1 00:00:00,090 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 00:00:10,690 continue to offer high quality educational resources for free. 5 00:00:10,690 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,270 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,270 --> 00:00:18,200 at ocw.mit.edu. 8 00:00:26,600 --> 00:00:27,350 HERBERT GROSS: Hi. 9 00:00:27,350 --> 00:00:29,970 Standing here waiting for today's lesson to begin, 10 00:00:29,970 --> 00:00:33,020 I was thinking of a story that came to mind. 11 00:00:33,020 --> 00:00:36,110 And that was the story of the foreman yelling down 12 00:00:36,110 --> 00:00:39,410 into an excavation, how many of you men down there? 13 00:00:39,410 --> 00:00:41,750 And the reply came back, three. 14 00:00:41,750 --> 00:00:45,110 And he said, OK, half of you come on up. 15 00:00:45,110 --> 00:00:47,780 Now with that funny story, I would 16 00:00:47,780 --> 00:00:52,750 like to launch today's lesson, which could be, I guess, 17 00:00:52,750 --> 00:00:54,860 sub called, or whatever word that you'd 18 00:00:54,860 --> 00:00:58,490 like to pick, realness is in the eyes of the beholder, 19 00:00:58,490 --> 00:01:02,600 the question being, of course, that when you talk about taking 20 00:01:02,600 --> 00:01:06,950 one half of three people, ah-- 21 00:01:06,950 --> 00:01:08,440 well, let's put it this way. 22 00:01:08,440 --> 00:01:09,610 I was going to say, you can't do it. 23 00:01:09,610 --> 00:01:11,402 Let's say, if you could do it, you wouldn't 24 00:01:11,402 --> 00:01:13,040 like to see the answer. 25 00:01:13,040 --> 00:01:16,150 On the other hand, to take three inches and say, 26 00:01:16,150 --> 00:01:18,400 let's divide that into two equal parts, 27 00:01:18,400 --> 00:01:22,510 there's a case where the answer does happen to make sense. 28 00:01:22,510 --> 00:01:24,340 Now today, you see we're going to talk 29 00:01:24,340 --> 00:01:27,970 about a new phase of our course called the complex numbers. 30 00:01:27,970 --> 00:01:31,300 The complex numbers happen to be a delightful topic, 31 00:01:31,300 --> 00:01:33,490 from the point of view that on the one hand, 32 00:01:33,490 --> 00:01:37,090 they offer a great deal of enrichment in pure mathematics, 33 00:01:37,090 --> 00:01:40,090 and on the other hand, they contribute a great deal 34 00:01:40,090 --> 00:01:43,100 to our physical understanding of reality. 35 00:01:43,100 --> 00:01:47,020 Now to launch into this, let's get right into the topic. 36 00:01:47,020 --> 00:01:50,530 I call today's lesson, The Complex Numbers. 37 00:01:50,530 --> 00:01:54,010 And going back to my opening hilarious story, 38 00:01:54,010 --> 00:01:56,620 what we're saying is if the only numbers that we knew 39 00:01:56,620 --> 00:01:59,560 were the integers and we were given the equation 40 00:01:59,560 --> 00:02:01,180 2x equals 3-- 41 00:02:01,180 --> 00:02:02,860 and notice that in terms of integers, 42 00:02:02,860 --> 00:02:05,800 the equation makes sense, because 2 and 3 happen 43 00:02:05,800 --> 00:02:09,490 to be integers-- we're looking for to solve this equation. 44 00:02:09,490 --> 00:02:13,090 And the question is, does this equation have a solution? 45 00:02:13,090 --> 00:02:16,600 The answer is it only has a solution provided 46 00:02:16,600 --> 00:02:20,380 that you want to invent numbers which are not integers. 47 00:02:20,380 --> 00:02:22,780 In other words, this does not have a solution 48 00:02:22,780 --> 00:02:25,280 if we insist that the answer be an integer. 49 00:02:25,280 --> 00:02:28,900 But if we're willing to invent a new batch of numbers called 50 00:02:28,900 --> 00:02:31,060 the fractions or the rational numbers, 51 00:02:31,060 --> 00:02:34,000 then this equation will have a solution. 52 00:02:34,000 --> 00:02:35,830 Do we want this to have solutions? 