1 00:00:28,630 --> 00:00:36,540 Money, everyone, we're on complex numbers, so just to the end of the last lecture, we were thinking about the arguments of a complex number. 2 00:00:36,540 --> 00:00:42,040 And I'd like to keep going from there a little bit. So sort of thinking about Cartesian coordinates, polar coordinates. 3 00:00:42,040 --> 00:00:52,890 And it's really important that it's convenient to pass between the two of those. So here is a useful thought. 4 00:00:52,890 --> 00:01:00,680 So we can pass between. 5 00:01:00,680 --> 00:01:11,570 Cartesian and polar coordinates. 6 00:01:11,570 --> 00:01:17,990 So here's my kind of picture of what I'm doing, so this is my all gang diagram, my complex plane. 7 00:01:17,990 --> 00:01:24,710 Here is some little complex number which I might think of as having Cartesian coordinates a comma b, 8 00:01:24,710 --> 00:01:31,190 or I might think of as having modulus off an argument theta. 9 00:01:31,190 --> 00:01:35,090 And I'd like to know if I know and b, can I find out and see if I feet to catch my flight? 10 00:01:35,090 --> 00:01:53,900 And B, that's the kind of point here. So if Z, which is a plus B or C, has modulus ah and argument beta, then we can find A and B. 11 00:01:53,900 --> 00:02:02,300 So maybe I'll throw this on here and another colour for you. So this is just a little bit of trigonometry. 12 00:02:02,300 --> 00:02:06,830 So this bit is our. This is a this is B sine Fita. 13 00:02:06,830 --> 00:02:22,620 Sorry, also in features. Try that again, which is B. So a is a closed beta and B is a sign theatre. 14 00:02:22,620 --> 00:02:30,900 So one kind of way of thinking about that then, is to think of it as being our times because it's plus I sign theatre, 15 00:02:30,900 --> 00:02:37,980 which is a representation that we find coming up in various contexts in the other direction. 16 00:02:37,980 --> 00:02:46,890 If that is a plus B and to find are in theatre, well, we know how to find oh, just by the definition, 17 00:02:46,890 --> 00:02:53,400 so are is the modulus of Z is just defined by root of a squared plus b squared. 18 00:02:53,400 --> 00:03:05,760 That's how we defined it. And also again, looking at the diagram time theatre is B over a, at least in the case when it's not zero. 19 00:03:05,760 --> 00:03:08,130 I'm very scared about the possibility of dividing by zero. 20 00:03:08,130 --> 00:03:14,070 I don't want to risk it, but as long as it's not zero, then B overall is going to give me time theatre. 21 00:03:14,070 --> 00:03:19,560 From that, I can kind of recover theatre. But as we know this, this sort of ambiguity about what theatre is, 22 00:03:19,560 --> 00:03:25,060 because if you have some argument, theatre will feature plus two pi or seats, plus it needs to. 23 00:03:25,060 --> 00:03:31,560 A multiple of two PI will also be a valid thing, and that's reflected in the fact that there's not a kind of unique inverse here. 24 00:03:31,560 --> 00:03:38,370 So I'll just put determining theatre is delicate. 25 00:03:38,370 --> 00:03:45,270 So in practise, this is not a problem. You just need to kind of have your wits about you a little bit. 26 00:03:45,270 --> 00:03:47,100 So I want to know what happens when we multiply. 27 00:03:47,100 --> 00:03:53,100 We saw that we could interpret addition of complex numbers geometrically with that kind of vector addition parallelogram diagram. 28 00:03:53,100 --> 00:03:58,380 What happens with multiplication when we know what happens to the modulus, when we multiply, that behaves nicely. 29 00:03:58,380 --> 00:04:13,860 What happens with the argument? So this is Proposition five. And this says take Z and W in the complex numbers, but not zero, then. 30 00:04:13,860 --> 00:04:23,550 So remember, the arguments are zero isn't defined. So I want to ignore that the argument of the Z Times W is equal to the arguments said. 31 00:04:23,550 --> 00:04:28,980 Plus the argument if w. So a couple of little comments here. 32 00:04:28,980 --> 00:04:32,520 One is the here we're working with arguments modulo two pi. 33 00:04:32,520 --> 00:04:39,060 So this idea that these two are equal up to some multiple integer multiple of two PI. 34 00:04:39,060 --> 00:04:52,360 So it's important that we remember that. So this is working with arguments modulates you PI. 35 00:04:52,360 --> 00:04:57,460 The other is that you've seen this kind of relationship before. This looks a bit like a logarithm, right? 36 00:04:57,460 --> 00:05:05,800 That's not a coincidence. There are no coincidences in mathematics, so maybe I'll just note here this looks a bit like log looks like log. 37 00:05:05,800 --> 00:05:09,700 That's very pleasingly illustrative. So let's think about how are we going to prove this? 38 00:05:09,700 --> 00:05:14,710 So this is not too bad actually, given this kind of thinking that we've been doing up here. 39 00:05:14,710 --> 00:05:25,420 So if I let this to be the arguments of Z and fi b the arguments, if w, I hope you're making good progress with letting the Greek alphabet. 40 00:05:25,420 --> 00:05:31,030 I'm not fantastic with every Greek letter of a feature for a really good Greek classes to know. 41 00:05:31,030 --> 00:05:37,900 So then what we've just said is that says it's going to be the modulus of Z Times because it's a plus. 42 00:05:37,900 --> 00:05:44,350 I sign Theta and W. It's going to be the modulus of W Times Code Phi. 43 00:05:44,350 --> 00:05:56,170 Plus I sign Phi so we can just multiply so that W is the modulus of Z times, 44 00:05:56,170 --> 00:06:08,800 the modulus of W times course theta plus i sign Theta Times because Phi plus I sign Phi and I can do some tidying up here. 45 00:06:08,800 --> 00:06:12,700 So one thing is that I know what happens with this multiplication for the modulus. 