hundred to the half, made a hundred. And you said, Oh is it the
[speaker002:] Oh of course that's right. Yes.
[John:] Is it the square root? And it was. So this time cos it's a third
[speaker002:] A cube
[John:] It's a third root.
[speaker002:] It's that.
[John:] Right. So taking E as about three, this answer
[speaker002:] Mhm.
[John:] that you've got is roughly one over In fact we can now go back to having E again. Erm so you can just put your E in and you can get to one over E to the and there's your cube root of E. Okay?
[speaker002:] Erm.
[John:] put three in. Put three in.
[speaker002:] Yeah.
[John:] Put three in, find it's cube root and then find one over that.... Was that square root twice?
[speaker002:] Mm.
[John:] Okay. Well well well why wasn't it right? I mean don't worry about, Oh well that wasn't right.
[speaker002:] Cos that'd be the the root to the fourth that.
[John:] That would be the fourth root. Right.
[speaker002:] Mm. Erm ah there is a cube root.
[John:] Oh. That's a shame.
[speaker003:] [LAUGHTER]
[John:] Ah.
[speaker002:] Saved.
[John:] Where's where's your cube root by the way?
[speaker002:] There a little tiny one. I hadn't it before.
[John:] Ah now not a lot of calculators have that.
[speaker002:] [LAUGHTER] Right er and
[John:] So hang on a minute right. You've got one point four four twoish. Okay, do it again
[speaker002:] Without using that. [LAUGHTER]
[John:] A and this time this this just this time just raise it to the one third and see if you get the same answer, to check it is the cube root.
[speaker002:] Er three. Mm.
[John:] Used three yeah, three.
[speaker002:] Three X Y point three.
[John:] Yeah.
[speaker002:] answer.
[John:] Good. That's good and can you t did you write. I would have had to change it
[speaker002:] Right.
[John:] if it hadn't been.
[speaker002:] Er
[John:] Right now Okay. That was that's raising it you know raise it to the one third, you find the cube root.
[speaker002:] Mhm.
[John:] But the actual answer you got wasn't raising it wasn't point three recurring, it was point nought three. So you start off again with your three. Put your three in. Okay and raise that to the power point nought three.
[speaker002:] X Y point nought three.
[John:] Which is not a lot.
[speaker002:] About one. [LAUGHTER] Roughly. []
[John:] Right and than, you get one over that.
[speaker002:] So basically just I'd be tempted to to inverse it with that.
[John:] Mhm.
[speaker002:] Yeah, the one over X would that do it?
[John:] Yeah.... So one over something a bit more than one, will come out to something a bit less than one.
[speaker002:] A bit less than it. Mm.
[John:] Okay? So that's great but we started off with three there and we should have started off with E. So through imme a this this is intensely repetitive so that you can you're building it up, you're sort of doing it with numbers and then you're doing it more
[speaker002:] Mhm.
[John:] and so you're seeing where you're going. So it was. The answer you got.
[speaker002:] Other sheet
[John:] Right other sheet. Other sheet. E so we start off with E. Rai it was on actually to the point three not point nought three. So I got that wrong. Raise it to the poi to the power point three.
[speaker002:] Point three.
[John:] And then take one over it.... And what's that come to?
[speaker002:] Point seven
[John:] Is that the answer they got in the book?
[speaker002:] Point seven four one..
[John:] Right.
[speaker002:] .
[John:] So you had got the right answer.
[speaker002:] Mhm.
[John:] But going from there
[speaker002:] To there.
[John:] evaluating that
[speaker002:] I loused up. [LAUGHTER]
[John:] on your your calculator, you probably really didn't know what on earth this means
[speaker002:] Yeah, I know where I went wrong there. I can see what I've done there.
[John:] Right. So there's a there's a few things come into this. One is obviously, being able to use the calculator. But before you can use the calculator, you need to understand what you're doing and you say, My calculator's got the cube root on it. I can do cube roots and things. Well great well what's the cube root mean? I don't know. No idea, just this thing on the calculator.
[speaker002:] Mhm.
[John:] So once you understand and my calculator can do sort of ten to the minus point four seven. So can mine but what's it mean? [LAUGHTER]
[speaker002:] Yeah.
[John:] When would you use it, how would you put things in, what order what would you be doing? So that's that's the basis really of your sort of higher calculations that you're
[speaker002:] Mhm.
[John:] doing in lots of in all sorts of places in the B Tec level come in. And you'll be trundling along quite happily saying, Yeah I understand this, we're doing that. and then he'll put some squiggles on the board and he'll say, Well of course this comes to erm seventeen point six. He m he might sort of put an answer down like this and erm he'll say, Well this this should come to something like seven teen point six times ten to the minus nought point four. Mm mm mm, what's happening? [LAUGHTER] now you know, you woul it's not standard form, don't confuse it With standard form, you always have integer indices here. So then you always get just the one thing before the decimal place. One digit before the decimal place. One point two three four, times ten to the minus twenty three or something.