53 00:02:35,830 --> 00:02:38,980 Well, sometimes there are meaningful situations in which 54 00:02:38,980 --> 00:02:40,990 it's meaningful to solve this equation, 55 00:02:40,990 --> 00:02:43,210 other times when it isn't. 56 00:02:43,210 --> 00:02:45,100 Again, the story of going into the bank 57 00:02:45,100 --> 00:02:48,790 and into the post office and asking for $0.03 worth of $0.02 58 00:02:48,790 --> 00:02:49,690 stamps. 59 00:02:49,690 --> 00:02:53,170 And we can think of all sorts of hilarious ways of embellishing 60 00:02:53,170 --> 00:02:53,980 this story. 61 00:02:53,980 --> 00:02:58,060 Going on further, though, let's take what ultimately became 62 00:02:58,060 --> 00:02:59,760 known as the "Real Numbers." 63 00:02:59,760 --> 00:03:02,500 And I'm going to put this in quotation marks. 64 00:03:02,500 --> 00:03:04,480 Because even though these are genuinely 65 00:03:04,480 --> 00:03:09,880 called the Real Numbers, they are no more real 66 00:03:09,880 --> 00:03:12,250 than the integers were real. 67 00:03:12,250 --> 00:03:15,390 By definition, a Real Number is simply any number 68 00:03:15,390 --> 00:03:18,240 whose square is non negative. 69 00:03:18,240 --> 00:03:20,820 Assuming that to be the definition of real numbers, 70 00:03:20,820 --> 00:03:24,060 we come to the equation x squared equals minus 1. 71 00:03:24,060 --> 00:03:27,430 We say, does this equation have a solution? 72 00:03:27,430 --> 00:03:29,630 And the answer is if we insist that the solution has 73 00:03:29,630 --> 00:03:32,730 to be a real number, the answer, obviously, 74 00:03:32,730 --> 00:03:34,650 is that this does not have a solution. 75 00:03:34,650 --> 00:03:37,300 Because the definition of a real number is that it's square 76 00:03:37,300 --> 00:03:41,430 cannot be negative, and this says that x squared is minus 1. 77 00:03:41,430 --> 00:03:45,990 So either we must say that this equation does not 78 00:03:45,990 --> 00:03:47,280 have solutions. 79 00:03:47,280 --> 00:03:49,470 Or if we want it to have solutions, 80 00:03:49,470 --> 00:03:53,160 we must invent an extension of the number system. 81 00:03:53,160 --> 00:03:56,310 We must extend the real numbers in the same way 82 00:03:56,310 --> 00:03:58,710 that if we wanted this equation to have a solution, 83 00:03:58,710 --> 00:04:00,840 we had to extend the integers. 84 00:04:00,840 --> 00:04:02,340 The extension of the integers that 85 00:04:02,340 --> 00:04:04,500 was necessary to solve this equation 86 00:04:04,500 --> 00:04:06,790 happened to be called the Rational Numbers. 87 00:04:06,790 --> 00:04:08,880 Let's talk about, first of all, whether we 88 00:04:08,880 --> 00:04:11,070 want an extension of the real numbers, 89 00:04:11,070 --> 00:04:14,790 and secondly, if we do, what shall we call it, 90 00:04:14,790 --> 00:04:16,680 and what properties shall it have? 91 00:04:16,680 --> 00:04:19,470 To motivate why we would like the equation, 92 00:04:19,470 --> 00:04:21,777 x squared plus 1 equals 0 to have solutions. 93 00:04:21,777 --> 00:04:23,610 And that, of course, is just equivalent to x 94 00:04:23,610 --> 00:04:25,230 squared equals minus 1. 95 00:04:25,230 --> 00:04:27,240 Let's go back to an example that we 96 00:04:27,240 --> 00:04:29,850 used when we first introduced exponents 97 00:04:29,850 --> 00:04:31,890 in part 1 of our course. 98 00:04:31,890 --> 00:04:34,440 We observed that one of the properties 99 00:04:34,440 --> 00:04:38,700 of an equation like y equals e to the rx 100 00:04:38,700 --> 00:04:41,610 where r is a constant is that if we differentiate this 101 00:04:41,610 --> 00:04:44,500 with respect to x, the r comes down. 102 00:04:44,500 --> 00:04:47,370 But the fact, the e to the rx remains. 