 We're good Okay, let me get organized here real quick. I do want to mention a couple of things about Some of the things I saw in the proofs that I would like you to modify Okay All I'm gonna say here is Okay, just a couple of things overall You guys did pretty well okay So I do expect over the course of the semester that you're gonna start to progress a little bit in your proof writing skills Here's one thing I noticed and this this is just this comes with time Let me Preface by saying that when I if I write something on your paper if I underline something and say meaningless or if I underline something And say nonsense. I'm not insulting you. I'm telling you this because you need to know it because I want you to get better Okay, I want to be very clear about that. I'm not being me Well, maybe you think I am I'm not trying to be mean Okay, it's just because I'm just trying to help you and that's why I spent a lot of time on the grading a lot of you'll see I have a written a lot of comments on your stuff I do that because I want you to get better this course for math majors a lot of you are math minors But for math majors, this is also another very fundamental course You need to get the proof writing down some of you are gonna be high school teachers And maybe a few of you are thinking God, this class sucks. I don't you know, I don't like writing proofs I just have to teach a high school You and you have trust me on this Part of being a good teacher Part of what goes into being a good teacher is having a good thorough understanding of mathematics It's absolutely true. Do you have a student that says why can't I divide by zero? Why can't why can't I just because you can't? That's not an and that's not a very good answer, you know or saying, you know, why does why does the vertical line not have any slope? Why is that true? Again, when you have a better understanding of why things work the way they do you can answer these things and you can you become a Better teacher. There's no question about that. So I'm trying to improve your your skills I'm trying to fill excuse me facilitate understanding of this stuff because it is important. It really is first thing I want to say and I noticed this on a lot of a Lot of your your papers. Here's one thing that I want you to try to do right Complete sentences Okay, this is something that a lot of you aren't used to in mathematics and I understand that And you several of you have seen on your paper that I've written this comment You're writing a proof. You're writing an argument, okay? It's not just you know, it's not the same thing is taking the derivative of tangent of e to the x We just write it down. You're trying to convince me of that. You know why something is true Okay, so what I expect you to do is Imagine that I have no idea why it's true imagine. I don't understand math very well Okay, and try to convince me that that is true Okay, don't assume that I know what how it works. I do but don't assume that That's not the point. Okay, you have to show me that you understand it and part of being clear is Having good exposition is writing things out completely. I posted solutions online, which I will if I can in a second I'll show you kind of what I'm looking for This I've seen I've seen a lot. This drives professors nuts Okay, and I'm gonna try to be very clear about this Don't in general don't use The conclude this is gonna look really stupid, but I'm just I'm just gonna write it this way because I want to Don't use the conclusion blah Equals blah to justify no, no, I'm not saying it's not true But I would you're making an argument and I'm gonna show you I have a point here. I want to make to justify Just because something's true doesn't mean it belongs in an argument right the cow is an animal that doesn't mean you should write in your proof right No, you guys you know that just because something's true doesn't mean that it belongs that it's relevant I've been writing proofs for a long time. I honest to God. I know how to do this stuff and I'm saying it for a reason, okay? Let me give you an example a lot of you were doing this You want to prove something so you just in your proof you just write it as if it's true And then you conclude for example k plus one equals k plus one and then you think that that justifies what you assumed and it may But it's not it's not that's not the best way to write the proof. I'm gonna give you an example Okay, so for example Suppose you want to prove This is gonna seem silly and it's hate to say this but this is gonna fly some of your heads You're gonna think this doesn't apply to me But it does Suppose you want to prove that one equals minus one Okay Well, let's just assume That one equals minus one That's what a lot of you are doing you're assuming what you're trying to prove Well, okay, then let's square both sides. We can certainly do that right one squared equals minus one squared So one equals one that's true. Therefore the original assumption is true. No, it's not Okay, what this this is relevant to this course if you think well, but I didn't square both sides Well, how do you know that what you did is legal? Okay, just so the point is circumvent the whole issue by not doing it Okay, that's what I mean by streamlining your proofs and writing them in a Logically beautiful way. I hate that sounds like a weird word use