[John:] we get more of you than me get plenty of me on these. Right that's better, okay? Now where's the question in the book, let's see what, what you made of it before you copied it down.
[Ian:] No actually it was just, it was just like a question erm find X. That
[John:] Okay.
[Ian:] was, that was all it was.
[John:] Okay.
[Ian:] It was just like a big list of questions.
[John:] So... right, so we've got X and two open brackets X add four close brackets equals minus four. And the problem is this brackets here, the X plus. What are we going to do with that? Erm before we do anything let's have a look and see what it would mean if, it would mean just as numbers. Let's say X is was ten, I mean it's not but let's say it was, we'd have ten add two brackets ten add four. Minus four, now we won't bother with that side, what would we do on this side? Would you add the two to the ten or what?
[Ian:] Yeah.
[John:] You sure?
[Ian:] No, I would multiply it.
[John:] Right, in there between the two and the brackets although they don't bother to put it in, there's a multiply. So that's the first thing to realize, we've got a multiply in there now we've got this two piggy in the middle here, between an add and a times, so the times win. Yeah? Now we've got multiply into the brackets, so we'll just leave the ten where it is and work out what happens to this, everything inside the bracket has got to be multiplied by?
[Ian:] By two.
[John:] So? Two times ten
[Ian:] Add two times ten, twenty. Two times four, eight.
[John:] Times ten add two times four, eight, That's got rid of the brackets, equal minus four. Now are you happy with that?
[Ian:] Yeah.
[John:] Okay, so you do it with the Xs then, I did it with the numbers, that's the easy bit.... So whenever you've got Xs and Ys and all sorts of strange things in there think just, it's only a number. We don't know what it is yet, we haven't found out, but it's only a number. And whatever we do, if it was a three, a seven, or a ten, a five just do the same but with the X....
[Ian:] Shall I?
[John:] Yeah that's fine carry on. What do you do next? It's going very well.... Are you sure about that?
[Ian:] Not really no.
[John:] It's right, it's correct it is minus twelve but I could see you were really not very sure what you should be doing about that, that's great, you've found, you've got that down to there, let's have a little look at the number line, zero, one, two, three and so on, minus one, minus two, minus three, minus four. And what have we got here? We've got minus four take away eight, so we start at minus four start there and then we take away, which means count along that way for eight, so we get minus five, minus six, minus seven, minus eight, minus nine, minus ten, minus eleven, minus twelve... Okay? Whenever you're not sure about what to do just look at that and just write it down or just think about it, visualize it, there's zero, then we can, adding a negative number or taking away, go that way up towards the negatives, adding a positive number, go that way. Okay, so you've got three X equals minus twelve... see minus twelve, a negative divided by a positive comes out negative. It's the same for addition and erm same for multiplication, division. If the signs are the same, both signs the same when you're multiplying or dividing, both signs the same will give you what?
[Ian:] Positive.
[John:] Right good, and if the signs are different?
[Ian:] Negative.
[John:] So you get th a negative divided by a positive be a negative cos the signs are different. Oh that's no problem is it?
[Ian:] No.
[John:] Erm now what were you doing when you were
[Ian:] Th
[John:] trying it? You were doing two times X add two times two X add eight
[Ian:] I was two there
[John:] Mm you were
[Ian:] And the two and the two X. I was leaving
[John:] Okay let's, let's have another little look at brackets... let's try multiplying say... a hundred and one... by... twenty three. Okay? And we'll do it with brackets. So what have we got? We'll, we'll, we'll do twenty three times a hundred and one, okay? Twenty add three, put the brackets round it show that's one number we have to work out what that is first. Times, which we don't bother writing in... a hundred add one, okay? Now we multiply everything that's in there by everything in here. So would you like to try that? How would you do it?...
[Ian:] You add those together.
[John:] Erm I want you do it without adding them together, okay I'll, I'll show you one. Erm you've expanded brackets before, like this, haven't you?
[Ian:] Yeah, I think s
[John:] Okay, so twenty times a hundred...
[Ian:] Oh yeah.
[John:] is two hundred. Twenty times one is how many?
[Ian:] Twenty.
[John:] Okay. Three times a hundred?
[Ian:] Three hundred.
[John:] And three times one?
[Ian:] Three.
[John:] Then if we add those up...
[Ian:] [whispering] hundred and twenty three [].
[John:] Right okay, do you want to check that on your calculator? It doesn't sound about right.
[Ian:] No.
[John:] Okay so have a look and what's gone wrong here?... Let's work backwards. Three times one, what's that?
[Ian:] Three.
[John:] That's okay. Three times a hundred?
[Ian:] Three hundred.
[John:] Okay. Twenty times one?
[Ian:] Twenty.
[John:] Twenty times a hundred?
[Ian:] Two thousand.
[John:] Right, not two hundred, two thousand. So that's th that's, that's my mistake there, but that's a very common error, yeah? Now if we add it up. It's going to look a little bit better. Two three two there.
[Ian:] Yeah.
[John:] Okay, we check it this way. Twenty three times one O one? How would you do it long division? L sorry multiplication, how would you do that?
[Ian:] What do you mean long?
[John:] Just normal multiplying, the way you multiply that out by hand. Have you tried that or you u have you always used the calculator?
[Ian:] Yeah, I've always used the calculator?
[John:] Okay the way you do it by hand is you just say, one times twenty three, is twenty three. No tens times it. A hundred times twenty three, will be twenty three
[Ian:] Twenty three
[John:] hundred. Add that up, two three two three, is that what you got on the calculator?
[Ian:] Yeah.
[John:] So when we're doing this normal multiplying we're really doing this.
[Ian:] Yeah.