103 00:04:47,370 --> 00:04:50,730 What this told us was that in taking derivatives, 104 00:04:50,730 --> 00:04:52,830 we would get powers of r. 105 00:04:52,830 --> 00:04:54,730 But e to the rx would remain. 106 00:04:54,730 --> 00:04:57,150 And we use that technique for solving 107 00:04:57,150 --> 00:05:01,050 certain types of second, well, differential equations. 108 00:05:01,050 --> 00:05:03,780 For example, looking at the equation y double prime plus y 109 00:05:03,780 --> 00:05:09,210 equals 0, one technique would be to replace y by e to the rx. 110 00:05:09,210 --> 00:05:12,630 When I differentiate twice, I have an r squared e to the rx 111 00:05:12,630 --> 00:05:13,390 here. 112 00:05:13,390 --> 00:05:14,990 Here is an e to the rx. 113 00:05:14,990 --> 00:05:16,980 The e to the rx cancels out. 114 00:05:16,980 --> 00:05:20,100 And I wind up with a polynomial equation for r. 115 00:05:20,100 --> 00:05:22,440 And we then solve for the values of r, such 116 00:05:22,440 --> 00:05:25,080 that this would be a solution of this. 117 00:05:25,080 --> 00:05:28,860 Now back in part 1, because we didn't know non-real numbers. 118 00:05:28,860 --> 00:05:31,920 We always fixed it up so that the resulting equation 119 00:05:31,920 --> 00:05:35,910 that we got here would have real numbers for values of r. 120 00:05:35,910 --> 00:05:38,790 By the way, I have chosen this particular equation 121 00:05:38,790 --> 00:05:40,570 for two reasons. 122 00:05:40,570 --> 00:05:44,520 One is, the values of r can't be real when I get through here. 123 00:05:44,520 --> 00:05:47,350 The other is, I already know an answer to this problem. 124 00:05:47,350 --> 00:05:49,800 Remember that if you differentiate sine twice, 125 00:05:49,800 --> 00:05:51,150 you get minus sine. 126 00:05:51,150 --> 00:05:55,440 If you differentiate cosine twice, you get minus cosine. 127 00:05:55,440 --> 00:05:57,450 Consequently, the second derivative 128 00:05:57,450 --> 00:06:00,180 of sine x plus sine x is 0. 129 00:06:00,180 --> 00:06:04,320 Second derivative of cosine x plus cosine x is 0. 130 00:06:04,320 --> 00:06:07,670 So I know at least two solutions to this equation, 131 00:06:07,670 --> 00:06:10,090 two real solutions to this equation, 132 00:06:10,090 --> 00:06:13,470 this equation has physical meaning because it essentially 133 00:06:13,470 --> 00:06:17,190 says, if you think of the parameter as being time here, 134 00:06:17,190 --> 00:06:20,010 look at the second derivative is equal to minus 135 00:06:20,010 --> 00:06:21,360 the function itself. 136 00:06:21,360 --> 00:06:23,640 That says that the acceleration is 137 00:06:23,640 --> 00:06:26,310 equal to minus the displacement. 138 00:06:26,310 --> 00:06:28,730 And that's a form of simple harmonic motion. 139 00:06:28,730 --> 00:06:30,480 You see, what I'm trying to establish here 140 00:06:30,480 --> 00:06:32,920 is that this does have a real solution. 141 00:06:32,920 --> 00:06:36,540 And the problem itself has real physical significance. 142 00:06:36,540 --> 00:06:40,020 At any rate, if we now try to solve this problem using 143 00:06:40,020 --> 00:06:42,900 this technique, and remember, again, 144 00:06:42,900 --> 00:06:45,360 I can give you problems where we won't 145 00:06:45,360 --> 00:06:48,100 know the answer intuitively, and the values of r 146 00:06:48,100 --> 00:06:49,600 will still come out to be imaginary. 147 00:06:49,600 --> 00:06:51,690 I simply chose this problem so that we 148 00:06:51,690 --> 00:06:54,810 could correlate a so-called imaginary answer 149 00:06:54,810 --> 00:06:56,670 with a real answer. 150 00:06:56,670 --> 00:07:00,230 The idea, again, is I differentiate 151 00:07:00,230 --> 00:07:01,400 each of the rx twice. 152 00:07:01,400 --> 00:07:03,090 I get this. 153 00:07:03,090 --> 00:07:05,580 So y double prime plus y is this. 154 00:07:05,580 --> 00:07:08,700 I factor out the e to the rx, which can't be 0. 