but that's really what you're trying to do here Don't do this don't do this don't end your proof with five equals five. Don't do that Okay, as this shows just because you end up with blah equals blah that does not justify your initial assumption counter example right here Okay, you guys with me on this Sorry, I'm getting excited again. I'm gonna start I want to start throwing stuff. I'm gonna knock my mic off and start breaking things Okay But overall You guys did pretty pretty well I was actually pretty happy with with your performance on this Okay, and so again take the the the know the meaningless the nonsense these comments just take them as face value Don't read some subtext text to these comments like you suck. You're terrible. I'm not saying that Okay, I'm really not that's not true No, I That's not to be inferred from the comments is what I mean Okay, so let me let me go ahead and move move on past this here Ah Geez, I don't know if I'm gonna have time to do this Solutions to the graded problems the proofs are all posted online on the course web page I've really strongly encouraged you to go online and look at this and you see how I'm writing the proofs Complete sentences. I'm not assuming what I want to prove in the beginning I only assume say the inductive hypothesis and I conclude That the statement's true friend plus one these kinds of things look at the proofs you if you read the proofs Which I encourage all of you to do you will look at and you'll say wow that's really nice That's really elegant. There's you know, you Really will be able to get better at writing these and you will start to approach this if you work hard And this will definitely help you with with your other more theoretical math classes for sure There's no question about it. So really I encourage you to look look this up online But again, just to be clear you guys overall you guys did a very good job. I was pretty happy okay, so Let's see. So yeah the on the syllabus you guys are all smart and on the syllabus I have the the website just go to the website the homework so when the solutions have been posted The homework, you know homework one homework two homework three etc. Those statements will be hyperlinked So you can whenever you can click on a homework. It's to the solutions. Okay? all right so Let's go ahead and move on So here's what we're going to do as I said last time today. We're going to finish up the division algorithm and then Yeah, so I spent way too much time on that but We're going to finish up the proof of the division algorithm, and then we're just going to do a few problems. Okay? So this is just a continuation of section 2.2 Okay, so again just to recap you don't if you were here, of course you don't need to write this down So if you recall this set s, I'm not going to redo everything but just to remember what we did a minus xb where sorry x is an integer and A minus xb is bigger than or equal to zero That's our set s What did we show so we proved Let me try to streamline this a little bit we found a Least element right I Think I called it R. Didn't I can somebody confirm that? Okay, we found a least element R in s by proving that it was not empty and then we use the well-ordering property, right? So a couple of things I think I finished off with this last time that we know about our first is that Since R is in s R is equal to a minus Qb right or some Inager K right. I think I use the same notation on Tuesday Right did I write this something like this? I think I use Q and so Shuffling this around A is equal to Qb plus R, right? Okay, so everybody see that Just add Qb to both sides. That's pretty easy Okay, and the second thing is by definition that zero is less than or equal to R Why is that because R is in s and by definition everything in s is bigger than or equal to zero Okay, they just follows from the definition of s that zero is less than or equal to R. Oops Okay, so Remember what it is that we want to prove Well, there's several things you want to prove one is that Okay, if you look back and for the sake of time, I'm not going to write it down again look back to the statement of the division algorithm Okay, what was it? We're trying to divide What a by b and The statement is that there exists unique integers Q and R satisfying a equals Qb plus R Which we have and the second is that zero is less than or equal to R and R is less than b That's the second part. This should be in your notes if you were here. So we have part. We have most of it We'd still need to show that R is less than b and then at least we have the existence part of the theorem You guys with me on this Okay, at least a couple of you are nodding. Okay Hopefully more of you are with me R is less than b. Okay, so Let's and this is a useful technique in writing proofs. Let's suppose That that's not true This is called a proof by contradiction, which you should have been exposed to then well far as not less than b What do we know and then R has to be what? Greater than or equal to b, right? Okay Thus and just try to follow this. It's I don't think this is too bad We put a star here a minus Q plus 1 times b And we've got a lot of letters floating around here, but I like said, I don't think this is too bad. This is equal to a minus Qb Minus b I'll pause for just one second you guys by that Distributing right a minus Qb minus b Okay