[speaker002:] Mm.
[John:] If you're dealing with erm atoms and stuff like this, this huge or tiny number will come in. One over ten to the twenty three. That but that though Watch out because he will
[speaker002:] [cough]
[John:] You should be drawn to that negative index as soon as you see one. But when
[speaker002:] Mhm.
[John:] you do see one, you leave it in because you're going to deal with
[speaker002:] Yeah, one over
[John:] these nice simple laws we've got for indices. Add them when you're multiplying, times them when you raise them to a power, so we don't want to start changing this to one over and things, because we'll lose that easy way of doing things.
[speaker002:] Mhm.
[John:] But when we're getting near the answer, or when we've first got the question and we're trying to think about it. Someone says, now is this a big number or a small number. W what is that, a big number or a small number. That's a number in standard form, is it big or small or what?
[speaker002:] It's very small.
[John:] It's minute. Right. If we make it something like one point
[speaker002:] It's nought point and twenty two
[John:] Right. One times ten to the minus six.
[speaker002:] [LAUGHTER]
[John:] Well what would one times ten to the six be?
[speaker002:] Nought point five nought [LAUGHTER] one.
[John:] W that would be ten to the six
[speaker002:] Oh.
[John:] plus six
[speaker002:] Yeah.
[John:] that's right.
[speaker002:] It's a million but erm
[John:] Now that's a million and that's a millionth. So this
[speaker002:] Mhm.
[John:] is
[speaker002:] Whatever you [LAUGHTER] call erm twenty four [].
[John:] That would be a million
[speaker002:] Erm
[John:] million million million.
[speaker002:] Yeah. [LAUGHTER]
[John:] Okay? And and and this which is a minus twenty four, is one million million million millionth.
[speaker002:] Mm.
[John:] Very small.
[speaker002:] Yeah.
[John:] And I think I think if I had something like that, I'd tend to write sort of zero
[speaker002:] Zero. [LAUGHTER] Yeah. []
[John:] But but there are times when that is not zero.
[speaker002:] Mhm.
[John:] If you're dealing with erm how many you know, how many atoms of Hydrogen do you get in erm well a litre of gas or something,
[speaker002:] Yeah.
[John:] sort of about ten to the twenty three. And erm so what would be the volume in in litres of one atom of Hydrogen. So what it would be
[speaker002:] Mhm.
[John:] this it would be something times ten to the minus twenty three. very small. Well we expect to be very small.
[speaker002:] Yeah.
[John:] It's not zero. Cos we're working atomic level.
[speaker002:] It's still there, yeah.
[John:] If you're working at erm how far is it from here to New York and someone gives you an answer like this like What is it? About two thousand miles is it? Someone gives you a sort of two point three four And you've got about twenty five decimal places of miles
[speaker002:] Mhm.
[John:] You'd say, rubbish I didn't even want to know whether it was say two thousand two hundred and twenty six. Two thousand miles is near enough.
[speaker002:] Yeah.
[John:] Yeah? Not bothered about that, just want an idea. Er but if you're working round the low end, then you are interested in the tiny tiny differences. Right, does that help with logs?
[speaker002:] Yes.
[John:] To understand what Logs are indexes. For everything that we've done there, all the way through, I should have been saying logs. If I had of done [LAUGHTER] you would have been thinking, Ooh ooh [LAUGHTER] ooh ooh []. like this. But with indexes it's easier to understand.
[speaker002:] Mhm.
[John:] Now did you draw I asked you to draw some graphs didn't I.
[speaker002:] I was going to and had no graph [LAUGHTER] paper [].
[John:] I see.
[speaker002:] [LAUGHTER]
[John:] You should have said.
[speaker002:] Yeah it only clicked as you'd gone, when I had to do it. I was halfway through,
[John:] Aha. I Yeah. Ah.
[speaker002:] got all these done and then bump.
[John:] I've got some graph paper [break in recording] car, when someone tries to describe it to you.
[speaker002:] Mhm.
[John:] You haven't got much chance of really understanding what they're on about,
[speaker002:] Mm, no.
[John:] saying, Well yeah sort of, I suppose or. But you can't really follow it. If they show you a picture, you've got a better chance. Erm or a a working model with the pistons going up and down and things.
[speaker002:] Where it's
[John:] What roughly what it did. And a real one fine, you can see what's happening then. Erm so if you're working with logs, if you're working wi I mean if you're working with say, Y equals X squared,
[speaker002:] Mhm.
[John:] and you want to tell someone something about it, you can give them lots of examples, three, Well it would turn that into nine. If someone gave you minus four, it would turn that into sixteen.
[speaker002:] Yeah.