155 00:07:08,700 --> 00:07:11,970 Therefore, it must be that r squared plus 1 is 0, 156 00:07:11,970 --> 00:07:16,177 or r is equal to plus or minus the square root of minus 1. 157 00:07:16,177 --> 00:07:18,510 Now you see, so far, I don't know what this thing means. 158 00:07:18,510 --> 00:07:20,550 Because there certainly is no real number, 159 00:07:20,550 --> 00:07:21,960 which has this property. 160 00:07:21,960 --> 00:07:24,390 Because the square of real numbers have to be positive. 161 00:07:24,390 --> 00:07:25,590 They can't be minus 1. 162 00:07:25,590 --> 00:07:27,420 Let me, for the sake of argument, 163 00:07:27,420 --> 00:07:31,110 invent a number, which is equal to the square root of minus 1. 164 00:07:31,110 --> 00:07:32,760 I'll call that number i. 165 00:07:32,760 --> 00:07:35,750 So r is equal to plus or minus i. 166 00:07:35,750 --> 00:07:38,520 And if I now remember what equation I'm solving-- 167 00:07:38,520 --> 00:07:40,830 it was y equals e to the rx-- 168 00:07:40,830 --> 00:07:44,250 I can replace r by either i or minus i. 169 00:07:44,250 --> 00:07:48,540 So the two solutions I would get would be y equals e to the ix 170 00:07:48,540 --> 00:07:50,700 or y equals e to the minus x. 171 00:07:50,700 --> 00:07:52,500 But this is bothering me. 172 00:07:52,500 --> 00:07:55,710 Because I really have no real feeling for what 173 00:07:55,710 --> 00:07:57,870 I means right now. 174 00:07:57,870 --> 00:08:02,760 What I do know, however, is this, that mechanically, 175 00:08:02,760 --> 00:08:06,050 this is a solution to a real problem. 176 00:08:06,050 --> 00:08:08,130 y double prime plus y equals zero 177 00:08:08,130 --> 00:08:10,680 is a real problem, not an imaginary one. 178 00:08:10,680 --> 00:08:12,960 And by the way, notice mechanically here, 179 00:08:12,960 --> 00:08:15,330 taking i to be a constant, if we still 180 00:08:15,330 --> 00:08:21,480 assume that the derivative of e to the x is re to the rx, 181 00:08:21,480 --> 00:08:23,910 even when r happens to be non-real, 182 00:08:23,910 --> 00:08:26,710 notice that if I differentiate this thing twice, 183 00:08:26,710 --> 00:08:28,700 the first time I differentiate, I bring down 184 00:08:28,700 --> 00:08:31,350 an i, the second time I differentiate, I bring down 185 00:08:31,350 --> 00:08:32,010 an i. 186 00:08:32,010 --> 00:08:35,159 That gives me i squared, which is minus 1. 187 00:08:35,159 --> 00:08:37,830 This is going to be minus e to the ix. 188 00:08:37,830 --> 00:08:41,460 And if I add on e to the ix, I really do get 0. 189 00:08:41,460 --> 00:08:46,200 Notice also, now, that I do know that a real solution 190 00:08:46,200 --> 00:08:50,670 of this equation is y equals cosine x, or also y equals sine 191 00:08:50,670 --> 00:08:51,390 x. 192 00:08:51,390 --> 00:08:54,570 So that somehow or other, I get the feeling 193 00:08:54,570 --> 00:09:00,840 that I should exist, and more to the point, that this expression 194 00:09:00,840 --> 00:09:06,190 e to the ix should somehow be related to sign x and cosine x. 195 00:09:06,190 --> 00:09:08,190 In other words, if I had never been lucky enough 196 00:09:08,190 --> 00:09:10,240 to invent the trigonometric functions, 197 00:09:10,240 --> 00:09:12,990 and I want to solve the problem, y double prime plus y 198 00:09:12,990 --> 00:09:16,710 equals 0 and wound up with this, then 199 00:09:16,710 --> 00:09:19,410 my feeling is, I must find a real interpretation, 200 00:09:19,410 --> 00:09:23,040 a physically real interpretation for each of the ix. 201 00:09:23,040 --> 00:09:25,860 Because I sense that my problem has a solution. 202 00:09:25,860 --> 00:09:27,930 Again, I can take the coward's way out. 203 00:09:27,930 --> 00:09:30,150 I can say, if it's going to be this much work, 204 00:09:30,150 --> 00:09:32,730 I don't want the problem to have a solution. 