[John:] Yeah? Now if we don't know, let's say we wanted, let's say we want to find out, what would erm what would a hundred and one times twenty seven be? Let's say I'm, I'm going, I give you this question, I want to find a hundred and one times twenty... six, right? A hundred and one times twenty seven, a hundred and one times twenty eight, twenty nine, we'll put twenty five in as well. In fact we'll ma we'll make it all of them, twenty three, twenty two, twenty one, right. Find all these. What's a hundred time twenty six, a hundred times twenty seven, sorry a hundred and one times all of these. Well we could work out a, a general thing we use brackets and see what happens. Are you happy with this?
[Ian:] Yeah.
[John:] Okay. So it'll be a hundred and one times... twenty something, twenty plus X. But we don't know whether we've
[Ian:] Yeah.
[John:] got twenty one, twenty two, twenty seven, twenty eight, twenty, twenty plus something, we could make that N if you like instead of X. We could make it N, N is just some number from about nought to nine. A hundred and one times twenty, what will that give us?... It'll be
[Ian:] Two thousand... and ten.
[John:] It'll be twenty hundred and twenty. Okay?
[Ian:] Oh. I see.
[John:] Yeah. Erm add, now what's N times a hundred and one? Well this might be th this might be too awkward this. It might be easier to split this hundred and one up a bit, hundred add one, times twenty add N, okay? That's a bit easier. A hundred times twenty?
[Ian:] Two, two thousand, two
[John:] Yeah
[Ian:] two yeah.
[John:] Two thousand or twenty hundred, same difference. Now a hundred times N, what will that give?
[Ian:] A hundred N.
[John:] Okay, a hundred. Now one times twenty?
[Ian:] That's one.
[John:] And one times N?
[Ian:] One N.
[John:] One N, so if we add those up, keeping the numbers and the Ns separate, we get two O two O add that
[Ian:] One O one N.
[John:] One O one N, okay so let's see what happens when we try a hundred and one times twenty seven, that just means that N equals seven. Now what could that come to? Well it should come to two two two O two O, plus seven times a hundred and one, which will just be seven O seven. Let's see what that comes to. You want to check that on the calculator, see if that one works?... No?
[Ian:] No.
[John:] No, oh wrong again, what's it, what it give then?
[Ian:] Two seven two seven.
[John:] It should always come to two seven two seven shouldn't it? We've got two seven, that's a two. Right, I put a nine in, that was a bit more than seven, okay. So it does come to two seven two seven. How about, we know it com it always come to that, two twenty plus a hundred and one N, so let's try twenty nine, when N is equal to nine we'll get two O two O add a hundred and one times nine, which'll just be nine O nine.
[Ian:] Nine O nine.
[John:] Add that up, nine is two nine two nine. I don't think there's any need to check it on that cos we can see the pattern
[Ian:] Yeah.
[John:] that's coming out, two seven two seven, two nine two nine, what would erm so what answer would you expect if we try one O one times twenty six, what would ma what would you expect the answer to be?
[Ian:] Three six two six.
[John:] So we expect that and we're pretty certain that'll come to two six two six. We can just put it in there, it's two O two add now this time we've got N equals six, six O a hundred and one times N added on, six O six, two six two six, okay? It's, all this is showing really is that when we do multiplication normally, you see if you, if you've done this sort of multiplication, you've seen the pattern and how things are working and how you're not, someone says to you multiply erm seventeen by a hundred and one. Don't try and do the wh whole lot all in one go, well that's simple George I happen to know my hundred and one times table,
[Ian:] [LAUGHTER]
[John:] as far as the seventeens. You work it out in stages, little bits, and then you add the bits together and then you add the bits together, so you say, well I could do a hundred times seventeen and then I could do ones times seventeen. And then I could add them together, or if that's a bit awkward, what I could do is I could think of the hundred and one as a hundred add one, I can think of the seventeen as ten add seven, and then I multiply them this way and I multiply all the bits and then add all the bits up at the end. So we're doing a hundred and one times, we'll make that erm twenty eight. We've done twenty seven up there, we'll do it again here, say it's a hundred and one times twenty seven. So a hundred times twenty is twenty hundred, a hundred times seven will gives us what?
[Ian:] Seven hundred.
[John:] One times twenty?
[Ian:] That's twenty.
[John:] And one times seven?
[Ian:] Seven.
[John:] And if we add all those up?
[Ian:] Two seven two seven.
[John:] Two seven two seven. Now work, this works with any number [cough] so if we've got, we want to find out erm sixteen times twenty five... there's an easy way of doing it, think of the sixteen as four times four okay? And then that's the sixteen the four times four, so now it's times twenty five. And then we could look at it that way. Four times twenty five is?
[Ian:] ... A hundred.
[John:] So the answer is going to be four times a hundred, four hundred. So it's an easy one to check, see what we do when we do it with brackets. We've got ten add six times twenty add five, okay? If you'd like to do that one?
[speaker003:] Hello John.
[John:] Thanks very much. Got
[speaker003:] Okay?
[John:] time to finish your tea tonight then? Sorry about that last week.
[speaker003:] I came in after you!
[John:] [LAUGHTER] Oh dear []!
[speaker003:] I, I, I was supposed to have a lift, and erm I was waiting like I say ten to five we knock off half one you see and the guy never turned up, the car was there but So er it was a case of where I worked er was G P T er buses
[John:] Buses Yeah.
[speaker003:] and there's no bus direct
[John:] No, how's it doing there, are they still sort of
[speaker003:] [sigh]
[John:] Yeah.
[speaker003:] It's er, well you see G E C took over er and that's the worse thing that could happen to anyone, er cos is an accountant and manufacturers.