[John:] Erm to someone who knows a little bit about maths,
[speaker002:] [clears throat]
[John:] a good way to explain it to them is draw the graph. Say, Well that's what it looks like, that's the whole
[speaker002:] Mhm.
[John:] function. They can look at it, in a glance, they've got a picture. Ah right, I can see what it does now. Just find a value anywhere on that, I can see the general trends, what's happening. So with logs, if you can draw a picture and Don't just sort of look at a picture in a book there's a graph of Y equals log X. Do it yourself. I mean
[speaker002:] Mhm.
[John:] It's only take you probably take you about twenty minutes, to do, Y equals log X, just use your calculator.
[speaker002:] Mhm.
[John:] And Y equals E to the X. And erm draw them on separate ones and then compare them and see what you get. And then you staring to understand what about. Now integration, did you have any with logs in?
[speaker002:] Erm let me think.... I can't remember off hand but I can soon tell you.
[John:] Okay.
[speaker002:] Cos if we have they'll be in here.
[John:] Right,
[speaker002:] Mm.
[John:] Erm
[speaker002:] Brrubububub. [vocalized pause]
[John:] Just here..
[speaker002:] [cough]
[John:] Here's a lovely... a lovely looking expression. D equals C log to the base E of one plus root L squared plus C squared
[speaker002:] blah blah blah.
[John:] Well it would wouldn't it? Yes. Oh I see.
[speaker002:] [LAUGHTER]
[John:] What's that mean? Let's let's see if we can decipher this one. The others haven't got logs in unfortunately. Let's start with one, let's start with an easy one. That hasn't got logs in.... Mhm.
[speaker002:] Erm... er Yes we obviously have had them, [LAUGHTER] there's one []. Three X bracket log E
[John:] Mm.
[speaker002:] five X.
[John:] I don't know what we're gonna do with that but at least now you
[speaker002:] Ooh God there's another one.
[John:] Right, so simultaneous why don't you i why don't you integrate this log Or why don't [LAUGHTER] you [] erm
[speaker002:] Because I don't know what it is.
[John:] What are we doing? Well let's let's just have a quick look at this one. What's this describing here? V equals large V or V nought or whatever you like,
[speaker002:] [cough]
[John:] E to the minus R T over L.
[speaker002:] Right, er
[John:] What would it's what would it's graph look like?... It was number ten somewhere.
[speaker002:] Yeah there you are, this one.
[John:] Mhm.
[speaker002:] no [LAUGHTER] actual description [] of what it was at all.
[John:] No, well these are all real, I mean there's one, number nine. Let's look at number nine,.
[speaker002:] Erm working out voltage across a capacitor in a... series.
[John:] Okay, so a capacitor is being charged up. Erm, have you seen the curves for a capacitor charging up?
[speaker002:] Once, a while back. [LAUGHTER] Yes I did.
[John:] Erm... Okay so there's
[speaker002:] It roughly goes, like that.
[John:] There's time and there's your potential voltage. It starts off at zero voltage at
[speaker002:] Mm.
[John:] zero time.
[speaker002:] It goes roughly like if you'll pardon the lumps [LAUGHTER] everywhere [].
[John:] Oh I'll pardon the lumps. In fact that's beautiful cos that's what it actually does. Ideally it doesn't have that wiggle in but actually
[speaker002:] Mm.
[John:] it quite often becomes unstable around there.
[speaker002:] Yeah.
[John:] Okay, that's lovely. That is exactly what it looks like. So
[speaker002:] That's when it reaches it's peak value
[John:] Sort of sort of never reaches
[speaker002:] Yeah it never quite gets there but.
[John:] some time over here it's it's getting. So that's it's peak value. Now from this equation, V equal Where are we on? This one. Sorry.
[speaker002:] [LAUGHTER]
[John:] Right, V equals one minus this lot.
[speaker002:] Right.
[John:] That doesn't affect the sign, that negative in the index.
[speaker002:] Mhm.
[John:] Right, so this is always a positive number, so it's always one minus a positive number.... That this this V here you can think of as V max.
[speaker002:] Yeah.
[John:] Right. What's the maximum value
[speaker002:] [cough]
[John:] That's
[speaker002:] [cough]
[John:] If that V max This describes how it's nearly getting there.
[speaker002:] Mhm.
[John:] It never actually gets there. Cos it's always one minus take away E to the Well let's say this thing here, T over C R, comes to about twenty three or something.
[speaker002:] Mhm.
[John:] One minus E to the minus twenty three.
[speaker002:] Which is
[John:] One over E to the million million million million sort of thing
[speaker002:] One over E to the twenty three. [LAUGHTER]
[John:] Zero by the time you get to about here
[speaker002:] Yeah, near as Yeah it's zero.
[John:] Forget it. Forget it, it's not making any difference. So here or here or here it is really zero We can never say it actually is
[speaker002:] Yeah, it's so small it's not worth it.