205 00:09:32,730 --> 00:09:34,830 But once I want the problem to have a solution, 206 00:09:34,830 --> 00:09:36,600 I must extend the number system. 207 00:09:36,600 --> 00:09:38,880 And the upside of that whole thing 208 00:09:38,880 --> 00:09:41,820 is that we now invent something called the Complex Number 209 00:09:41,820 --> 00:09:42,730 System. 210 00:09:42,730 --> 00:09:44,880 First of all, the complex numbers 211 00:09:44,880 --> 00:09:48,510 are defined to be the set of all symbols-- 212 00:09:48,510 --> 00:09:50,770 let's just call them Symbols for the time being-- 213 00:09:50,770 --> 00:09:55,020 all symbols of the form x plus iy, 214 00:09:55,020 --> 00:09:57,430 where x and y are real numbers. 215 00:09:57,430 --> 00:09:59,420 And just to refresh our memories here, 216 00:09:59,420 --> 00:10:02,280 i is the symbol which numerically represents 217 00:10:02,280 --> 00:10:05,490 the square root of minus 1, OK? 218 00:10:05,490 --> 00:10:08,330 Notice that the X and the y are real numbers, though, 219 00:10:08,330 --> 00:10:09,780 where by Real, we mean what? 220 00:10:09,780 --> 00:10:12,030 The squares are non negative. 221 00:10:12,030 --> 00:10:17,880 Now you will recall, that in any mathematical system, 222 00:10:17,880 --> 00:10:19,590 the thing that you're dealing with 223 00:10:19,590 --> 00:10:22,530 is more than a set of numbers, it's a set of numbers 224 00:10:22,530 --> 00:10:27,630 or a set of objects together with a structure, certain rules 225 00:10:27,630 --> 00:10:31,230 that tell us how to work with these elements that make up 226 00:10:31,230 --> 00:10:32,350 the set. 227 00:10:32,350 --> 00:10:35,130 Remember, the difference between a set and a system 228 00:10:35,130 --> 00:10:37,560 is a set that's just a collection of objects. 229 00:10:37,560 --> 00:10:40,050 The system is the collection of objects 230 00:10:40,050 --> 00:10:43,500 together with how we combine these objects to form 231 00:10:43,500 --> 00:10:45,330 similar objects, et cetera. 232 00:10:45,330 --> 00:10:48,580 And what is the structure in this case? 233 00:10:48,580 --> 00:10:54,930 Given the two complex numbers, x1 plus iy1 and x2 plus iy2, 234 00:10:54,930 --> 00:10:59,430 we define equality to be sort of component by component. 235 00:10:59,430 --> 00:11:02,220 The real parts x1 and x2 must be equal. 236 00:11:02,220 --> 00:11:05,310 And the imaginary parts-- and by the way, just a word of caution 237 00:11:05,310 --> 00:11:09,090 here, the imaginary part of a complex number 238 00:11:09,090 --> 00:11:11,520 is, by definition, the coefficient of i. 239 00:11:11,520 --> 00:11:15,090 Notice that the imaginary part is itself a real number. 240 00:11:15,090 --> 00:11:18,630 The coefficient of i is the real number y1. 241 00:11:18,630 --> 00:11:22,230 What we're saying is we define the definition of equality 242 00:11:22,230 --> 00:11:25,290 for complex numbers to be that two complex numbers are 243 00:11:25,290 --> 00:11:28,530 equal if and only if their real parts are equal 244 00:11:28,530 --> 00:11:30,720 and their imaginary parts are equal. 245 00:11:30,720 --> 00:11:35,850 Secondly, we agree to add two complex numbers, component 246 00:11:35,850 --> 00:11:37,720 by component, so to speak. 247 00:11:37,720 --> 00:11:40,170 In other words, to add two complex numbers, 248 00:11:40,170 --> 00:11:44,790 we add the two real parts, and we add the two imaginary parts, 249 00:11:44,790 --> 00:11:45,600 OK? 250 00:11:45,600 --> 00:11:47,190 That's all this says. 251 00:11:47,190 --> 00:11:50,490 And thirdly, to multiply a complex number 252 00:11:50,490 --> 00:11:54,330 by a real number, we multiply the real part 253 00:11:54,330 --> 00:11:57,090 by the given real number and the imaginary part 254 00:11:57,090 --> 00:12:00,630 by the given real number, which is what this third rule says. 255 00:12:00,630 --> 00:12:02,430 And all I want to see from this is 256 00:12:02,430 --> 00:12:04,920 that by the very definition of these three 257 00:12:04,920 --> 00:12:08,640 structural properties, we have made the complex numbers 258 00:12:08,640 --> 00:12:10,690 a two-dimensional vector space. 