well What is this equal to? This is Okay, I'm gonna be a little sloppy here So you forgive me for this but just to make sure everybody's with me on this What's a minus Qb equal to we have another expression for that if you look up above This is all right. You see that right up there a minus Qb is R. So this is R minus b okay, and Somebody tell me why this is true R minus b is less than R This goes back to an assumption that we made that's not on that was actually back on Tuesday But remember we're dividing by b. What was the assumption on b is that b is bigger than zero? That was the assumption that's in your notes. It's there b is bigger than zero So if we're subtracting something positive from our we have to end up with something that is smaller than R right So this is true since B is positive Okay, so We can conclude that so a minus Q plus one times b Let's see What else do I want to say here? Okay, so let me just put this together first. So this is R minus b Which is less than R? Okay, that's the first thing I want to say Also a minus Q plus one times b equals R minus b which is bigger than or equal to zero Why is this? since R is bigger than or equal to b. Okay, let me call this one Let me call this two First thing we showed was that a minus Q plus one times b is less than R, right? And the second thing I want to I want to be clear about this a minus Q plus one times b We just of course we just figured out out that that's R minus b This is bigger than or equal to zero because we're assuming that R is bigger than or equal to b Okay, so R minus b just subtract b from both sides R minus b is bigger than or equal to zero Okay So what what went wrong here? There's something that bad that has occurred here that will allow us to conclude that our assumption was not correct And that R actually does have to be less than b But then Okay, look at look at this and think about it everyone once you guys to think about this you buy that this is an s a minus Q plus one times b is an s Why is how do we know it's an s? Look at the definition of s up here s is the set of all The set of all integers of the form a minus xb where x is an integer and a minus xb is bigger than or equal to zero It certainly has that form right? This is a minus an integer times b, but that's not enough to conclude that it's an s We also needed to be bigger than or equal to zero Right, but we have that from from this right here You see that I'm not going to write all that down, but you can just see it from what I have on on the screen Now there's a contradiction here Okay This does contra this information we have at the bottom of the screen does contradict something Can anybody tell me what the contradiction is when we know it's an s? There's also something else we know about it It's less than R Okay, listen what I'm saying R was chosen to be the least member of s Here now we have a member of s that's smaller than our contradiction because ours the smallest Okay, therefore It can't be the case that ours bigger than or equal to be R has to be less than b So let me write that down you guys mostly have this now Can I move on to another screen? You guys okay, you good so this and you can just keep writing this in your notes this contradicts that Yeah You said Hang on one second, let me just finish writing this I'll answer that okay, so we've this this I'll answer that in a second this so now we've proven the existence part of the theorem right we've we got our least element R We know that a is equal to qb plus R and we just proved that zero is less than equal to R Which is less than b that's exactly the existence part of the statement of the division algorithm now We have to prove that that these q and R's are unique So the answer your question the So you are you just sort of asking where did the q plus one come from or? Okay, well here's the here's the idea. Let me let me go back to this for a second. Okay, I mean just this just roughly I mean the the q plus one well, why do we have the q plus one? Well because it It makes the proof work. I mean that's a very unsatisfying answer, but that's really the case The idea though is that if our so think of R as a remainder here's the rough idea Okay, suppose that R is bigger than or equal to be B is in your head what you were dividing by okay? Then the point is then the quotient can actually be increased by one So if you can imagine right dividing through if you divide 17 by 5 and you say it's 2 Well, they have a remainder of 7 but then really you can you can extract 5 out of that 7 and increase the quotient by 1 That's the idea of where this comes from Okay Does that make sense? Well, no the point is though. We wanted to choose We don't know I mean the idea is just knowing that R is bigger than or equal to B Then we know that Well, the quotient can be increased by at least one for sure But we don't know what we don't know anything other than R is bigger than or equal to B So that's really about all we can say and that's all we need to say because it gives us a contradiction Okay Okay, so All right, are we okay with this first part any other questions? No, okay This thing is all checked up. All right. I don't know what's going on here second part Seems to be okay now. All right and This is the last part of the proof uniqueness Okay, so the sake of time I'm not gonna go back and write everything down