[John:] cos we've just got that ti but you could never show it on a graph,
[speaker002:] Mhm.
[John:] very accurate. So... Now here when T is small, let's let's say let's put let's put C R equal to one, they're only
[speaker002:] Mhm.
[John:] a constant in your circuit, in your electrical circuit.
[speaker002:] Yeah. resistance and capacitance.
[John:] So let's say. Curr capac capacitance Yeah I for current, C for capacitance. Okay?
[speaker002:] [cough]
[John:] So.
[speaker002:] [cough]
[John:] Let's rewrite the formula as something like V equals V max times one minus E to the make that one, minus T.
[speaker002:] Mhm.
[John:] Yeah? I mean we could make that one, we could choose the R the R
[speaker002:] [sniff]
[John:] so that C times R came to one.
[speaker002:] Yeah.
[John:] Okay? What happens here? well if T is nought, we've got one minus E to the
[speaker002:] Zero.
[John:] zero. And what's E to the zero come to?
[speaker002:] Er Brdubudubum [vocalized pause] one.
[John:] Right. So that'll be one minus one, comes to zero. That's good, that works.
[speaker002:] Zero. Which is right. Yeah.
[John:] What about if T goes on for about erm one second, when it's going to be equal to V M one minus E to the minus one. What does E to the minus one?
[speaker002:] One well, E to the one over one.
[John:] Good. One minus one over E
[speaker002:] E to the one.
[John:] to the one. We don't bother about the one, we just leave it one over E. And E is about three.
[speaker002:] So
[John:] So one minus a third
[speaker002:] A third
[John:] So after about sort of, after about one second, it's come up to about two thirds. So about
[speaker002:] Mhm.
[John:] after one second it's come up to two thirds of V.
[speaker002:] Mhm.
[John:] And what happens after two seconds? Well minus E to the minus two. And what does that mean, E to the minus two?
[speaker002:] Is one over E squared.
[John:] Right one over E squared. One over E squared, take E as about three.
[speaker002:] So it's about one over
[John:] One minus one minus one ninth.
[speaker002:] Mhm.
[John:] Which is about eight ninths.
[speaker002:] Yeah.
[John:] So after two seconds, it's got up to about
[speaker002:] Eight ninths of [LAUGHTER] the way [].
[John:] Eight ninths of the way it's going. And the rest of the times it's you know, it's taking If we could keep going for
[speaker002:] It's just dwindling more and more and more away.
[John:] We keep we keep going on for longer and longer but it's taking a lot longer for us to get any closer. You get most of your stuff done, sort of pretty instantly. So after three seconds, we On graph paper I mean that that's a good one for you to do on graph paper actually. Do
[speaker002:] Mm
[John:] that one on your graph because it's a real application that will apply to your work and it's the sort of thing you're going to be plotting anyway. So you can still understand what's going on with these. Rather than just putting these numbers into your I mean you will put these numbers into your calculator, but do it with that, leave leave your C R out of it and just keep it as T.
[speaker002:] Mm.
[John:] So if you kept C R fixed and you happened to fiddle them so it came to one, you could just You You're working out your graph you see you see then. Erm and when we get E to the minus three?
[speaker002:] Then it's one minus one over E cubed.
[John:] Okay. One over E cubed and E is about three so twenty seven. So we've got about twenty six, twenty sevenths.
[speaker002:] Which is getting pretty close to near as dammit now [LAUGHTER].
[John:] Right and if you take E as about say ten seconds
[speaker002:] [LAUGHTER] Yeah.
[John:] don't have to go that far, six seconds say.
[speaker002:] it's what, it's about ninety nine hundredth of the way.
[John:] Well i is it? After six seconds
[speaker002:] W after six?
[John:] One minus one over E to the six.
[speaker002:] Six. Eek. Erm
[John:] well have a go see what three to the power os six
[speaker002:] [cough]
[John:] Three to the power of six comes
[speaker002:] [cough] It's seven hundred and twenty nine.
[John:] Seven hun and then do an inverse of that.
[speaker002:] Erm seventy nine
[John:] Inverse.
[speaker002:] Erk. No that ain't right.
[John:] It might be. No okay.
[speaker002:] Seven seventy nine It is. Inverse,
[John:] Yeah so we've got by by
[speaker002:] It's it's something hideous.
[John:] six seconds, we've got seven hundred and twenty eight, seven hundred twenty ninths of the way
[speaker002:] And twenty ninths Yeah
[John:] so I mean, after a few seconds forget it
[speaker002:] You can say forget it.
[John:] There's hardly any change.
[speaker002:] Mhm.
[John:] So that's a that isn't a straight logarithmic curve, but that's a thing that occurs in nature a lot and in electrical circuits, that's a growth curve. The way it gets most of its growth in the first I mean,
[speaker002:] Few seconds.