46 00:06:12,700 --> 00:06:17,110 So that's the modulus of that w. I'll write it that way round. 47 00:06:17,110 --> 00:06:20,260 And I guess we're using Proposition three here. 48 00:06:20,260 --> 00:06:25,120 I'm not going to record every time you use Prop. three for the rest of all time because it's just going to come up so much. 49 00:06:25,120 --> 00:06:34,090 But I'll just note it here to remind you. And if I multiply out here, I get constater to cause Phi minus sign C to sign Phi. 50 00:06:34,090 --> 00:06:47,230 That's coming from these terms. Plus all times cause it's a sign PHI plus sign theta calcify. 51 00:06:47,230 --> 00:06:52,270 And hopefully you might recognise these two will kind of compound annual formulae and trig. 52 00:06:52,270 --> 00:07:03,640 So this is where my jealousies at W times cause of C so plus Phi plus i times theta plus phi. 53 00:07:03,640 --> 00:07:14,530 And so we can read off immediately that the argument is that W is c c plus phi, which is the argument of Z Plus the arguments of W. 54 00:07:14,530 --> 00:07:24,070 Of course, it's not a coincidence that the trick here works out nicely, right? So we can use this to make sense of multiplication in a geometric way. 55 00:07:24,070 --> 00:07:44,690 So remark. We can interpret multiplication and see geometrically. 56 00:07:44,690 --> 00:07:54,020 So multiplying by a non-zero complex numbers that's multiplying by zero, it's not so exciting, 57 00:07:54,020 --> 00:08:02,300 multiplying by some non-zero complex number Z that rotates the complex plane. 58 00:08:02,300 --> 00:08:14,510 I'll move over here. Anti-Clockwise. 59 00:08:14,510 --> 00:08:23,760 Why the argument is that this is a consequence of that proposition, we just proved, and it enlarges. 60 00:08:23,760 --> 00:08:31,650 By a factor, the modulus of that and that sort of centred on the origin. 61 00:08:31,650 --> 00:08:40,200 So there is a nice way to make sense of multiplication without geometric view of complex numbers as well as algebraically. 62 00:08:40,200 --> 00:08:44,190 So we're going to find ourselves using this proposition. Lots. 63 00:08:44,190 --> 00:08:48,780 OK, I want to introduce you to some particularly nice complex numbers there called the inner circle. 64 00:08:48,780 --> 00:08:58,170 So definition the inner circle. 65 00:08:58,170 --> 00:09:04,080 And see, I'm defining unit circle. It's kind of what you expect, 66 00:09:04,080 --> 00:09:10,860 but I'm also telling you this because I want to tell you that it gets called s one and s one is defined to be the set of all z in C, 67 00:09:10,860 --> 00:09:16,620 such that the Modulus Z is equal to one. 68 00:09:16,620 --> 00:09:21,420 So geometrically, it's a circle of radius one centred on the origin. 69 00:09:21,420 --> 00:09:26,640 And what we've just noticed is that we can write that's in another way. 70 00:09:26,640 --> 00:09:29,520 So here are a few little remarks about the unit circle. 71 00:09:29,520 --> 00:09:42,960 So we've just seen essentially that S1 is also the sex of all cos theta plus I sign feature by feature as well. 72 00:09:42,960 --> 00:09:48,150 So that would be another way of thinking about it. 73 00:09:48,150 --> 00:10:02,490 Here is an amusing and really surprisingly useful facts about numbers in the unit circle, so if I take Zs and ask one, what is the inverse of Z? 74 00:10:02,490 --> 00:10:06,900 Well, we know that Z Times Z bar is the modulus of z squared. 75 00:10:06,900 --> 00:10:09,090 If you're in the inner circle, the Modulus Z squared is one. 76 00:10:09,090 --> 00:10:16,920 So Z times that bar is what in this case, so the inverse of Z is its complex conjugate kit. 77 00:10:16,920 --> 00:10:22,050 Also useful, and this is also going to lie in the unit circle. 78 00:10:22,050 --> 00:10:32,760 So the inner circle has all sorts of nice structure within itself, and in fact, S1 is a group under multiplication. 79 00:10:32,760 --> 00:10:36,660 This is not the time to tell you what a group is. There's a course on that. 80 00:10:36,660 --> 00:10:45,480 It's called groups. So in February, some person will introduce you to groups in the course. 81 00:10:45,480 --> 00:10:50,950 In groups and group actions, you'll meet as one as an example. So this thing about the inverse is relevant. 82 00:10:50,950 --> 00:10:57,000 There are some other things that go into that. The thing to remember here is the inner circle is a kind of important and interesting object. 83 00:10:57,000 --> 00:11:07,350 It has kind of structure. So here's a really super exciting results about numbers in the inner circle. 84 00:11:07,350 --> 00:11:21,540 This is a theorem, and it's called the maths theorem. And this is going to be, I imagine, familiar to quite a few of you. 85 00:11:21,540 --> 00:11:24,330 But I might not be phrasing it necessarily in the way that you're used to. 86 00:11:24,330 --> 00:11:38,220 So for Z in the unit circle and any integer and remember, this just means any integer we have. 87 00:11:38,220 --> 00:11:46,540 That the argument of Z to the N is end the times the arguments is that. 88 00:11:46,540 --> 00:11:51,760 So this proposition is a very special kind of it's kind of related to this. 89 00:11:51,760 --> 00:11:55,960 We're going to be using this proposition to feed into it. But this is kind of helpful. 90 00:11:55,960 --> 00:12:11,190 So you might be more familiar with equivalent formulation equivalently for feta in R and and in Z, we have. 91 00:12:11,190 --> 00:12:19,410 That cost data, plus I signed Visa to the end is cause of and. 92 00:12:19,410 --> 00:12:23,820 Plus I sign of the feature. 