again But the point is and I will write part of this down the point is the quotient and the remainder are unique And what is that? What does that really mean? Here's what it means and I'll write it down I just want to say it first it means that if You take a and you divide it by B and you get that a equals QB plus R and R is between 0 and B And if you also divide it and you get that a is equal to say Q prime B plus R prime And our prime is between 0 and B then the those two quotients Q and Q prime and the two remainders are the same They both match up Okay Yes, yes, okay, so I'm just gonna write this down Now suppose that we have one a equals QB plus R and Zero is less than or equal to R, which is less than B and Let's assume the same thing is true for some other integers Q prime and R prime. So let me call this one prime a is equal to Q prime B plus R prime and 2 prime will be That zero is less than or equal to R prime, which is less than B Okay, so here Just to be clear of course Q R Q prime and R prime are all integers Okay Now maybe it's a little clear what I mean by uniqueness now Basically what we're doing is we're assuming that we've divided a by B and we've gotten one quotient and one remainder on one hand But then on the other hand we've divided a by B and we've got another quotient and another remainder We're going to show that the quotients and the remainders have to be the same in other words Q and Q prime are equal and R and R prime are equal Okay, that's the uniqueness part. So there are a couple ways you can do this Okay, we're actually almost done and this is the last theoretical thing I'm going to do and then we're going to Just talk about some homework This is not this part's really not that bad By one and one prime we get this shouldn't be too much of a stretch QB plus R Equals Q prime B Plus R prime right Okay, I would encourage you to try to try to follow along I know it's very tempting to just kind of daydream and just write this down and not think about it But we'll help you to actually think about this as we do it You see where this is coming from right a is equal to QB plus R But a is also equal to Q prime B plus R prime so therefore since of course a is equal to a QB plus R is equal to Q prime B plus R prime right You guys see what I'm getting this they're both equal to a so they have to be the same right thus our prime minus R is equal to B times Q minus Q prime Okay, think about this for a second. This is this is not anything too complicated Okay, our prime minus R. We're just going to subtract the R off and then we're just going to take the Q prime B And we're going to subtract that off to the other side We're going to factor out a B on the left to get Q minus Q prime Okay, just basic algebra here Okay We're going to use this fact here in a bit hence Okay, let me put a A couple of stars around this The absolute value of R prime minus R is Equal to B times the absolute value of Q minus Q prime Christina you have a that's your name right? Yeah, okay. You have a question Not the way I have it because I've taken the R over here and I'm taking the Q prime B over here So this will be this so I'm taking this over and then this over so this will be QB minus Q prime B So when the B gets pulled out, it's Q minus Q prime Okay, so we can certainly take the absolute value of both sides Right, so now the question is well, how much do you have to prove and how much do you do not well? This is this well, I mean You don't Things going back to the foundations of the real numbers. You don't actually have to prove these things so the point is the absolute value is Multiplicative in the sense that well certainly this is true The absolute value of R prime minus R is certainly equal to the absolute value of B times Q minus Q prime That's certainly true. They're the same so the absolute values of course have to be the same Why can I do this? Well because B is assumed to be positive That's why The absolute value B is B because B remember don't forget that hypothesis B is positive okay, and The absolute value of X Y is the absolute value of X times the absolute value Y you don't have to prove that So let me just put this parenthetically since B is positive. That's why we can we don't need the absolute value signs around the B because it's positive Okay Now we're gonna look at a couple of inequalities and we will be done very soon so from See I keep saying that and I just keep going and going and going you keep thinking all it'll be done So I can listen So okay, well we'll be done kind of soon from to Okay, let's see. I'm going to go through this kind of slowly Minus B is less than minus R. Okay, do you guys buy that? From this second inequality up here Subtract R from both sides and then subtract B from both sides. You can certainly do that and You see what I you see what I did okay? Subtract subtract R. You get zero is less than B minus R now subtract B. You get minus B is less than minus R Okay, do you buy this Minus R is less than or equal to zero right? Okay, because zero is less than or equal to R just add minus out of both sides and minus R is less than or equal to zero okay, okay, and now from Two prime. We're not gonna monkey with this. We're just gonna list it The way it was I think anyway, let's see Zero is less than or equal to R prime Which is less than B. Okay, so and we'll see