[John:] human growth looks like that as well.
[speaker002:] Yeah.
[John:] Erm you get most of your you know you The amount of weight and height you put on from say, nought to six months and then six months to two months and by the time you've got to about eight or something.
[speaker002:] Mhm.
[John:] Your half well your whatever age it is
[speaker002:] Yeah.
[John:] measure kids at eight and say, Oh right you're going to be six foot two, you're going to be five foot ten.
[speaker002:] Mhm.
[John:] Er because they know that on average unless things change a lot, that they they're following this curve this growth this growth curve. But try try E to the X as well. Y equals
[speaker002:] [cough]
[John:] E to the X and Y equals log to the base E X. And and that wi with just one in. Erm... and I think you'll I It's not you know, it's not about following all all this stuff all the technology stuff in Tech
[speaker002:] Mhm.
[John:] engineering, science in general, physics, it's not Anyone can do the maths because you just put it in your in the calculator and if it
[speaker002:] Yeah.
[John:] says erm he wants seven to the minus nought point six, well you just put you know, when you're doing X to the Y you put in seven and you put minus nought point six and you get some answer out.
[speaker002:] Mm.
[John:] Well okay, fair enough. But then when you get things like that,
[speaker002:] Yeah.
[John:] And this doesn't tie up with what's in the back of the book. You don't know whether this was right or not.
[speaker002:] No.
[John:] But if you can look at that and think, Well minus so it's one over. It's one over E to the point three. What's point three, well it's about the cube root of three, or you could sort of try cube root of three
[speaker002:] Mhm.
[John:] on your calculator. And then try one over that. Oh that's about the answer in the back of the book, so I must have just gone wrong on the last bit here. And you could you could go back and start working through all this again. If you So it's knowing where you are, having a good feel, like if erm if I say, multiply two hundred by thirty. Well let's say I multiply two hundred and twenty two by three hundred and thirty three. And you try it on your calculator and you get about fifteen billion.
[speaker002:] Mm.
[John:] Or you get six. You say, That's obviously rubbish I've done
[speaker002:] Yeah.
[John:] something wrong here. So you know to check it and go back and do it differently and it's the same with this moved up a level now, that you you keep in keep in charge of it. So let's have a quick look at your log integrals.
[speaker002:] Right.
[John:] How do you feel about integrals and differentials?
[speaker002:] Erm the differentials I'm getting the hang of eventually. And getting there slowly but surely.
[John:] Are you are you leaving it for a day and then trying to integrate them?
[speaker002:] Yes I am.
[John:] Well what have you what have you done in term Have you got some you've done in terms of differentiating and then integrating back?
[speaker002:] Erm to be honest with you I think I've binned them now cos I cleaned up me room.
[John:] Okay.
[speaker002:] I've only got a couple that I did last week.
[John:] Doesn't matter. Erm try and do a few each week. Erm which are the most awkward things to differentiate?
[speaker002:] Erm...
[John:] Don't worry if you can't find any cos I have a bag full here of [LAUGHTER] awkward differentials.
[speaker002:] [LAUGHTER]
[John:] I've only got a few books on differentiating. about we're probably due to finish, well we'll just have a quick look at this. Erm well there we are, here we are. How about this...
[speaker002:] The awkward ones for me are when they start throwing stuff like er
[John:] Like what, pick an awkward one out of that lot.
[speaker002:] An awkward one out of that lot. Is I wish Erm to be honest with you, I find them as awkward as hell. Where you've got a division in them. They do
[John:] Okay.
[speaker002:] stump me. I must admit. They really do throw me
[John:] Right so... A quick sort of look at different patterns. to di to differ I mean it's it's it is no good sort of trying to go in in depth into the integration until you're really happy with the
[speaker002:] Mhm the basics.
[John:] the differention
[speaker002:] Mm.
[John:] differentiating. Because then, you just look at it and say, Ah I know how we got here, that reminds me of that one I did the other week where it came out looking like this. So I'll work backwards from that.
[speaker002:] I'll work backwards. Yeah.
[John:] And you you you have to sort of attack it in that way. X squared over cos X plus sine X. You like you like the trig functions then?
[speaker002:] Oh I adore them.
[John:] Erm Y equals... X squared over X.
[speaker002:] I'll just close this over.
[John:] Okay, could you differentiate that?
[speaker002:] Erm...
[John:] Write write the formula out.
[speaker002:] I'm thinking about
[John:] If you're using the formula.
[speaker002:] Erm minus one so it's two X... to the power of one, not very nice is it. Erm
[John:] Do what would you do to get that?
[speaker002:] To get which bit?
[John:] To get the whole answer.
[speaker002:] The whole thing?
[John:] Yeah.
[speaker002:] Erm what I do is erm... bump
[John:] Okay so you differentiated the top part.
[speaker002:] Mhm.