93 00:12:23,820 --> 00:12:31,740 So definitely when I first learnt about troops there in my last question, this form, but this form is a really helpful way of thinking about it. 94 00:12:31,740 --> 00:12:36,660 So I encourage you to think about why these are saying the same thing, because if you understand why they're saying same thing, 95 00:12:36,660 --> 00:12:40,200 that's a really good kind of sign that you are understanding the theorem. 96 00:12:40,200 --> 00:12:52,650 So let's prove this. So I'm going to fix some Z and S one and show that the theorem applies to the Z and four and greater than equal to zero. 97 00:12:52,650 --> 00:12:58,020 So for non-negative and I'm going to use induction on N. and I'm going to write that down because it 98 00:12:58,020 --> 00:13:03,240 helps the reader to know that we going to use induction so far and great ones are equal to zero. 99 00:13:03,240 --> 00:13:14,610 We use induction on N. So my base case is energy zero, and that's just a quick check. 100 00:13:14,610 --> 00:13:19,200 So Z to the zero is equal to one. 101 00:13:19,200 --> 00:13:24,330 So the argument is that's the zero zero is end times. 102 00:13:24,330 --> 00:13:29,910 The argument is that in this case. So that was important, although not super exciting. 103 00:13:29,910 --> 00:13:46,470 More interestingly, let's think about the inductive step, so let's suppose the result holds. 104 00:13:46,470 --> 00:13:53,700 For some and greater than or equal to zero, so we've got some fixed value then where we're supposing it holds. 105 00:13:53,700 --> 00:13:59,550 So we're saying that the all humans have said to the end is equals and times the 106 00:13:59,550 --> 00:14:03,810 arguments of Z and that we want to think about the arguments is that to the end +1. 107 00:14:03,810 --> 00:14:12,720 So let's do that over here. So then the argument of Z to the N +1. 108 00:14:12,720 --> 00:14:18,150 I want to relate to the arguments of Z, but I can do that using Proposition five, 109 00:14:18,150 --> 00:14:24,270 which tells us that that's the argument is that the N plus the arguments of Z so right down, that's why. 110 00:14:24,270 --> 00:14:33,670 Proposition five. And then we know about the occupants Z to the N, because that was our induction hypothesis. 111 00:14:33,670 --> 00:14:50,410 So this is enzymes. The argument is Z plus the arguments of Z by the induction hypothesis, and that's equal to, of course. 112 00:14:50,410 --> 00:15:01,390 And plus one times the arguments of Z. So we say that the results holds for and plus one. 113 00:15:01,390 --> 00:15:07,480 So that's an inductive proof that shows us that this relationship works for any greater than or equal to zero. 114 00:15:07,480 --> 00:15:20,070 No, I wouldn't think about negative ed and. One possibility would try to be to do a kind of induction downwards that could be a useful strategy, 115 00:15:20,070 --> 00:15:28,920 but actually I think we can just use the results for positive values so far and less than zero, we use the results for positive. 116 00:15:28,920 --> 00:15:36,090 And so this can be a nice strategy to kind of use work you've already done to save yourself a bit of effort. 117 00:15:36,090 --> 00:15:43,080 So if I fix some negative end, I'm going to let m b minus n. 118 00:15:43,080 --> 00:15:56,250 So of course M is then positive. So by our previous result, so we know that the argument of W to the M is m times the arguments. 119 00:15:56,250 --> 00:16:06,750 If W for any W in as one that's using the results for M, which is opposed to values. 120 00:16:06,750 --> 00:16:12,960 So that follows from our previous pod. So we just need to think about how are we going to apply this? 121 00:16:12,960 --> 00:16:19,590 Well, let's think about Z to the end. Is that to the end is z inverse to the power M? 122 00:16:19,590 --> 00:16:28,140 That would be another way of rewriting it. And Z Inverse I notice in my kind of fun fact over there was the complex country, this of Z. 123 00:16:28,140 --> 00:16:33,810 So we going to be able to apply this result with WB complex, conjugative z. 124 00:16:33,810 --> 00:16:38,250 I quite like to know what the arguments of the complex conjugate subset is, 125 00:16:38,250 --> 00:16:43,260 but if you ponder that for a moment, you'll see that it's minus the arguments of Z. 126 00:16:43,260 --> 00:16:56,370 Again, everything is modulo two pi here. So the argument is that the N is the argument of the Z bar to the M, which is m times. 127 00:16:56,370 --> 00:17:07,290 The argument is that bar, which is m times minus the arguments of Z, which is end times the argument of Z. 128 00:17:07,290 --> 00:17:13,740 So that proves the results for negative values using the result for positive values. 129 00:17:13,740 --> 00:17:18,030 So it's one theorem. It's just kind of somehow utterly fundamental. You get to use lots. 130 00:17:18,030 --> 00:17:36,570 Here is a specific example of a kind of classic way in which it gets used, namely trig compound angle formulae. 131 00:17:36,570 --> 00:18:01,770 So by dint of for any feature in R, we know the cause of three feature plus I sign of three Fita is close to plus I sign C to keep it. 132 00:18:01,770 --> 00:18:04,620 That's from that kind of equivalent formulation. 133 00:18:04,620 --> 00:18:12,390 My goal here is to find some nice formula for close to three three two in terms of close to four sign of three three two in terms of sine theta. 134 00:18:12,390 --> 00:18:17,340 So I'm going to expand this out. So this is a binomial theorem and kind of tidying up, 135 00:18:17,340 --> 00:18:24,300 and I think I'm going to get close cubed theta minus three cause features sine squared 136 00:18:24,300 --> 00:18:40,710 thetr plus i times three core square features science beta minus sign keeps the beta. 137 00:18:40,710 --> 00:18:47,300 So that's just by multiplying out quite a lot of practise and multiplying out keeps. 