if I can squeeze all this in here And I'm going slowly here so that I don't lose anyone. So what do we have we've got? minus B is less than minus R Which is less than or equal to? Okay, this I should say something about minus R is less than or equal to R prime minus R because R prime is bigger than or equal to zero. I want you guys to look at this inequality right here And I this is really sloppy. Sorry about that. Let me try to fix this. Okay Minus B is less than minus R. Well, that just comes from this Why is minus R less than or this is horrible? Sorry This might yeah, okay, that's even worse. Okay, let me erase it Getting no, I don't want to do that. Okay. I'm not gonna worry about it's fine Okay, I could go back to the white, but why is this true? Well our prime is bigger than or equal to zero Okay, so what's the difference between these two sides of this inequality? This is just minus R is less than or equal to R prime minus R Well, this has to this can only get bigger if we add something non-negative to it. That's the idea Okay, make sense Okay I'm running out of room here. Okay. This is less than Yeah, well Sloppy and small is gonna be really bad Okay Why is this true? Okay, this isn't this is this Last while we have one more This inequality right here Well, we already know that our prime is less than B from this So what happens if we just subtract R from both sides of this inequality? We get our prime minus R is less than B minus R, which is exactly that Okay Okay, and So I'm gonna squeeze this on here Let's see. This is less than or equal to B This is terrible. I'm sorry since R is bigger than or equal to zero Right, so the same this is kind of the same idea We're subtracting something that's bigger than or equal to zero if we're subtracting something bigger than or equal to zero That's only gonna make it smaller than what it was Right, that's where that comes from Okay, so I'm gonna write this on the next slide. So what have we actually done now? Yeah, this okay does everybody have this down I'm gonna write this on the next slide. Here's the here's the idea This is the main point. I want to extract from this minus B is Less than R prime minus R, which is less than B The point is that our prime minus R is squished between minus B and B That's the main point to take away from this. Can I go to the next one? Okay, so minus B is Less than R prime minus R Which is less than B. That's just from that last nasty sequence of inequalities so We're gonna go back to the absolute value here Think about this. I think some of you have taken analysis. No, no about this R prime minus R. It's just it's just think of it in your mind. It's just one number Don't get confused with the R prime in the R. You have a number that's strictly between minus B and B There's something you can say about the absolute value of it Okay, anybody tell me what the next step should be then What can we say about the absolute value of? What can we say about the absolute value of R prime minus R? It's right. It's less than B. Okay I'm not gonna mess up the Lecture with this, but if this seems a little strange you just think about just think about Suppose you've got a number that you know is strictly between minus five and five Okay, well if it's If it's positive the absolute value is equal to itself and it's certainly less than five But because it's between minus five and five if it's if it's negative Maybe it's minus four, but the absolute value is four. That's still less than five Maybe it's minus four point nine the absolute values four point nine It's still less than five because it's squeezing me these between these two numbers the absolute value has to be less than This guy on the right. Okay, so now we literally are almost done. Okay, so recall that Okay, I think I denoted this with two asterisks the absolute value of R prime minus R It's equal to B times the absolute value of Q minus Q prime so what does that tell us? I'm gonna use so again. Okay Yeah, B is just shorter and I don't like writing on this thing The triangular dots it's it's it's just a yeah, I mean, it's it's like an upright one one dot on the top, but that's okay You guys see where I'm getting this now We know that the absolute value of R prime minus R is the absolute value of R prime minus R is less than B But the absolute value of R prime minus R is B times Q minus Q prime and absolute value So we can just replace this with that you see that that's where I'm getting this last inequality Can we cancel the B's? Well, yes, we can because what we can do and this is really the proper procedure Which I'm not gonna write it right out, but we can multiply both sides by 1 over B because B is positive The inverse exists and it's and it's positive. So we have no problem with this Well, yeah, but the problem is if you didn't know then you wouldn't know if you had to switch the sign or not Because you wouldn't know if it was positive or negative But I mean I think what you're asking is suppose you knew it was negative then you would flip the sign Yes, that's that's right Okay, are we okay with this? okay, so All right, I will not say so again thus Dots whatever You guys believe this zero is less than or equal to The absolute value of Q minus Q prime Which is less than one right Would I get to zero from well absolute value can't be negative That's where the zero is