[John:] So he just looked at it and says making a big enough profit.
[speaker003:] That's right and if, that's right, and even if you're making a profit
[John:] Yeah.
[speaker003:] if he thinks he can a er
[John:] Make a better profit.
[speaker003:] make a better, by selling the assets and literally stripping it, he'll do so. Er he couldn't give a damn about er
[Ian:] About er people.
[John:] No.
[speaker003:] I mean first thing he did was er redundan we used to get four weeks for every year redundancy payment, bang, no forget it. And er they're going through the courts now.
[John:] It, I mean it now just it just ruins the morale doesn't it? So you're
[speaker003:] That's right.
[John:] not getting good work out, people
[speaker003:] That's right.
[John:] won't put themselves out for the firm and it's, it's you know
[speaker003:] Yeah.
[John:] He can't see that doesn't help in the long run if,
[speaker003:] Yeah, mind you his, his short er term policies so it doesn't matter in the long run.
[John:] Right. [door closing]
[Ian:] Ten times twenty is two hundred. Ten times five is fifty, then we go down here and six times twenty is one hundred and twenty. Six times five is thirty.
[John:] Right. Your, your arithmetic has really improved you know, you can do these things in your head now whereas not all that long ago you'd have been reaching for the calculator thinking, oh I can't do that. And now you have a go, and get it right. Good.
[Ian:] [whispering] []... Is it four?
[John:] Erm let's have a look, all zeros down the end and then we've got three and two that's a five
[Ian:] Yeah.
[John:] Five and two is seven and three makes?
[Ian:] Ten.
[John:] Ten, so that's a nought and carry one, right. And carry one and then we've got two and one is three and the one you carried makes four. Actually you're right with your four there, it was just you snuck an extra ten in somewhere, okay? And that's what we got doing it that way. So you could, you reckon you could do sort of any number of those? If I gave you lots of those to do you
[Ian:] Yeah.
[John:] reckon you could do them no problem? Yeah? So try one with the Xs in now. Now what's the difference? What's the difference? No difference at all is there? Do you want to do another one with the numbers in?
[Ian:] No.
[John:] No? No okay. Try one with the Xs. on the back of that, save my paper a bit,. Right.... Erm... we'll put. We'll go right for the awkward ones and put an X and a Y in as well. What does that come to?
[Ian:] ... Hundred times ten
[John:] Okay.
[Ian:] is a thousand.
[John:] Right.
[Ian:] Hundred times Y, a hundred Y
[John:] Okay.
[Ian:] X times ten, ten X. X times Y, X Y.
[John:] Okay. And then adding all these up, well there's none of them that add up that we can just add in to any of the others, so we'll have to leave them all like that, so the answer is just a, a thousand, plus a hundred Y plus ten X, plus X Y. Now if we wanted to multiply, let's say we want to multiply a hundred and three by seventeen, that just means that X is seven, sorry X is three and Y is seven. So this should come to one thousand plus, what's a hundred times Y?
[Ian:] ... Hundred thousand seven hundred.
[John:] Good, and what's ten times X?
[Ian:] ... One thousand
[John:] Right, and what's X times Y? That's three times seven. So we've got a thousand
[Ian:] Twenty one.
[John:] Right, we've got a thousand and we've got a hundred times Y, hundred times seventeen, okay. And we've got tens times X, ten times three, Hang on, ten X, X is three... okay, we add those up, one, two and three is five seven so see if that looks anything like right. A hundred and three by seventeen.... Is it wrong again?
[Ian:] Yeah.
[John:] [whispering] Oh no, never works this, does it []? So is this right? A hundred times ten is a thousand.
[Ian:] Yeah.
[John:] A hundred times Y is a hundred Y. X times ten is ten X, X times Y is X Y. So that's okay, the bit you were worked out is fine. So we're doing a hundred and three which is a hundred add X, X being three. Times ten add Y and Y is seven. So let's see if we've got these, we've got one thousand, yes. A hundred Y, Y is? No Y isn't seventeen.
[Ian:] Seven.
[John:] Y is seven so that should be seven hundred. Okay? Erm ten times X, which is the thirty
[Ian:] Yeah.
[John:] and then X times Y which is twenty one. Is that right now, are we still one seven five one?
[Ian:] Yeah.
[John:] Okay. Now you can see how easy it is with the numbers to make mistakes especially if you're multiplying by tens or hundreds or thousands or looking at, trying to just look at a bit of it, oh that's just, oh hang on is that seven or is that seventeen? It's actually easier when you're using all Xs and Ys.
[Ian:] Yeah.
[John:] It really is, this is the thing that people won't believe. They I'd much rather be doing a hundred and three times seventeen. But it's easier if there are no numbers in it at all, if it's all letters. When you pop it in you're not gonna, you don't make these mistakes. So try this one, that was a good, that was good that. Try this, A add B times X add Y.... See what that comes to....
[Ian:] [whispering] []... A times X
[John:] That's it.
[Ian:] A X, A times Y.
[John:] Yeah, you normally write them in a straight line but when, when they were numbers I was writing them under each other cos it was easier to add. So yeah, A times Y is?
[Ian:] A Y.
[John:] Right.
[Ian:] B X, erm B Y.
[John:] Okay, and they're all, they're all added together because there was. Well what did you think of that compared to doing this with the numbers in?
[Ian:] That's a lot easier.
[John:] It was an awful lot easier, wasn't it. There's no chance of making all these errors that I was making, oh no what's this? It's seventeen hundred or seven hundred or what's, what's a hundred times twenty, is that two hundred? Oh no it should have been two thousand. It's easier, it's the easiest thing is when there isn't a number in sight, when it's all letters.