259 00:12:10,690 --> 00:12:12,360 In other words, it means that we can now 260 00:12:12,360 --> 00:12:16,470 visualize the complex numbers, geometrically, 261 00:12:16,470 --> 00:12:18,240 in the same way that we could visualize 262 00:12:18,240 --> 00:12:20,100 the real numbers geometrically. 263 00:12:20,100 --> 00:12:23,220 Remember, geometrically-- when we define the real numbers 264 00:12:23,220 --> 00:12:25,830 to be those numbers whose squares are non negatives, 265 00:12:25,830 --> 00:12:30,210 we do not need any picture to visualize that. 266 00:12:30,210 --> 00:12:32,730 It just happens that the number line, the x-axis, 267 00:12:32,730 --> 00:12:35,220 is a very convenient way of visualizing 268 00:12:35,220 --> 00:12:36,840 real numbers pictorially. 269 00:12:36,840 --> 00:12:40,290 What we're saying is that since complex numbers seem 270 00:12:40,290 --> 00:12:43,440 to indicate a two-dimensional vector space, namely, 271 00:12:43,440 --> 00:12:46,680 real and imaginary parts which are independent, 272 00:12:46,680 --> 00:12:49,950 it would appear that the analog for visualizing 273 00:12:49,950 --> 00:12:53,640 complex numbers pictorially would be to use the plane. 274 00:12:53,640 --> 00:12:55,560 And that's exactly what we do. 275 00:12:55,560 --> 00:13:00,280 The diagram is called the Argand Diagram. 276 00:13:00,280 --> 00:13:01,900 The idea is this. 277 00:13:01,900 --> 00:13:06,550 Given the complex number x plus iy, which we'll call z, 278 00:13:06,550 --> 00:13:09,400 we visualize z as either the point 279 00:13:09,400 --> 00:13:12,980 x comma y in the xy plane-- in other words, 280 00:13:12,980 --> 00:13:15,830 notice that the x-axis is the real axis, 281 00:13:15,830 --> 00:13:19,270 meaning it denotes the real part of the complex number. 282 00:13:19,270 --> 00:13:22,150 The y-axis is called the imaginary axis, 283 00:13:22,150 --> 00:13:25,900 because it denotes the coefficient of i, 284 00:13:25,900 --> 00:13:29,460 the imaginary part, OK? 285 00:13:29,460 --> 00:13:34,450 And what we're saying is we can visualize the complex number x 286 00:13:34,450 --> 00:13:37,240 plus iy to be the point x comma y. 287 00:13:37,240 --> 00:13:39,130 Or for that matter, the vector that 288 00:13:39,130 --> 00:13:41,260 goes from the origin to the point 289 00:13:41,260 --> 00:13:46,270 z, whose components are x and y, meaning what? x plus iy. 290 00:13:46,270 --> 00:13:49,450 The components are the real and the imaginary parts here. 291 00:13:49,450 --> 00:13:52,930 We can translate this into polar coordinates, 292 00:13:52,930 --> 00:13:55,870 meaning that we can measure the point z by its r 293 00:13:55,870 --> 00:13:57,610 value and its theta value. 294 00:13:57,610 --> 00:13:59,770 But one important thing to remember 295 00:13:59,770 --> 00:14:02,920 is that in polar coordinates, it's always 296 00:14:02,920 --> 00:14:05,360 assumed that r is positive. 297 00:14:05,360 --> 00:14:08,080 Unlike the usual polar coordinates, 298 00:14:08,080 --> 00:14:10,780 where r could be either positive or negative, 299 00:14:10,780 --> 00:14:14,320 the idea is that we would like to identify 300 00:14:14,320 --> 00:14:17,380 the absolute value of a complex number 301 00:14:17,380 --> 00:14:19,570 with being its distance from the origin. 302 00:14:19,570 --> 00:14:22,390 Since we want absolute value to be non-negative, 303 00:14:22,390 --> 00:14:25,300 we simply say that r is the positive square root 304 00:14:25,300 --> 00:14:26,930 of x squared plus y squared. 