popping in from right by definition. The absolute value is not negative You guys buy this this is an integer for sure right certainly not negative integer, but it's I mean it did certainly an integer The absolute value of an integer is an integer What's our conclusion Look at this previous inequality Okay, so I'll write this out. Yeah, so what is the absolute value of Q minus Q prime have to be? Has to be zero right There's no other possibility for it says because it's an integer and if it wasn't zero there would be an integer trapped strictly between zero and one which That's not possible right What do you know about the absolute value of a number if suppose you know the absolute value of a number of zero What can you say about that number? Itself has to be zero right and Voila Q equals Q prime that was one of the two things we had to prove right the quotients are the same Now my question is looking and actually you can do this just looking at what I have on the screen right now How can we there's another part we have to prove right? We have to prove that the remainders are the same too How do we know that? Yes, yeah, I mean that's mostly you have the right idea. Yeah, I mean the point is that this is zero You said a little more than you than you had to but this is zero so therefore B times it doesn't matter what B is doesn't matter that it's positive this thing is certainly zero because This guy is zero Therefore the absolute value of our prime minus R is zero and for the same reason our prime equals R Okay, so does that make sense? Okay Yes, I am good. Okay So recall that our prime minus R is B times Q minus Q prime Thus I'm going to say thus again minus R is B times zero which is zero hence Our prime minus R equals zero so R equals R prime. Okay That's the end of the proof. That's uniqueness Okay, yes, so that took a while Well, I ask you to write this proof on the exam. No Do what I expect you to state the division algorithm precisely on the test. Yes The definition what the division algorithm says that means a theorem really, but yes, you I may ask you that No, as long as it I mean as long as it's yeah I mean you don't have to use the same letters necessarily, but but it should be you should be saying the same thing Okay, can I move on here? Okay Not insulting me. I don't have to start bringing rocks to class start throwing in with people. I've died been known to do that No, I'm just kidding. I don't bring rocks. I'll bring vodka to class. I've been doing but I won't share it I just drink it myself and then I just get really silly Okay So Let's see what else we want to do. Okay. Well, that's that is it for this section. So like I said I want to Spend the remaining 25 minutes or so on a problem. So I feel since this is this is due on Tuesday I definitely want to do a problem using this division algorithm Luckily the idea behind most of the problems you're going to have in this section is this is it's the same idea for almost every problem So I'm going to do one in fact I'm going to do one of your handling problems for you Okay, and then that that will give you the rough idea of how you proceed with the rest of the stuff. Okay, so Let's this this is still 1.2. Okay You definitely want to pay attention to this because you're going to be handing this in Okay, you're going to have a lot of cases here. So this is going to be a little bit tedious the square of Any integer is either of the form 3k or 3k plus 1 This is exactly how the book has worded the problem. They don't tell you what K is. It's understood that K is an integer, okay? So let's see why this is true, and you might think well the division algorithm I learned this when I was 7 How could you possibly use it to do anything interesting? Well, you'll be you might be surprised I don't think this is that interesting of a problem, but the point is here's here's what the problem is saying If you square an integer and you divide by 3 the remainder is either 0 or 1 it can't be 2 That's that's really what this is saying. Okay This is the way you want to interpret some of these things the 3k and the 3k plus 1 You want to think about division by 3 and the division algorithm says okay. I'm going to say it first and I'll write it down The division algorithm says that we can take a we can certainly divide by 3 3 is positive So it fits the hypothesis of the intermediate. Sorry of the division algorithm, and so we know that 3 that Any integer a is equal to 3 q plus r where q's an integer and ours between 0 and 2 Right okay zero is less than or equal to r which is less than 3 right. It's always less than what you're dividing by And then when you square you're going to see that a remainder of 2 is not possible And then I'll write that that down for you, but what you should be thinking about using the division algorithm Okay, so how you would prove this would be to do something like this you would just say okay let a be an integer by The division algorithm. We know that a is equal to 3 q plus r for Some integer q some integer r Satisfying 0 is less than or equal to r which is less than 3 that's exactly what the division algorithm says right Dividing by 3 quotient and remainder Okay, so we need to prove something about a squared though right the square of any integer has this form So a was just an arbitrary integer We're going to square it and we're going to