[Ian:] Yeah.
[John:] Okay? Now a more interesting one, try this. A plus B times A plus B....
[Ian:] A times A, A squared and A times B, A B.
[John:] Right.
[Ian:] B times A, B A.
[John:] Right, normally write it A. If we get an A, a B A we'd write it A B. We keep the letters in alphabetical order when we multiply just so we can see what's going on. See why when you've done them, finish off the last one then.
[Ian:] B times B is B squared.
[John:] Right. A B, what does that mean?
[Ian:] A times B.
[John:] And what does B A mean?
[Ian:] B times A.
[John:] So it comes to the same thing. So instead of those two, we'd, we'd do that first, as you've done it and then instead of those two we'd just write two A B. So it'll come to A squared plus two A B plus B squared. So if you want to just write that in.
[Ian:] So it's A squared
[John:] A squared underneath that one, now these two, we just add the A B and we, that's also an A B, we've written
[Ian:] Yeah.
[John:] it as B A so we've got two A Bs.
[Ian:] So it's like doing the A B and then double them?
[John:] Yeah, so it's two A B, two times A times B.
[Ian:] Yeah.... Plus B squared.
[John:] Right, brilliant. Try this one. What do you think of that compared to the numbers?
[Ian:] It's a lot easier,
[John:] It's, it's, it's e once, once you get over this shock of, aargh I haven't got a number, oh what am I going to do, nothing to cling on to it's all letters, ooh I can't do it, I can't do it. You know,
[Ian:] Yeah.
[John:] once you get over that [whispering] a doddle []. This is easier than doing it with numbers, any day. So let's try A minus B times A whoops A minus B. See what you get from that. Now it's just what you've done there, right, but you've now got to start thinking about signs a bit, haven't you? We've got a plus times a minus or a minus times a plus or what? What does it give?...
[Ian:] A times an A
[John:] Right.
[Ian:] go, that's A squared.
[John:] Okay.
[Ian:] And then erm... A minus B
[John:] Yeah.
[Ian:] times A minus B
[John:] Well hang on. Follow, follow the normal erm... oh okay do it, go on, do it your way. A minus B times A minus B, that'll give you what?
[Ian:] ... B squared.
[John:] Yeah, and is that plus or minus?
[Ian:] Plus.
[John:] Good, good. Cos the signs are the same. So it's A squared plus B squared and now what about the other bits?
[Ian:] B minus B times A is...
[John:] Now when you were doing over here, you did a B times A is B A. Okay so it's gonna be B A, all you've got to decide now is it plus B A or minus B A, so you can put your B A down while you're thinking about it. And what have you got?
[Ian:] Minus.
[John:] Good. So cos it hasn't got a sign in front of that A so it's a plus A. We've got a plus times A minus which is A minus. And what about the last one then?
[Ian:] B times B erm minus B times minus B.
[John:] No, you've done that already.
[Ian:] Yeah, so it's
[John:] Which is why it's better to sort of hey now, hang on, don't put another B squared in, cos you've done that. Let's, let's just put it here, A minus B times A minus B. Now it's tempting to go for the easier ones in it as you did, so we'll do the A squared, okay that's no problem. Done the A squared, and you said, well, let's do the B squared cos that gives a positive, then you got a little bit sort of not sure of which ones you'd done and which ones you hadn't. So it's probably better to stick to the system of A times that one, that gives us the A squared okay. Now A times minus B, what does that give us? That gives us the minus B A or minus A B.
[Ian:] Yeah.
[John:] Okay? So we need to carry on from there, so I've done the A times A and the A times minus B. Now we start with the minus B times each other, so what does give?
[Ian:] Minus B ti minus B times A
[John:] Will give you?
[Ian:] ... minus A B.
[John:] Good, brilliant yeah? Gives you another minus A B there, now erm
[Ian:] And then B Minus B times minus B
[John:] Gives you A?
[Ian:] A B squared.
[John:] Right.
[Ian:] A plus B squared.
[John:] A plus B squared. Put that one in, put the plus B squared at the end there. Right so when we add those up then we've just got A squared and how many minus A Bs have we got?
[Ian:] Two.
[John:] So we've got to finish off with minus
[Ian:] Two minus A
[John:] So we finish off with minus two A Bs. Okay, so you finish up with A squared minus
[Ian:] Minus two A B.
[John:] That's it, good.
[Ian:] Plus B squared.
[John:] Plus B squared. Now those two, you probably won't come across it much, but the level you're doing but those, those two expressions, A plus B times A plus B and A minus B tems times A minus B come into algebra a lot, they crop up again and again. And there's a n there's one more that comes in for you to have a look at. A plus B times A minus B.... What does that give then?
[Ian:] ... [whispering] A times minus B []
[John:] Erm do, do the first one of this into the first one.
[Ian:] Wasn't it? Oh that's right.
[John:] Okay. So A times A.
[Ian:] A times A, A squared.
[John:] Right.
[Ian:] A times minus B
[John:] Right.
[Ian:] is minus A B.
[John:] Okay, good.
[Ian:] ... B times A is B A.
[John:] Right, or A B.
[Ian:] [whispering] A B []. And B times minus B is minus B squared.
[John:] Good, right. Minus B squared. So you've got a plus A B and a minus A B there. They just cancel out.
[Ian:] Yeah.
[John:] So what does that come to altogether then?
[Ian:] A squared minus B squared.