138 00:18:47,300 --> 00:18:51,780 And I'm just trying to tidy up the real and imaginary parts as I go along. 139 00:18:51,780 --> 00:19:03,750 So comparing real and imaginary parts. 140 00:19:03,750 --> 00:19:18,720 Gives that cause of three Fita is cause keeps B to minus three cost to sign squared theta. 141 00:19:18,720 --> 00:19:22,710 And then I got a formula for sine square theatre in terms of cost pizza, right? 142 00:19:22,710 --> 00:19:33,000 So I could kind of keep going and tidy up, and I could write out some kind of corresponding formula for sign three seater and again, tidy that up. 143 00:19:33,000 --> 00:19:37,680 So I'm going to let you do that tidying up while we have a short pause, 144 00:19:37,680 --> 00:19:43,290 I also have one other thing for you to do in this short pause at the end of each of your lecture courses. 145 00:19:43,290 --> 00:19:49,050 This year will give you a paper questionnaire towards the end of the course so that you could give some feedback on how you found the course, 146 00:19:49,050 --> 00:19:51,900 how you found the lectures and so on so that we can then, as a department, 147 00:19:51,900 --> 00:19:56,340 use that to improve teaching for next year, even though it's only Thursday if week one. 148 00:19:56,340 --> 00:20:02,610 This is in fact the last lecture of this course officially, which means it's question time. 149 00:20:02,610 --> 00:20:09,090 So I'm going to put some piles of these in strategic places if you could pass them along so everybody gets a questionnaire, that would be great. 150 00:20:09,090 --> 00:20:11,160 If you want to fill it in during the lecture, you can. 151 00:20:11,160 --> 00:20:15,060 And if you just need two piles, one at the top there and one of the top that, I'll collect them at the end. 152 00:20:15,060 --> 00:20:21,660 If you prefer to fill it in later, you can also return it directly to the academic admin team who are the people who process them. 153 00:20:21,660 --> 00:20:26,950 So this is your opportunity to give some feedback on the very short, complex numbers course. 154 00:20:26,950 --> 00:20:32,560 OK. The question is making their way round, so if you can if you can concentrate on keeping the momentum go passing those round, 155 00:20:32,560 --> 00:20:38,080 that would be great so that everybody gets them. I will try to remember a couple of minutes to check that everybody's got them, 156 00:20:38,080 --> 00:20:41,380 but hopefully that's giving you a little bit of a chance to think about these trick formulae, 157 00:20:41,380 --> 00:20:44,380 especially if this isn't something that you've thought about before. 158 00:20:44,380 --> 00:20:51,400 Let's keep going over here because excitingly, it's time for endless routes of unity. 159 00:20:51,400 --> 00:21:07,740 So here's a definition. So if this is a complex number and is a strictly positive integer, which you might call a natural number, 160 00:21:07,740 --> 00:21:11,490 but somehow we can't decide whether or not there is an actual number and I don't have an argument about it. 161 00:21:11,490 --> 00:21:17,190 So I'm just trying to be unambiguous. And that to the end is equal to one. 162 00:21:17,190 --> 00:21:40,590 Then we say the Z is a writ of unity or more precisely on the spirit of unity. 163 00:21:40,590 --> 00:21:44,850 What I learnt about this stuff, I think I remember thinking all routes of unity that sounds really cool, 164 00:21:44,850 --> 00:21:48,780 and then unity turned out to be one sort of slightly less dramatic than I expected. 165 00:21:48,780 --> 00:21:53,640 But reach of one is still pretty good. There are lots of interesting things to explore. 166 00:21:53,640 --> 00:22:00,600 For example, Proposition seven tells us which numbers are, in fact, routes of unity. 167 00:22:00,600 --> 00:22:05,130 So we can we can pinpoint where these things are. Without too much difficulty. 168 00:22:05,130 --> 00:22:12,180 So let's take Z in C then. 169 00:22:12,180 --> 00:22:20,990 Z is an end threat of unity. 170 00:22:20,990 --> 00:22:31,730 If and only if. And I've got two conditions here, one on the modulus is that one of the arguments is that so I need the modulus said to be one. 171 00:22:31,730 --> 00:22:49,830 And the argument of that? So we took PI over, and for some integer, okay, so this precisely describes what the merits of unity look like. 172 00:22:49,830 --> 00:22:53,250 So let's prove Proposition seven. 173 00:22:53,250 --> 00:22:58,560 And you'll notice the Proposition seven is an if and only if statement, so it's really two statements packaged as one. 174 00:22:58,560 --> 00:23:03,040 Let's prove the two separately and try to help you see what one we're doing. 175 00:23:03,040 --> 00:23:07,440 I find it helpful to write these little kind of implication arrows in brackets when I'm doing this for my own benefit. 176 00:23:07,440 --> 00:23:11,640 I do this not just for you. So we're going to see the left to right first. 177 00:23:11,640 --> 00:23:22,950 So we're going to suppose that is and through it of unity. 178 00:23:22,950 --> 00:23:28,800 So we're supposing that the end is equal to one, and then we wanted to do kind of interesting things. 179 00:23:28,800 --> 00:23:33,810 Well, we can say something about the modular straightaway. The module, as opposed to the power end, is the modular. 180 00:23:33,810 --> 00:23:43,890 So Z to the N is one and the modular is that is strictly positive. 181 00:23:43,890 --> 00:23:49,230 So the modular z better be equal to one. 182 00:23:49,230 --> 00:24:02,040 So much of this is that it's just a real number here and bytes and one of which conveniently we've just proved and come up to of comma. 