show that in fact when we when we do the algebra What we get is three times an integer or three times an integer plus one and then you'll be done Yes, so we've got a equals 3 q plus r. We're going to square both sides, and this is just a pretty simple algebra 9 q squared plus 6 q r Plus r squared All right you guys by that Well, there are three possibilities And you're going to have to do these cases separately R is an integer satisfying that 0 is less than or equal to r which is less than 3 So what are the possibilities for our are 0 1 or 2 You're going to consider all three of these cases, and you're going to show that no matter what a Squared is going to have to have one of those two forms And that's it then you're done Well the two that the two could okay. No the two r could be two, but the point is when you square You you end up being able to sort of shuffle the algebra around so that you just have three times an integer plus one Remember the r is a division is what you get as a remainder when you divide a by three So are certainly could be two right so you divide five by three right the remainder is two But when you square everything Yes, when you square everything the two falls away. It's not possible. Yes, that yes, that's exactly right. Yes okay So case one and these I you know I'm sorry to say these these are a little tedious but If you see kind of what the mechanics are these Several of these problems aren't that hard then if r is zero then a Squared equals you guys by that a squared sorry a squared equals nine q squared Okay, because a squared it when we right was nine q squared plus six qr plus r squared when r is zero Those other two terms are going to go to zero. So we just get a squared equals nine q squared Don't forget and here's where I think a lot of you are going to fumble around with with the proof Don't forget what it is. You're trying to show you're trying to show that a Sqa on this case in particular. We're trying to show that a squared is three times an integer. We haven't explicitly shown that yet But how do we do that? We just factor out of three then we have three times an integer I expect you to write that in your proof, okay? so This is equal to three times three q squared and setting K equal to three q squared We get a squared equals three k And that's Satisfactory because we want to prove that it's either 3k or 3k plus one. We've got that as 3k. So we've got what we want it Okay, case two R equals one then Maybe you can see how this is going to go a squared equals nine Q squared plus six q plus one you guys by that Okay, because the r is one so we're just replacing We're putting in one for r in that equation Now what do we want to do so now? Yeah, so we Right exactly so if we pull out a three Then we have three q squared plus Plus two q Yeah, so when you pull out the three we've got three q squared plus two q left over right and then we saw the plus one on the Very end. Okay. I'm really sorry for being sloppy. I Because of time here. I know I should be right If we call this k right this is equal to 3k The three q squared plus two q. So if we call that k then we've got queues that we've got um, sorry I keep okay this I have to be careful about okay my cues and the a's are running together here No, you don't you don't have to write that. I mean that that is true Sure, right, but you don't have to you don't have to even worry about trying to prove that that comes back to the Basic foundations of set theory and it's way beyond the scope of what we're going to be doing in here Yes, yes, that's right Okay, did you guys yeah? No, I mean every case we're sort of erasing the variables and starting over again I mean if you want you could say we'll let k be three q squared and then the next case We'll let k prime be three q squared plus two q and the next case will let k double prime be Yes, yes k is just going to be we know that ace we want a squared to be three in this case We want it to be three times an integer plus one and by writing it this way We know that since this is an integer because q is an integer we have it in the form three times an integer plus one So we're just going to let k be that thing Well, but remember what it is that you're trying to prove for case one you're trying to prove that a Squared is equal to 3k or 3k plus one So you have to remember what you're trying to get to I mean so you have this I mean yes, sir You could you could take the square root of both sides But it doesn't get you where you want to go No, I mean no, no, we know that we know that a squared is 9q squared Okay, we know that because we're assuming that are zero that we already have the division algorithm So we already know that a squared is this in case one is r is equal to zero So therefore we do get this okay, but the point is you're you're going in a different direction than where you need to end up You have to always remember where you're trying to end up and that is to show that a squared is three times something or three times Something plus one so it's very tempting when you see squares to sort of just go back to You know high school mode and just take square roots and all that stuff But that doesn't mean that it's relevant to the problem you need to get to where we want to go And this is how you how you do it, okay? Or that yes Okay, do we have this we have this down Okay, case