[John:] Right... Erm so those ar those are the, I mean that that one is the most useful actually, A plus B times A minus B comes to A squared minus B squared. Cos the A B, the plus A B and the minus A B cancel out. So if we wanted to do some, I mean if you, if you can use that to do mental arithmetic to impress your mates, that's. Or maybe even the teachers if they're not teachers. Let's say we wanted to do erm... twenty one times nineteen. Well let's rewrite it as A plus B times A minus B, so if A is twenty we've got twenty add one times twenty minus one. Okay? And we know that the answer comes to A squared minus B squared. Well A is twenty so that comes to twenty squared minus one squared. So twenty squared, two squared is how much?
[Ian:] Two squared?
[John:] T just two squared would be four. And ten squared?
[Ian:] A thousand.
[John:] And ten squared is?
[Ian:] Ten s
[John:] Ten
[Ian:] Four hundred.
[John:] Is a hundred. So it's four hundred minus one squared and one squared is
[Ian:] One.
[John:] just one. So it should come to?
[Ian:] Three hundred and ninety nine.
[John:] Erm have we got that right? Twenty squared. Is that not bigger than that? It should be very nearly twenty, twenty times twenty.
[Ian:] ... Yeah.
[John:] So could you try [cough] what I said, could you do erm... what's a big number for you? A hundred and one times ninety nine.
[Ian:] Yeah....
[John:] No.
[Ian:] What's wrong?
[John:] A add B comes to a hundred and one.
[Ian:] Mhm.
[John:] And A take away B comes to ninety nine. So what, what would A be?
[Ian:] A plus B... s... two. It'll be two.
[John:] Th th that two is very important, where did you get that two from?
[Ian:] Cos ninety nine plus two, a hundred and one.
[John:] Right, so the difference between these two, right so let's say that's A add B and that's A minus B and if we take that one away from this one we get two. So if we had A add B, that's one number, take away A minus B it comes to two. Now let's, what d what does that lot come to on the left hand side? A A plus B minus brackets A minus B?
[Ian:] ... Shall I write the arrows
[John:] Ah, now we're not multiplying, we're not multiplying here. We're just working out what this comes to,wh with the minus going in to it. Now what did you mean by write the arrows, did you mean from here?
[Ian:] Yeah.
[John:] No.. Let's write it slightly differently, let's write it as A add B... add minus one times A minus B. Are you happy with that?
[Ian:] No this is all going
[John:] Right.
[Ian:] and I don't know anything
[John:] That's what I thought, right. You do know quite a lot but you're thinking, phworgh where does he get that from. Well it's back to the old confusion again, that we've got this stupid sign here which can mean it's a negative number or it can mean take away, and sometimes it doesn't really matter which way we look at it. So let's put some numbers in, let's put some numbers in. Let's say we're doing, let's look at the one that I did. Which w erm twenty add one take away twenty take away one. What does that come to? Well it's, leave that as it is, this bit comes to mi think of a minus one. What do we do, you must take away everything that's in the brackets, so we take away a twenty, so that's the same as a minus twenty and then we'll take away a minus one, signs are the same so it's add one. Right take away a minus one is the same as add one, so we've got twenty add one, we can get rid of these brackets now, twenty add one take away twenty add one. Twenty and take away twenty cancel out and it comes to two. Right, so what happens here? With this lot, we've got A add B, that's one number, take away A minus B. That's going to come to A add B, A take away A or A minus A and then take away A minus B, taking away A minus B is the same as adding a B. So the A take away A go out and this comes to two B. Now... this is not using a long way, a long way round to do it. Erm it is, but we're l we're seeing quite a lot about how brackets work in algebra. Now brackets weren't invented for algebra, they were invented for these normal numbers that we play with. They work for those and algebra just follows, so it just follows the normal rules that we're using for the numbers that we know. So we've looked, we've looked at one lot of brackets times another lot, which is the most difficult thing to do really, and you can do that, no problem. Now we need to look at one lot of something in brackets add another lot, well let's, let's forget about the first one being in brackets, let's say we're doing ten take away six. What does that come to?
[Ian:] Four.
[John:] Okay, so I'm going to do ten take away, now I'll put some brackets where that six was I don't want to write a six, I'm going to write it as five add one. Now that should give me the same answer as if I do ten take away five so it's take away a plus five, Yeah? That just comes to take away five, and then take away a plus one, take away one. Ten take away five, how many does that come to?
[Ian:] Five.
[John:] Take away one?
[Ian:] Four.
[John:] Right, so that works okay. Now I should be able to say ten take away, I'm going to write six in here, instead of six I'm going to put it as nine take away three. So we do, ten take away a plus nine, that's just take away nine.
[Ian:] Yeah.
[John:] Take away a minus three, is the same as adding the three. Don't forget the same with, with take-aways as well, if the signs are the same you get a plus or you get an add, erm so that's, I mean I can see this is the bit you're not too happy with, but we'll just see if it works. So take away a minus three is the same as adding a plus three, and what would that come to? Ten take away nine
[Ian:] One.
[John:] Add three?
[Ian:] Four.
[John:] Ah. So it does seem to work this method, and it wor this is the method we have to use for all the numbers including the As and Bs and Xs and Ys where we don't know what the number is. So that that bit you, you got no problem with it, if I keep to positive numbers inside there, have you? So ten take away, if we did something like ten take away A add B, well that's the same as ten take away A and then take away B.
[Ian:] Yeah.