[John:] And then you drew your line and you differentiated the bottom part?
[speaker002:] Yeah. I'm not happy with that.
[John:] Right. Is that what you do?
[speaker002:] No.
[John:] Right what what
[speaker002:] Definitely not.
[John:] Well if I gave you something like this erm Y equals... X times... X squared plus two.... Three X two. How would you differentiate that?
[speaker002:] As a function of a function.
[John:] Mm.. Now you could integrate it as a function of a function. If I said, integrate that
[speaker002:] Mm.
[John:] Then you'd say, Ah D X is a differential of X squared.
[speaker002:] Mm.
[John:] If I differentiated this I'd get two X cos the two would disappear.
[speaker002:] That's right
[John:] right order. So I could integrate that. That's a that's that's one that might even be easier to integrate than differentiate. But how would you differentiate it. What what rule what methods do you know for differentiating?
[speaker002:] Erm... well there's a few. [LAUGHTER]
[John:] Okay, there's a few, so name a few.
[speaker002:] Well I've got to the first one
[John:] Right, that's that's just that's just the basic one
[speaker002:] Mm.
[John:] for when you've got a simple straightforward of X. Like if I said, Y equals X squared plus two, you just differentiate it straight off by doing each term like that.
[speaker002:] Mhm.
[John:] Right, and then some that we were looking at there was a function of a function. Right, now what other ways are there of of differentiating something that's awkward?
[speaker002:] Erm ch erm... Well they have different techniques of differentiation like differentiation of part. Erm differentiation by substitution is it? No?
[John:] Mm. Have you heard of products and quotients?
[speaker002:] Oh yeah. Oh yeah
[John:] Right?
[speaker002:] of course. Sugar.
[John:] Right?
[speaker002:] Yeah.
[John:] Okay. So how would you differentiate that as a product then?
[speaker002:] Mm....
[John:] Okay. Right and what does that come to?...
[speaker002:] Right then. So it's X plus two... three X plus three plus... three X...
[John:] Okay and you want What does that come to?... Good. Yeah by D X otherwise it looks as if it's just D Y
[speaker002:] Yeah... Erm...... [cough]... Right er X squared... squared.. Yeah...
[John:] X squared plus X.
[speaker002:] Mm.
[John:] Okay. Now I'll show you how I would differentiate that.
[speaker002:] Okay.
[John:] Okay. You may prefer your method, if so, you're very welcome to use it.... Three X times X squared
[speaker002:] Three X squared.
[John:] Three X times two.... Now I want to differentiate that. Differentiate X cubed, nine X squared. Sorry three X squared.
[speaker002:] Mm.
[John:] Three X squared times three... X squared
[speaker002:] X squared plus six. Mm.
[John:] Which proves
[speaker002:] It's quicker. I'll give you that much, it's much quicker.
[John:] right. Now [cough] this is is a really good way for a product. It works very well.
[speaker002:] Mhm.
[John:] But sometimes they give you something like this and you say, Oh it's one times Especially
[speaker002:] Yeah.
[John:] give you something like this, they might say erm,... say, Four X squared minus one times two X squared plus three. can do it this way.
[speaker002:] Yeah.
[John:] Or you could just multiply them out if you liked.... Yeah? And then differentiate that. It doesn't matter, it depends on how big it is. If this was sort of a cubic or something then you'd have all sorts of terms and you could chance of making a mistake. But erm watch out for that cos sometimes it's easier to multiply out Hi. Sometimes it doesn especially if it's just one thing outside there, like it might have been sort of a a three X or
[speaker002:] Yeah.
[John:] five X cubed or something. Just multiply it. Erm so right, if you get something then where you're integrating I'd like to sort of have a little practice this week of Have you done much practice of differentiating a product?
[speaker002:] Erm I did, but I haven't lately.
[John:] Okay do a couple of products. I mean that was fine, you did that alright.
[speaker002:] Mm.
[John:] Erm that was great. So just do a couple two of those and then Do you know the formula for differentiating a quotient?
[speaker002:] Erm... It's a sim similar except it's minus over V squared.
[John:] Wow brilliant. That was great.
[speaker002:] The only way I can remember it, that way [LAUGHTER] round [].
[John:] Yeah erm most people just get lost in this and sort of finish
[speaker002:] Yeah.
[John:] up looking it up.
[speaker002:] Well, what happened with me was, when we got taught it, he
[John:] Yeah.
[speaker002:] started off doing them like this and it was U D V by D X
[John:] Mm.
[speaker003:] I'm slipping out Graham.
[speaker002:] Mm erm Okay.
[speaker003:] leave these coppers for this gentleman here.
[speaker002:] Mm.
[John:] Oh yes.
[speaker002:] Okay. We got it
[John:] Thanks very much.
[speaker002:] this way round.
[John:] Okay, what did he give you?.
[speaker003:] Are you sure? No that's fine.