three. Oops. I gotta go back to this thing Okay Let's go. Oh, it's on white. It's on white. Yep. Yep. Sorry No, I refuse to do that on principle Let's see, okay, do we have class I'm done. I'm not gonna be here. I mean I have like 17 dates lined up There's no way I'm coming to class No way Let's see. Okay, so just kidding. I'm there's class this class Yeah, you think of course he doesn't have 17 days. He's a math professor. I'm Yeah, I'm a little cooler than you might you might think I've got some stories you would not believe but I'm not gonna go into that I'm not kidding. But yeah, okay, not like that. But um, okay The next case the last case is r equals two, right? Then Okay, I try to get through this quickly and maybe I can talk about one of the problems from the other section then a squared equals nine q squared Plus Okay, you've got this in your notes If you go back to your notes and you replace the r with two you guys buy this Okay, so a squared is in this case when ours to a squared is nine q squared plus 12 q plus 4 Okay So How do we get now? We want it to be of the form three times something plus one, right? Or well, that's that's what we're gonna you might think well Maybe it's three times something because that's another possible option. It's not I'm just telling you three times something plus one Is where we want to go here. How do we do that? Yes, exactly rewrite the four is three plus one then you can pull out of three and then you've got your plus One on the end. Yep. Yep, that's right. Okay Hopefully you guys believe that Okay, good good Yeah, I'd rather not I know I would do it of course, but if you know if you if you got it then it's fine. Okay Okay, so there it is. There's the proof All right, so we're running out of time, but I do and I think this will well Be useful to you this is I'm gonna talk about one of your homework problems They're gonna hand in on on Tuesday from the other section. I'm gonna say something really quickly about this Okay, so this goes back to what was it 1.2? I think right 1.2 Number 4c. I'm not gonna have time to say a whole lot about this, but I do want to mention this prove that n choose n equals n choose r plus 1 if Sorry, this is getting sloppy Can you guys read this? n choose n oh Wait a minute. Yes. Sorry. Sorry about that. Okay The ink was bleeding on my on my paper Let me erase this. Thank you for that Okay, and choose r is equal to n choose r plus 1 if and only if n is odd and R Equals one half times n minus 1. Okay make this a big huge period. There we go. Okay r plus 1 Okay, here's what I want to say. I'm not gonna have much time to do a whole lot here But if and only if statements those of you that took to scream maybe are aware of this There are two statements you have to prove and if and only ifs, okay It's a it's a sort of a double implication if and only if means the first thing implies the second and the second implies the first That's what if and only if means so when you're writing this it's sort of a two You have a sort of two sub problems here The first thing you have to do is prove that n choose r is to do this assume N choose r equals n choose r plus 1 and then prove the other side of this that n is odd and r is one half times n minus 1 That's the first part of the proof The second part of the proof is assume n is odd and r equals one half times n minus 1 prove that those two binomial coefficients are the same So there's really a part a and a part b here that you need to do So what I will say I'm not going to have time to go into this But how do I do how would you do the first direction well assuming that n choose r equals n choose r plus 1? We want to prove we want to prove this What you do is you say okay? Well, we have n choose r equals n choose r plus 1 Write out the definition right n factorial times r factorial times n minus r factorial equals blah blah blah what your goal is is to get down to that Okay, what does that mean think about it? We've got a bunch of factorials to begin with this has no factorials So you should use the inductive definition of the factorial to try to cancel things out And if you do it right you will end up with that That makes sense. I had to say it quickly, but how do you know that n has to be odd? How do you know that? This is the last thing I'm going to keep you here until someone Yes You're going you're going way off in an area that you don't need to not that what you're going to say isn't true It's simpler. It's much simpler than that What's that? But yeah, okay, so when you so that's one thing you could do yeah, and that's well Yeah, that's that's one perfectly reasonable way to do it Hang on one second hang on there There's other things you can say to but the point is once you've got to this point Okay, solve for n you're going to get n is equal to 2r plus 1 so by definition n is odd That's it. All you have to do is solve for it Okay, another way of thinking of it is r for this day even have meaning r has to be an integer And if n was even then you'd take an odd number divided by 2 and that's not an integer Okay, so either way, but that's the idea so I Think I'll just we have a few minutes by thinking when to stop here I'm just not going to be able to get very far if I if I don't so That is that is not true I do I do not maybe one maybe two but that's about the extent of it Okay Yeah, sure