183 00:24:02,040 --> 00:24:10,590 The argument is that to the end is end times the argument is dead, but we know the Z to the N is equal to one. 184 00:24:10,590 --> 00:24:15,750 So the arguments of Z is the arguments of one over N, 185 00:24:15,750 --> 00:24:21,300 and then we have to concentrate because there's a little subtlety here because we're working with arguments modulo two pi. 186 00:24:21,300 --> 00:24:27,120 So if we to say the arguments of one is zero, therefore this is zero and that wouldn't be correct. 187 00:24:27,120 --> 00:24:31,920 All we know is that the argument is zero plus an integer multiple of two PI. 188 00:24:31,920 --> 00:24:44,370 So what we know is that this is 2kay pi over n for some integer K, and that gives us what we were looking for the other direction. 189 00:24:44,370 --> 00:24:54,780 Let's get rid of these smudges, really the other direction. So now we're going to suppose that the modulus ZS is equal to one. 190 00:24:54,780 --> 00:25:05,730 And the arguments is Z is two pi over MN for some integer K. 191 00:25:05,730 --> 00:25:10,200 So then what can we say about the modulus? Is that to the end? Well, that's the modulus of Z. 192 00:25:10,200 --> 00:25:19,720 So the ED is one we like. Modulus modulus works out really nicely and also. 193 00:25:19,720 --> 00:25:29,140 The argument is that to the end is end times, the arguments have said, is to party. 194 00:25:29,140 --> 00:25:36,120 So what's telling us? It's telling us that Z to the N has the same. 195 00:25:36,120 --> 00:25:48,400 Modulus and argument as one. So in fact, to the end is equal to one. 196 00:25:48,400 --> 00:25:54,870 So is that is an end to of unity and that finishes are proof. 197 00:25:54,870 --> 00:26:01,710 So if I carry on over here, so one small observation from Proposition seven, 198 00:26:01,710 --> 00:26:17,880 so vermaak Proposition seven shows, but roots of unity lie in as one they lie in the unit circle. 199 00:26:17,880 --> 00:26:22,110 And that's the kind of handy thing to be aware of occasionally. Right? 200 00:26:22,110 --> 00:26:27,530 How are we doing on questionnaires? Could you please put your hand up if you do not have a questionnaire? 201 00:26:27,530 --> 00:26:33,190 Interesting. Could you please put your hand up if you have some spare questions? 202 00:26:33,190 --> 00:26:45,570 OK. The people who didn't have questionnaires were over here somewhere. 203 00:26:45,570 --> 00:26:54,340 Oh, thank you. My question is why if if you need a questionnaire? OK, so I got to send some this way and some this way, 204 00:26:54,340 --> 00:27:07,420 and it would be great if you could sort them out amongst yourselves and hopefully there are enough. 205 00:27:07,420 --> 00:27:14,200 So cute consequence of Proposition seven is that we can count and threats of unity, 206 00:27:14,200 --> 00:27:20,770 so I think maybe I said on Monday a corollary is a quick consequence of something you've already proved. 207 00:27:20,770 --> 00:27:27,790 So Corollary eight says four and basically are equal to one. 208 00:27:27,790 --> 00:27:37,660 There are exactly NW and three acts of unity. 209 00:27:37,660 --> 00:27:43,030 And that's quick to prove from Proposition seven because we know what the end streets of unity are. 210 00:27:43,030 --> 00:27:46,870 So by Proposition seven, 211 00:27:46,870 --> 00:28:06,790 the end roots of unity are so one way of writing them would be as cause of took pi over and plus I sign of two pi over n for any K in the integers. 212 00:28:06,790 --> 00:28:13,000 That makes it look like there are lots of them. But of course, there's lots of duplication here. 213 00:28:13,000 --> 00:28:38,290 So there are. All and such distinct values, e.g. we could choose K to come from zero one up to and minus one. 214 00:28:38,290 --> 00:28:42,460 So that was a handy consequence, it's good to know not only where they intrude c.a.r., 215 00:28:42,460 --> 00:28:46,600 but that we've got and and threats of unity, that's a good thing to know. 216 00:28:46,600 --> 00:28:53,500 There are some end suites. You're going to see that even more exciting than others. There are some threats of unity, the zone and threats of unity. 217 00:28:53,500 --> 00:28:59,800 But there aren't smaller groups of unity aren't threats of unity for smaller ebb and that particularly important, so they get a special name. 218 00:28:59,800 --> 00:29:14,890 So this is a definition. So let that be an ensuite of unity. 219 00:29:14,890 --> 00:29:25,090 If Z to the M is not equal to one for one less than or equal to m less than equal to n minus one, 220 00:29:25,090 --> 00:29:45,320 then we say that Z is a primitive and threats of unity. 221 00:29:45,320 --> 00:29:51,800 And these kind of have Connexions with things from group theory and other other things that you will see those very strange noise, 222 00:29:51,800 --> 00:29:59,270 other things that you will see this year. So a primitive and through to you, this is like end is the first power of said where you get one. 223 00:29:59,270 --> 00:30:06,740 So here is a proposition that is useful and certainly applies to primitive roots of unity, 224 00:30:06,740 --> 00:30:20,210 although it applies a little bit more generally than that as well. So if that is an end of unity? 