[John:] If I say to you, I want you to do this sort of ten take away three add four in brackets, so you'd add the three and four, get seven, ten take away seven, three. Or you could do, you could take them away separately. take away the three then take away the four. So you're okay with that, ten take away A minus, ten take away A add B will be ten take away A and then take away B. But if we do ten take away like we've been doing here, A minus B and it comes out the same as this, it'll be ten take away plus A, just take away A, and then take away minus B. Well taking away a minus B is the same as adding B. Erm do you remember the table?... For when we're adding and taking away and things. That if we have, let's have a look at some of the easier ones. If we're adding a positive number, so if I say add plus three to a number, it's the same as just adding them, okay? If I. so these are say those are the, that's a positive number and that's a negative number, and this is, it's, that means just add the positive version. Now how about if I take away? If I take away a positive number of course that's taken as normal take away.
[Ian:] Yeah.
[John:] Good. What about if I add a negative number? I'm going to add some money to, how much money, how much money have you got? You've got ten pounds say, I'm going to add some money to that good. So I'll add two pound to it, I'll add a positive amount to it and you've got more, you've got twelve. My, next day I come in and say oh I'm going to add some money to what you've got, and you think oh that's good. How much have you got, put it on the table. So you put your ten one pound coins on the table and I say well I'm going to add minus eight pounds to it tonight, [unidentified noise] help yourself to eight pounds,. I've added. [LAUGHTER]
[Ian:] [LAUGHTER]
[John:] [LAUGHTER] Very much like taking away to me that did, very much like no no no no []. I was adding minus eight pounds to what you had. I've got eight pounds left, well it's the same. Adding a negative number is the same as taking it away. Okay? So
[Ian:] It's minus.
[John:] That's, that's not minus that's take away, right. The ones with the rings around them are, are positive or, or negative. So adding the positive number that's just normal adding, adding a negative number that's the same as taking it away. And you're
[Ian:] Yeah.
[John:] going, where's me eight pound gone? He's told me he was going to add to my money and he said, oh the bad news is I'm adding a negative amount of money to it, and he's taken away. Okay, now the good news the next night is that I'm gonna take away some of your money, and you think, oh wow he's taking. Last night he said he was adding and he took away so what's he gonna do tonight when he said, really admits he's taking it away. I say, I'm gonna take away minus eight pounds... Let's say so I add it. Let's say erm we forget about all that ten pound early and you owed me a hundred pounds, right, and I say erm forget that, just, just take it away, forget it it's gone, you're a hundred pounds better off. We've taken away, let's say erm have you got a bank or a building society account or anything?
[Ian:] Yeah.
[John:] Right. So let's say you're, let's say I'm the building society and you can have your put in your book and that. You make a withdrawal, they take it away and let's say you can go overdrawn, so you've got eighty pound in the building society, and you take a hundred out, how much have you got left now?
[Ian:] None.
[John:] Less than none.
[Ian:] Yeah.
[John:] You've got minus twenty.
[Ian:] Minus twenty.
[John:] Okay, you're twenty pounds over, you owe them, you owe me the building society, twenty pounds. Now if I say, oh well as it's I mean Easter we'll forget about that. Right? I've got your balance here and I say what have you got? You've got minus twenty, well I'll take away, let's put the sign round that. Usually they put debit after it or something, or put it in red. That's a minus twenty. If I take away the minus twenty well I'm taking it from the same thing aren't I? I'm taking a minus twenty from
[Ian:] Yeah.
[John:] a minus twenty so it's got to be zero, the answer, taking away a negative, it would be the same if I added, if, if it was minus twenty and I added twenty come to zero, they'd just cancel out. So if I take away a negative number it's the same as adding. You come into the building society and you're, you've got no money in the world and you owe twenty pounds and I say oh we'll take that away, we'll take that minus twenty away. You walk out twenty pound richer. You've got no more money but twenty pounds so taking away a negative number is the same as adding, yeah? And again if the signs are the same, I mean I know they're not exactly the same this is and add and that's a positive, it's the same as adding a positive number. A negative and a positive it's the same as taking away. A positive and a negative, it's the same as taking away. this was, this was the night I added minus eight pound to your money, and walked off with eight of it, [LAUGHTER] right []? And then this, this time I say well I'm going to take away some money and I took away a negative amount and you've finished up with more. So all you need to do is remember that table which is the same, the same as when you, this is for adding... adding or subtracting, and the same table for multiplying or dividing, yeah?... A positive times a pos... [tape change]
[Ian:] It's not no.
[John:] And it's no good just looking at little bits of it. You want to sort of look at the whole picture together. Erm do you think you got the whole picture?
[Ian:] Yeah.
[John:] Okay, you can guess what comes next don't you?
[Ian:] questions.
[John:] questions first to build up.... Erm... seventeen take away ten take away three. How would you, if you had to do that if you do it the easy way, don't do it the hard way, how would you do that?
[Ian:] Ten minus three, is seven.
[John:] Is seven.
[Ian:] Seventeen minus seven.
[John:] Is ten. So if we get it right the answer should come to ten.
[Ian:] Yeah.
[John:] Now I want you to do it the hard way, I mean you wouldn't normally do it this way but when you've got letters in you've got no choice. So we'll do it while the numbers are there to get you familiar with it. So what we'll do is we'll put that, that means take away, right? That's a positive ten there and this is a negative three here, if you like.... So take away a positive... I'm going to take away a positive ten pounds away from you now.
[Ian:] It's take away, minus.
[John:] So that comes up, that comes out as a minus yeah as a minus ten.... Now I'm gonna take away a minus three.
[Ian:] Minus seven. Er it's a seven.
[John:] Just. Well we're just looking at the
[Ian:] I these minus and pluses and that.
[John:] Well there's two, there's two, there's two things, there are four things going on here, and it's, it was designed to be confusing. It's not, it's not just you and everyone going what on earth is going on here. Because we've got, and you've seen this before, we'll just have a look, get this right if you can. What's that? No that's a plus. What's that?
[Ian:] Take away.