[John:] Thanks.
[speaker002:] Oh.
[John:] Well I think you've done very well actually. We've covered a lot tonight that I was saying other people erm could take three lessons to cover it pick it up very well. Good.
[speaker003:] Thanks very much. [LAUGHTER]
[John:] How how do you feel about it?
[speaker002:] I feel fine ploughing through it though yeah.
[John:] Did you you know, did you get it Mm. you're not sort of ploughing through you're you know, you're picking it up and your bringing it I like the way you the way you're assimilating it.
[speaker002:] Mhm.
[John:] That you're not sort of, Oh just I'll memorize this lot. But you're understanding it and you're fitting it in with what you already know. Which is the only way to do it.
[speaker002:] Mhm.
[John:] You're building up a knowledge base, you're not just building up lots of odd facts like you know, how many people live in China, or
[speaker002:] Mm. You
[John:] try to memorize them you're you're making them mean something. So if we g let's say if I gave you something top integrate.
[speaker002:] Mhm.
[John:] Right, if I gave you er something like that. Let's say we'd started off with... three X
[speaker002:] [cough]
[John:] X squared there.
[speaker002:] [cough]
[John:] Right. let's say let's say I just gave it to you like that.
[speaker002:] Yeah.
[John:] Right. I gave you that and I said, Y equals that, now integrate it. And you sort of spot this connection here.
[speaker002:] Yeah two X
[John:] And
[speaker002:] and the X
[John:] And
[speaker002:] squared.
[John:] And that.
[speaker002:] Mhm.
[John:] Say, that one's Well if that one was if that one was... U that would be D U or if this one was V, that would be D V and
[speaker002:] Mm.
[John:] D V by D X
[speaker003:] Sorry to be a nuisance.
[John:] No you're no problem.
[speaker003:] We're off Graham okay, we'll see
[speaker002:] Right.
[speaker003:] you later, we won't be long.
[speaker002:] Okay.
[John:] And do hang on a minute don't go don't go. Do you want to go out with them? Or is it just to do the shopping?
[speaker002:] Oh no no
[John:] No? Okay. I mean we're about finished anyway. Erm so if I gave you something like that to differentiate
[speaker002:] Mhm.
[John:] you'd say, Oh that's yeah. That three if you differentiated the three X you'd get the three. If you differentiated the X squared, you'd get the two X.
[speaker002:] [clears throat]
[John:] So if I gave you something to differenti to to integrate
[speaker002:] To integrate.
[John:] If I gave you something to integrate that looked like erm say it had a sine X, I know you like those.
[speaker002:] Yeah, right.
[John:] Right. So instead of the three, let's let's leave the two X exactly as it is
[speaker002:] Mhm.
[John:] and leave that. But here instead of the three X,
[speaker002:] [cough]
[John:] had
[speaker002:] [cough]
[John:] X squared plus two plus two. So X squared plus two times something add something times two X. Okay
[speaker002:] Mm.
[John:] differentiate the two X, you get that. And in...... in there if I had sine X,
[speaker002:] Mm.
[John:] and in there,
[speaker002:] Cos.
[John:] cos X. Now you say, Oh okay, maybe er Now don't forget you're integrating so you're going to have problems with the sine. Er when you differentiate a sine you get a cos. Differentiate a cos you get a minus sine. So when you integrate one And the best way to
[speaker002:] Mm.
[John:] do it the best way to do it if you're trying to say, let's integrate cos X. cos cos X. Trying to integrate that you say
[speaker002:] Mhm.
[John:] well just forget trying to integrate it. It came from a sine didn't it? If I differentiated a sine, I would get plus cos right okay.
[speaker002:] Mhm.
[John:] So it does integrate straight back to that. But if I was integrating a sine, it would have come from a cos and the cos should have given me a minus sine.
[speaker002:] Minus sine.
[John:] So integrating sine will give me minus cos. And
[speaker002:] Yeah
[John:] you can get th I've seen people spend sort of quite a few minutes going round in circles and give up on that when they're nearly at the answer. [LAUGHTER]
[speaker002:] Mm. The easiest thing is if you can just remember one.
[John:] Just one.
[speaker002:] Just one of them it sticks.
[John:] And and and work from it instead of trying to remember the whole rule. So to integrate.
[speaker002:] It's cos X brackets X squared plus two. Cos you just pull them out of the first brackets of each one.
[John:] Right.
[speaker002:] There's U and there's V.
[John:] Right, so unless you have practised doing a few, integrating U times V
[speaker002:] Mhm.
[John:] you won't spot that because this is
[speaker002:] You won't spot that. [LAUGHTER]
[John:] this is about, Ah I've seen something like that before. And it would be absolutely laid out on a plate for you like that.
[speaker002:] Oh no.
[John:] It might be the other way round, it might be
[speaker002:] You have to twist it about a bit.