225 00:30:20,210 --> 00:30:25,550 For some and greater than or equal to two. 226 00:30:25,550 --> 00:30:39,830 And that is not equal to one, then Z two, the power and minus one plus that's the power and minus two plus plus z plus one is equal to zero. 227 00:30:39,830 --> 00:30:43,190 This is the kind of thing where if I were working on these lecture notes this afternoon, 228 00:30:43,190 --> 00:30:47,420 thinking about the might be thinking, Oh, why is Vicki excluded the case when Nichols won? 229 00:30:47,420 --> 00:30:53,360 What was special about that? That kind of why the hypotheses that that's the sort of question that you could be asking yourselves. 230 00:30:53,360 --> 00:31:03,770 So this is not too tricky to prove because we know that zero is said to the N minus one. 231 00:31:03,770 --> 00:31:09,500 We're assuming the Z is an each of unity. Oh, we have factories that we have minus one. 232 00:31:09,500 --> 00:31:12,560 If this factorisation is not second nature to you, 233 00:31:12,560 --> 00:31:19,760 I invite you to think about it because it's a really useful factorisation and kind of generalise this difference of two squares. 234 00:31:19,760 --> 00:31:31,050 And in fact, it's this. And that is not equal to one. 235 00:31:31,050 --> 00:31:37,370 And that's the end of the proof. So this is not super difficult, but it's kind of useful. 236 00:31:37,370 --> 00:31:45,840 It's it's a nice result. So this starts to have some nice consequences geometrically, for example, when you're thinking about routes of unity. 237 00:31:45,840 --> 00:31:52,200 OK, I want to think about another way to represent complex numbers. 238 00:31:52,200 --> 00:31:57,180 We sort of thought about much of this argument to be thought about this Cartesian coordinates kind of way. 239 00:31:57,180 --> 00:32:09,240 There's another very useful way of writing them, which comes from the following fact which gets called Euler's formula, at least sometimes. 240 00:32:09,240 --> 00:32:25,470 May, says Fifita, and oh, we have E to the I theatre is equal to cost, plus I sign the theatre. 241 00:32:25,470 --> 00:32:30,270 While you take 10 seconds to just absorb how lovely that is, 242 00:32:30,270 --> 00:32:35,660 how we do a question questionnaires, please put your hand up if you do not have a question at. 243 00:32:35,660 --> 00:32:43,520 Results forget night, you will notice that I have labelled this as facts rather than theorem or any of those kinds of things. 244 00:32:43,520 --> 00:32:47,540 This looks a lot like a thing that needs proving right and it does need proofing. 245 00:32:47,540 --> 00:32:51,440 But in order to prove it, we first need to define the exponential in the cosine in the sign. 246 00:32:51,440 --> 00:32:55,400 This is, we haven't done any of those things. We're not in a very good place to prove this. 247 00:32:55,400 --> 00:32:58,820 Defining the exponential is something you will do an analysis this year and also 248 00:32:58,820 --> 00:33:02,810 cosine and sine once you've learnt how to add up infinitely many things in a safe, 249 00:33:02,810 --> 00:33:07,700 controlled manner. It's very important to be careful when adding up infinitely many things. 250 00:33:07,700 --> 00:33:10,640 Once you have kind of got your licence that you are safe to do that, 251 00:33:10,640 --> 00:33:13,910 you can think about the exponential and then you'll be able to think about why this is true. 252 00:33:13,910 --> 00:33:17,990 So I just put a little bit more in the online note. So if you want to kind of read a little bit more about that. 253 00:33:17,990 --> 00:33:23,570 Have a look at the online notes, but we're just going to kind of accept this as a fact for our purposes. 254 00:33:23,570 --> 00:33:46,160 So here is a remark. This gives a very convenient way to represent a complex number. 255 00:33:46,160 --> 00:33:52,850 So if that has the modulus is that is ah. 256 00:33:52,850 --> 00:34:03,630 And the arguments is said is theatre, then that is r e to the I theatre. 257 00:34:03,630 --> 00:34:09,750 When I started learning about complex numbers, I definitely thought about the most in the A-plus B form. 258 00:34:09,750 --> 00:34:12,750 Then I think when I was doing A-level kind of studying a bit further, 259 00:34:12,750 --> 00:34:22,860 I thought about the more in the times cos that's a plus I sign thesis form and I probably think about the more in this form now. 260 00:34:22,860 --> 00:34:32,100 So I went to stress that this is very convenient, so I put flashing neon lights to remind you. 261 00:34:32,100 --> 00:34:38,190 So if this is relatively new, it feels a little bit uncomfortable, it feels natural to go back to your old ways of thinking about it. 262 00:34:38,190 --> 00:34:42,660 That's that's what we do when we meet something new is kind of try to see what can we keep go with the old way. 263 00:34:42,660 --> 00:34:44,250 This has lots of advantages. 264 00:34:44,250 --> 00:34:53,160 I really encourage you to look for opportunities to practise working with this way of thinking about it so you become more fluent, for example. 265 00:34:53,160 --> 00:35:15,890 Proposition 10. So Proposition 10 says Take Z a non-zero complex number and you'll see why I wanted to be known zero z in C, but not zero. 266 00:35:15,890 --> 00:35:29,300 That's a zero, then Z has exactly NW and three. 267 00:35:29,300 --> 00:35:34,220 So for any and greater than or equal to one. 268 00:35:34,220 --> 00:35:43,040 So we thought about risks of one right proposition and a corollary eight, so he said that exactly and and three to one. 269 00:35:43,040 --> 00:35:49,460 This is generalising to the exactly average of any non-zero complex number, which is kind of nice. 270 00:35:49,460 --> 00:35:54,610 So let's prove this. 271 00:35:54,610 --> 00:36:00,520 So I had to do this in three parts, and I got to kind of bullet point what the sections are to try to help you see what the structure is clearly. 272 00:36:00,520 --> 00:36:08,980 So the first part is that there's at least one and threw it right at the start. 