[John:] Erm no that's a negative. You can't tell they're the same the only way you can tell is if they're, if they're working with number, I mean... that's a plus or a positive that's a positive three. That means take away, if it's between two numbers it means take away. So this one is take away.... And what's this one?
[Ian:] Take away.
[John:] This is a take away, but we can treat it, take away three is the same as add minus three. Let's write this out again seventeen take away ten add minus three. Yeah? I'm going to add some money tonight. [LAUGHTER] The bad news is I'm gonna add minus three pounds to what you've got there. Well we know how to, how to work when it's add. We just do the first thing and then we do the second thing. You see this, this sign outside the brackets let's change this as well. Let's make this seventeen erm add... minus whatever this number is. Is this getting more confusing and you can think of that as seventeen add minus one times and then we're multiplying into the brackets. Seventeen add this big number here which is minus one times whatever we've got in here... minus one times ten, what will that give us?... Minus one times ten?
[Ian:] ... Ten.
[John:] Are the signs the same or different?
[Ian:] Different.
[John:] So I'm gonna say?
[Ian:] Positive. Negative.
[John:] That's it. Signs are different
[Ian:] Okay.
[John:] will give me minus and so a minus one times a ten gives us a minus ten. Now a minus one times a minus three?
[Ian:] Is a positive three.
[John:] A positive. Plus that's the same plus from there, plus a positive three. That's what they'll come to there, so we've got seventeen add minus ten add plus three. That's seventeen as well. Seventeen add a minus ten the same as take away a positive take away a positive ten, and add a positive three. We've got all positive numbers now. Hurray.
[Ian:] [LAUGHTER]
[John:] So we can drop the signs and we can forget about mentioning whether they're positives or negatives, and get now a nice straightforward, seventeen take away ten add three. Seventeen take away ten?
[Ian:] Seven.
[John:] Add three.
[Ian:] Ten.
[John:] Ten, okay? So... same as, same as you got here. What's what di do you get here?
[Ian:] Ten. What I've done was three minus ten seven.
[John:] Yeah.
[Ian:] Seventeen minus seven.
[John:] Good. The big thing is we've got two things going on here, two different types, one of them is a, a f. One, one is an operation add or subtract. What does add mean?
[Ian:] Adding the two values to together.
[John:] What does it mean, you're still using adding, what does add mean?...
[Ian:] Put them together you know?
[John:] Okay, what does it mean in terms of the number line?... the number line there, and we want to do something like two add three, what does that mean?
[Ian:] You start at two positive to
[John:] Right, good. So we've left a bit of this out, we're, we're a bit sloppy in our notation, we should be saying let's put a sort of a ring round the whole thing, to show that's the number positive two. Add positive three, see this thing of using the same symbol for add and for positive is very confusing, very confusing, it's as though, you know you're speaking a different language where one word has about fifteen different meanings and you can't understand what they're talking about most of the time. So... positive two add positive three, what does that mean?
[Ian:] Start at the two.
[John:] Start at positive two okay? What does add mean?... What does g g go
[Ian:] Yeah.
[John:] along is right, erm which way?
[Ian:] Positive
[John:] Go, go that's it go along towards the positive numbers. So you better count along that way for how many?
[Ian:] Three.
[John:] One, two, three. So the answer is positive five
[Ian:] Yeah.
[John:] is that right? Okay. Now what would this mean? Minus two add positive three. Don't tell me, I can guess. What does this minus two mean?
[Ian:] Start in the negative.
[John:] That's where we start. So we start negative two, minus two, and I know what add means cos you told me last time. Count along towards whoops. Count along for how many?
[Ian:] For three.
[John:] For three. One two three. So the answer should be one.
[Ian:] One. [whispering] Positive one [].
[John:] Is that right? Positive one. Yep. Okay, now. You didn't think adding up or taking away was this hard did you?
[Ian:] No. [LAUGHTER]
[John:] Because it's very, I mean you know mathematicians have written books about this, erm and yeah kids of five or six are quite happily adding and taking away until they get to negative numbers, until someone says ah I'm going to add to that money you've got there, I'm gonna add minus seven. You can get a tantrum [LAUGHTER] you won't get away with explaining [].
[Ian:] [LAUGHTER]
[John:] So, okay. A minus two add a positive three, now what would minus two add so that's a negative two, add a negative three mean? Well minus two is where you start, okay. Add, that means go along towards the positive numbers, and I'm going to count, how many am I going to count?
[Ian:] Minus three.
[John:] Oh. I count in that direction though, so minus means, minus three means count the other way. So I start from them and I count this way.
[Ian:] That's a minus.
[John:] One two three, and I finish up there, minus three, minus four, minus five. The other way of looking at it is you can say I'm going to count... you start from nought and you count along that way until you get to this number, so I'd go nought minus one, minus two, minus three. So altogether I'd still count three that way. So negatives go that way. Now how about if I do four... add minus five. So it's plus four add minus five. Okay that's always the number where you start isn't it? So we start at positive four add, add to count in the positive direction, oh
[Ian:] Minus.
[John:] we're adding minus five, whoops switch round count the opposite way. So from four we count five backwards count minus five just means count down the opposite way, so we go one, two, three, four, five, so it should be minus one. Yeah? Negative one. Now the tricky one, let's do plus four make it plus five, plus five a a takeaway a negative three. Okay? Where do we start?
[Ian:] Plus five.
[John:] Plus five, no problem there, start where the first number tells us to start. Take away, what does that mean?
[Ian:] Count negative
[John:] Count that way. So if we said take away three, let's do this first it's bit easier, plus five take away plus three, you start at plus five, take away means count that way, count to the left towards the negative numbers, so we count three. Plus five, count three, one two three.