[John:] minus one, it might be you know erm for example that might be. It wouldn't be given as a cos X two X, it would be given two X cos X.
[speaker002:] Mm.
[John:] So you'd have to start looking for these patterns, but you need to build up to those, build up your experience so that you've seen them. Do a couple of products.
[speaker002:] Mhm.
[John:] probably do a few. Do a few products and then, leave them for a few days and go back and integrate them.
[speaker002:] And try and Mm.
[John:] And you t you'll find sort of quite a few examples of products erm
[speaker002:] [cough]
[John:] So one day, say, do a few normal type products, next say, put some sines and coses and and gets
[speaker002:] Mm
[John:] boring after a while doesn't it if you don't have sines and coses. So put some sines and coses in.
[speaker002:] Mhm.
[John:] Differentiate those. And then sort of set those two on the side and then go onto products Sorry, quotients.
[speaker002:] Quotients yeah.
[John:] And do the same. Differentiate a few quotients, get the hang of it, throw in a few sines and coses. Okay, and then by the time you've done that it'll be time to go back to the start
[speaker002:] Yeah.
[John:] and start integrating your products. Integrate your simple products, then integrate the products with your sines and cos and
[speaker002:] Mhm.
[John:] then integrate your quotients. The an the result of getting the quotient and by the time you get to looking at those you'll integration's hard and you will not ever say again what you said to me a long time ago that you thought integration was a lot easier than differentiation.
[speaker002:] Mm.
[John:] I thought, There's something you're missing there if you think that. [LAUGHTER] What do you think now then, which would you rather do, differentiate or integrate?
[speaker002:] I think the differentiation now I'm beginning to
[John:] Especially especially
[speaker002:] [LAUGHTER] understand it better [].
[John:] Especially if I'm going to sort of give you things like Y equals erm cos E to the minus K X. I mean it's only a simple one. It's only a simple one.
[speaker002:] Yeah.
[John:] It's not a real thing. And I could be throwing but it's the sort of thing that's gonna come into your circuit stuff. Yeah?
[speaker002:] Yeah.
[John:] If you've got phase diagrams and you've got V and things varying. And some fi sine functions and it's not a straight
[speaker002:] .
[John:] It's not a straight sine thing, it's erm another curve
[speaker002:] [cough]
[John:] a lot.
[speaker002:] [cough]
[John:] A logarithmic decay, sort of thing, a rat's tail cap
[speaker002:] Oh I know what you mean, yeah.
[John:] Capacitor discharging over over running the opposite way and then that's not drawn properly but you know what
[speaker002:] I can see what you mean, yeah.
[John:] You've seen it before you know what's
[speaker002:] It's going down and every time it goes down it's equal on both sides yeah.
[John:] Yes. Gradually gets dies down to nothing. It might be a very tight thing that looks almost like this.
[speaker002:] Yeah.
[John:] down to down to
[speaker002:] Gone in no time.
[John:] So you've got plenty to do, now
[speaker002:] Right.
[John:] we I try and cover what we can in the lessons Not so much what we can but what I think is the right amount for you.
[speaker002:] Mhm.
[John:] Erm and it's up to you then to plenty of time through the week I know you've got nothing
[speaker002:] yeah.
[John:] else to do. [LAUGHTER] No You've got ab there's a lot of work in this
[speaker002:] Mhm.
[John:] erm but the more understanding you have, there will almost certainly come a point in in your course when
[speaker002:] Mhm.
[John:] people start throwing stuff at you, developing a whole new topic and you're lost from the start. I mean you know, there's some some concept of a model which just escapes you. It it happens to everyone.
[speaker002:] Yeah.
[John:] It happened to me when I was sort of looking at some sort of electronic engineering that I wasn't supposed to be doing I was just looking at it for interest like Well I'd like to have the time to spend on that but I'll leave that thank you [LAUGHTER] very much. You know []. You can keep that. So... you can you can get lost because you don't understand the notation, cos you don't understand the model or because your maths can't keep up with it because you're not you know, E to the
[speaker002:] Yeah.
[John:] E to the minus nought point three. So the more the more that you understand, the less you're gonna be
[speaker002:] Yeah.
[John:] So you've got plenty to do. [LAUGHTER]
[speaker002:] [LAUGHTER] Mhm. []
[John:] I'll leave you to get on with it and I'll
[speaker002:] Okay.
[John:] get on.
[speaker002:] [LAUGHTER]
[John:] rushing about doing things today. next week.
[speaker002:] [LAUGHTER]
[John:] That's yours that's your pen too isn't it.
[speaker002:] Yeah.
[John:] Right. My pen. I lost a pad somewhere the other week with some It wasn't important notes but some notes in. print out again..
[speaker002:] not a bad idea.
[John:] Mm. [break in recording] Graham, Monday or Thursday preferred. [end of recording]