273 00:36:08,980 --> 00:36:19,810 It's not completely clear that there is at least one end throughout. So if I let all be the modulus of Z and seem to be the arguments of Z, 274 00:36:19,810 --> 00:36:30,190 so z is r e to the fitter I can sort of stare at a set in this form and guess what an end through might be. 275 00:36:30,190 --> 00:36:41,390 So there is. A in fact, unique, positive real. 276 00:36:41,390 --> 00:36:50,930 As such that as to the act as to the power is equal to R. 277 00:36:50,930 --> 00:36:55,550 So this is since R is greater than zero. How do we know this is true? 278 00:36:55,550 --> 00:36:59,580 By analysis, so I'll just put C prelims analysis. 279 00:36:59,580 --> 00:37:05,030 You will think about why you can do this in that course. 280 00:37:05,030 --> 00:37:14,870 So that's going to be the modulus of my proposed rate. And now I'm going to let Phi equal theta over N. 281 00:37:14,870 --> 00:37:17,600 Why did I start over here? Yeah, OK. 282 00:37:17,600 --> 00:37:30,950 So if I let w be equal to the s e t i phi, then double t w to the end, when you do a quick calculation, you c is equal to Z. 283 00:37:30,950 --> 00:37:36,230 So this definitely at least one right now, I'd like to show that there are, in fact, 284 00:37:36,230 --> 00:37:44,450 at least and and it's how did you show that they're exactly and if something you show that at 285 00:37:44,450 --> 00:37:48,560 least and if something you say the most and if something that could be a useful strategy. 286 00:37:48,560 --> 00:37:53,540 So I would say that there are a list and so I'm going to take W as above. 287 00:37:53,540 --> 00:38:03,740 So W is a fixed and through it, I could get a whole bunch more and through it by multiplying by then three of one. 288 00:38:03,740 --> 00:38:11,770 So if I let Alfa be an M3 of unity. 289 00:38:11,770 --> 00:38:18,880 Then you do a quick calculation and you realise the alpha times the power and is also equal to Z. 290 00:38:18,880 --> 00:38:30,830 So Alpha W. is another Richard Z. And by corollary eight, there are and such alpha. 291 00:38:30,830 --> 00:38:45,070 Giving a well, at least, and distinct and fruits of that. 292 00:38:45,070 --> 00:38:52,090 So this says I can think of and distinct sense roots is that we haven't shown that there aren't some or maybe some more that have some other form, 293 00:38:52,090 --> 00:38:54,340 of course, their arms, that's what we're about to prove, 294 00:38:54,340 --> 00:39:00,850 but that's why we're kind of structuring it like this if I carry on over here for the third bullet point. 295 00:39:00,850 --> 00:39:10,030 The third bullet says most and and through it. 296 00:39:10,030 --> 00:39:16,270 So if I can take W as above, so it W is my fixed and three z, 297 00:39:16,270 --> 00:39:23,110 I want to show that any end through to Z must be a very tough one time W that would do it. 298 00:39:23,110 --> 00:39:38,620 So if I let you be an end through to Z, then my secret aim, if you like, is to show that you must be w times some root and threats of unity. 299 00:39:38,620 --> 00:39:42,310 Well, we know that W to the end is that. 300 00:39:42,310 --> 00:39:47,230 And that's also due to the N. That wasn't very good and we try again. 301 00:39:47,230 --> 00:39:53,770 And that is not zero. So W not zero. 302 00:39:53,770 --> 00:40:00,010 So I can do you divide it by W two power n? And that's not one, right? 303 00:40:00,010 --> 00:40:12,400 Having got to use the N and W to the end in the same, if I just rearrange this says that you over W is an m three to one. 304 00:40:12,400 --> 00:40:18,950 And by corollary eight, there are. 305 00:40:18,950 --> 00:40:32,150 And of these, so there are mice and choices of you. 306 00:40:32,150 --> 00:40:37,100 So that completes the proof. So it's a little bit fiddly and you might want to kind of think about that in your own time, 307 00:40:37,100 --> 00:40:43,570 but hopefully the structure at least makes some kind of sense. 308 00:40:43,570 --> 00:40:48,310 So that was thinking about routes of particular complex number, 309 00:40:48,310 --> 00:40:55,180 and that means that we've sort of thought about solutions to the polynomial equation W to the N equals Z, 310 00:40:55,180 --> 00:40:59,860 we've just counted solutions to that equation. We'd like to know about solutions polynomials more generally. 311 00:40:59,860 --> 00:41:04,720 I said on Monday we could think about solutions quadratic, so we we'd like to be able to generalise. 312 00:41:04,720 --> 00:41:10,840 And that's what we're going to do now. So the first thing is to show that if you've got a polynomial with complex coefficients, 313 00:41:10,840 --> 00:41:15,670 a complex polynomial of degree n that you can't have too many roots. 314 00:41:15,670 --> 00:41:27,570 So this is Proposition 11. And this says. 315 00:41:27,570 --> 00:41:47,030 A complex. Polynomial, meaning a polynomial with complex coefficients of degree n has most and root. 316 00:41:47,030 --> 00:41:52,400 I mean, maybe it has no roots, maybe it has some roots. I'm not making any claims about that in this proposition. 317 00:41:52,400 --> 00:42:02,660 All I'm saying is there are most and complex rules. So I mean and roots in C, so let's prove this on. 318 00:42:02,660 --> 00:42:10,040 My strategy is going to be to use induction on RN. 319 00:42:10,040 --> 00:42:23,000 The degree of the polynomial so and is one, for example, is a base case. 320 00:42:23,000 --> 00:42:31,190 That's a quick check. I'll let you make sure that you're happy to see that a linear polynomial with complex coefficients has almost most one root. 321 00:42:31,190 --> 00:42:41,840 I feel confident that you can handle that. So let's do the inductive step. 322 00:42:41,840 --> 00:42:57,550 So I'm going to let P be a complex polynomial with degree. 