[Ian:] Two, plus two.
[John:] So. It should be plus two. Now what does this one mean, you tell me? Plus five take away minus three.
[Ian:] Start at plus five
[John:] Okay.
[Ian:] going that way.
[John:] That take away means we're going in that direction, and how many are we going to go?
[Ian:] Three.
[John:] Three or minus three?
[Ian:] Minus three.
[John:] Minus three. When we were taking away three we went in this direction, for three. But to take away a minus three
[Ian:] Go the other way.
[John:] go in the opposite direction, turned your pen round and you go the other way. So A take away a negative, those two signs together is the same as an add. A positive. When we did our table... this is for adding and taking away erm that's a positive number and that's a negative number, okay? Add a positive number is the same as a these are all the things you do with positive numbers, this is add a positive number is the same as add a positive number, right. So that's add a positive number, these are all for the positive numbers, add a positive number, okay? Take away a positive number, well okay we
[Ian:] Take away.
[John:] take away, a positive number... right. Add a negative number, this is when I said I'm going to add minus eight pounds to your money tonight, add a negative number is the same as take away a positive number. Add take away a negative number is the same as add the positive number. So whether it means add or take away or subtract or negative we can sort of forget about it and just look at the signs so if get... if we get take away plus three, that will be the same as take away plus three, yeah? Take away three. If we got add minus three, that's the same as take away plus three, just take away three. The interesting one is when we've got take away... a minus three, that's the same as add the positive three, add plus three. Now, what I'd like you to do... this, this is not straightforward. [LAUGHTER]
[Ian:] [LAUGHTER]
[John:] It's er anyone can do this you know. People, people always think that until you start throwing in the double negatives and they think, whoa, don't want to know!
[Ian:] [LAUGHTER]
[John:] No no no no this is all rubbish you're making this up as you go along [LAUGHTER] yeah? [LAUGHTER] It doesn't se it's not natural is it? Not sort of, it's not obvious.
[Ian:] Especially when I could do this one yeah, erm positive five going that way
[John:] Mm.
[Ian:] and then turning back this way.
[John:] Yeah.
[Ian:] You know, when all I've ever known is plus is this way, negative this way, it's either that way or that way, you know.
[John:] Right. Now I'd like you to just have a look at these for next time erm... and work them out on the number line.
[Ian:] Yeah.
[John:] Erm because that's what gets you to accept it and stop thinking this is a load of rubbish, this is a con he's just making this up. Yeah I am making it up, it's not me making it up, mathematicians had things like erm you, you, you met this sort of earlier on in school. What's seven take away three? Okay, what does that come to?
[Ian:] Four.
[John:] No problem, one number take away another number gives us a number, wow that's great. Okay, what's three take away seven? Infants
[Ian:] Minus
[John:] infants school and you say can't do it. It's obviously a load of rubbish. You ask some kid, he'll say there you are I've got three pens on the table now take away seven. think he's off his chump you know.
[Ian:] [LAUGHTER]
[John:] What's he on about, and then just forget about it and they go and play in the sand or something and he's crackers that bloke. And they're probably quite right. But cos mathematicians are a bit crackers, we want, we don't like saying I can't do that. we'll, we'll, we'll find a way. You know we'll change the rules so we can do it, we'll move the goal posts slightly
[Ian:] [LAUGHTER]
[John:] We'll, we'll make sure we can do that. That's what they say well we have this silly game that we play on the number line, going up and down the thing and when you do three start from three, count seven in the opposite direction we finish up at this number called minus four. Well what on earth is minus four? You show me minus four in anything, there's mi minus four houses out there. There's mi minus four people coming to the party tonight, they're all crackers, okay but it works, it's very useful it solves an awful lot of problems like you know getting rockets to the moon and things like that that we couldn't do otherwise. It comes in handy with your bank statement your temperature, oh it's minus six. It's, tonight it's, think of these as a sort of temperature numbers that, try, try these, it starts at plus three tonight and then it drops five degrees, plus three and it drops five degrees, what does it come to? Okay, it's minus seven and it drops minus three. It's erm minus fourteen and it drops by minus six.... fourteen take away six. Er oh keep these numbers small so you can sort of play about on the number line more easily, and ten minus, minus ten take away minus four. Six add minus three, three add minus six. And minus six take away minus six. Any number take away itself, what should it come to?
[Ian:] Nought.
[John:] Yeah. So minus three take away minus three should come to nought.... Minus six take away minus seven. And minus six take away minus five and then X take away minus one. X take away minus X. Take away X, which is just plus X, X take away X. And three X take away minus two X.... Hey now on the number line you can't it's difficult to do X take away X but you can work out roughly where they'd be and what you would do and what the system would be. So if you have a look at those and play with them for next time erm and make sure that you learn those two rules about if the signs are the same
[Ian:] Yeah.
[John:] it's always then a plus sign, whether it means an add or a plus. Al also the way we talk about it is very very sloppy, we often say A minus B when we should be saying plus A take away plus B. We start talking like that people he's off again. [LAUGHTER] [LAUGHTER] He's flipped []! Erm... it's, it's, negative numbers aren't a natural thing, fractions are actually easier and that was what happened in the history of mathematics, fractions were developed a long long time before negative numbers, the old Greeks used to play about with fractions quite a lot. Negative numbers came later. Anyway I must dash because I've got
[Ian:] I'll do that for
[John:] So you have had before I try and don't, I won't go off with your homework hone honestly I won't. Erm now next week er where are we, right. You still want a lesson next week?
[Ian:] Yeah.
[John:] Usual time, okay. That's good.