323 00:42:57,550 --> 00:43:08,290 And I'm going to suppose the results hold. 324 00:43:08,290 --> 00:43:18,970 For polynomials with degree less than or equal to and minus one. 325 00:43:18,970 --> 00:43:28,590 That's our inductive hypothesis. So I want to show that the number of roots of PE can't be too big. 326 00:43:28,590 --> 00:43:36,090 Dividing through by the leading coefficient and the coefficient of Z to the N, 327 00:43:36,090 --> 00:43:51,980 if you like by the leading coefficient X two, that maybe doesn't change the rates. 328 00:43:51,980 --> 00:44:01,320 And what that means is that we may assume. 329 00:44:01,320 --> 00:44:15,960 The P is a monarch polynomial monarch means it has leading coefficient one, so p looks like P of X is X to the N. 330 00:44:15,960 --> 00:44:40,730 It has leading coefficient one because its monarch. Plus a and minus one x minus one, plus the dot plus A1 x plus a zero for some A0, A1 dot dot dot. 331 00:44:40,730 --> 00:44:46,820 And minus one in C. I would like you to notice two things about this. 332 00:44:46,820 --> 00:44:50,630 One is this I very carefully introduced to zero up to and minus one. 333 00:44:50,630 --> 00:44:55,070 Or if you turn off as a party and you take your friend along and they don't know the people that you introduce, 334 00:44:55,070 --> 00:44:59,030 them writes Really not naughty because otherwise people can't have a kind of sensible conversation. 335 00:44:59,030 --> 00:45:01,490 We can't talk about these things without knowing what they are. 336 00:45:01,490 --> 00:45:07,280 Also, notice how I've carefully matched the coefficient with the power so that it's dead easy to do. 337 00:45:07,280 --> 00:45:17,180 I didn't want to start with A0 A1 here. Right? This is a top tip. This would make your life easier if he has no roots. 338 00:45:17,180 --> 00:45:50,270 Then we're done. If P has a roots, say Alpha in C, then P of X is equal to X minus alpha times f of X for some complex monarch polynomial f. 339 00:45:50,270 --> 00:46:00,490 With degree and minus one. If you haven't seen how to prove this or you're not. 340 00:46:00,490 --> 00:46:04,270 What's the point, you're not confident you could do yourself. I did put the details in the online notes. 341 00:46:04,270 --> 00:46:12,420 If you want to think about that, you can't. What this tells us is that any roots of PE is either alpha or at a rate of F. 342 00:46:12,420 --> 00:46:27,600 So by the inductive hypothesis. F has most and minus one roots. 343 00:46:27,600 --> 00:46:38,100 And any rate of p is alpha or a rate of F. 344 00:46:38,100 --> 00:46:42,900 So P has most. 345 00:46:42,900 --> 00:46:52,650 OK, i need to carry on over here. P has at most and root. 346 00:46:52,650 --> 00:46:56,010 And that completes the induction argument. 347 00:46:56,010 --> 00:47:03,120 So proving that a complex polynomial of degree and has at and roots is not too difficult, I mean, I've gone through that fairly swiftly. 348 00:47:03,120 --> 00:47:09,060 You've got to want to look at that in your time. But it's very doable. What we'd really like to know is, does it have any roots? 349 00:47:09,060 --> 00:47:12,240 Does it always have at least one? Does it always have any roots? 350 00:47:12,240 --> 00:47:22,620 And the answer is given by Theorem 12, which is super important, and it's called the fundamental theorem of algebra. 351 00:47:22,620 --> 00:47:29,520 You should draw some conclusions about the importance of this theorem from the fact that it's called the fundamental theorem of algebra. 352 00:47:29,520 --> 00:47:42,420 And this says any complex polynomial of degree n. 353 00:47:42,420 --> 00:47:53,430 Has exactly and returns counted with multiplicity, so repeated routes, 354 00:47:53,430 --> 00:48:00,900 you have to count the appropriate number of times which when you sort of think about it, is a relatively natural thing to do. 355 00:48:00,900 --> 00:48:16,680 So that is. If he is well, we may as well focus on a monarch complex polynomial. 356 00:48:16,680 --> 00:48:23,830 With dignity and. 357 00:48:23,830 --> 00:48:49,450 Then we can fix rise p of X is x minus alpha one, two x minus alpha n four, some alpha, one two alpha and in C, they don't have to be distinct. 358 00:48:49,450 --> 00:48:54,640 This theorem takes quite a bit more work to prove that Proposition 11. 359 00:48:54,640 --> 00:49:03,790 It's not in this course, you're going to want some extra ideas, so maybe some ideas from complex analysis or topology. 360 00:49:03,790 --> 00:49:09,430 So you will see proofs of this theorem in the next couple of years that you can be looking out for. 361 00:49:09,430 --> 00:49:14,560 For now, you may assume this theorem, but only if you really, really have to write. 362 00:49:14,560 --> 00:49:20,740 If you can avoid this theorem, you want to avoid it because you haven't seen a proof. And that means you're sort of not morally on top of this there. 363 00:49:20,740 --> 00:49:28,120 So if you possibly can avoid using this theorem but know that it's there if you need it, the history of this theorem is really interesting, 364 00:49:28,120 --> 00:49:33,640 and I put some links to where you can read a bit of an introduction to that on the online notes. 365 00:49:33,640 --> 00:49:36,610 That is the end of the complex numbers course on Monday. 366 00:49:36,610 --> 00:49:42,490 Excitingly, we will be here for linear algebra if you want to leave your questions at the top on that side or that side. 367 00:49:42,490 --> 00:50:04,012 I will collect them up. Otherwise, please return to academic opinion. See you on Monday.