305 00:14:26,930 --> 00:14:29,890 And that names the magnitude of the complex number. 306 00:14:29,890 --> 00:14:35,590 The angle theta is called the Argument, abbreviated Arg, 307 00:14:35,590 --> 00:14:39,340 not to be confused with this funny coincidence of Argand 308 00:14:39,340 --> 00:14:40,180 diagram. 309 00:14:40,180 --> 00:14:45,310 This comes from the word Argument, argument. 310 00:14:45,310 --> 00:14:48,340 The angle is called the Argument, all right? 311 00:14:48,340 --> 00:14:51,460 And if you elect to write the complex number 312 00:14:51,460 --> 00:14:54,880 x plus iy as r comma theta, because 313 00:14:54,880 --> 00:14:59,470 of the connection between polar and Cartesian coordinates, r-- 314 00:14:59,470 --> 00:15:02,070 see, x is what? x is r cosine theta. 315 00:15:02,070 --> 00:15:03,830 y is r sine theta. 316 00:15:03,830 --> 00:15:06,610 So the complex number is r cosine theta 317 00:15:06,610 --> 00:15:08,230 plus ir sine theta. 318 00:15:08,230 --> 00:15:10,510 That's what r comma theta denotes. 319 00:15:10,510 --> 00:15:14,230 Now using this as background-- so we now have what? 320 00:15:14,230 --> 00:15:17,590 We've invented, at least abstractly, a system of numbers 321 00:15:17,590 --> 00:15:19,400 called the Complex Numbers. 322 00:15:19,400 --> 00:15:22,990 And by the way, let me point out here that don't use the word 323 00:15:22,990 --> 00:15:24,640 Imaginary too strongly. 324 00:15:24,640 --> 00:15:26,710 There is certainly nothing imaginary 325 00:15:26,710 --> 00:15:29,140 about this geometric interpretation. 326 00:15:29,140 --> 00:15:31,150 What may be imaginary is I may not 327 00:15:31,150 --> 00:15:33,730 have a place where I want to use these kind of numbers. 328 00:15:33,730 --> 00:15:36,550 But believe me, we wouldn't have made a whole block of material 329 00:15:36,550 --> 00:15:38,530 about these numbers if there weren't 330 00:15:38,530 --> 00:15:42,080 real interpretations of the so-called complex numbers. 331 00:15:42,080 --> 00:15:45,260 This is a perfectly real geometric interpretation. 332 00:15:45,260 --> 00:15:47,320 So far, we have established the fact 333 00:15:47,320 --> 00:15:51,190 that this interpretation gives us a vector space. 334 00:15:51,190 --> 00:15:55,180 But the complex numbers have an additional structure as well, 335 00:15:55,180 --> 00:16:01,350 namely, if I saw the two complex numbers, a plus bi times-- 336 00:16:01,350 --> 00:16:04,850 and c plus di, and I multiplied them-- remember now, a, b, c, 337 00:16:04,850 --> 00:16:06,660 and d are real-- 338 00:16:06,660 --> 00:16:10,350 I would like to believe that the rules of ordinary algebra 339 00:16:10,350 --> 00:16:12,300 still apply here. 340 00:16:12,300 --> 00:16:15,150 To multiply this, I would want this times this, 341 00:16:15,150 --> 00:16:19,650 this times this, this times this, and this times this. 342 00:16:19,650 --> 00:16:22,290 Notice that a times c as ac. 343 00:16:22,290 --> 00:16:26,790 bi times di should be bd times i squared, 344 00:16:26,790 --> 00:16:29,280 if the ordinary rules of arithmetic are to apply. 345 00:16:29,280 --> 00:16:34,560 Since i squared is minus 1, this term becomes minus bd. 346 00:16:34,560 --> 00:16:37,380 Since a, c, b, and d are real numbers, 347 00:16:37,380 --> 00:16:41,310 this is a real number, right? 348 00:16:41,310 --> 00:16:44,310 Correspondingly, the coefficient of-- the multiplier 349 00:16:44,310 --> 00:16:47,640 of i, the coefficient of i is what? bc plus ad, 350 00:16:47,640 --> 00:16:49,810 which is also a real number. 351 00:16:49,810 --> 00:16:53,010 So if we make up this definition of multiplication 352 00:16:53,010 --> 00:16:55,680 for complex numbers, we have made sure 353 00:16:55,680 --> 00:16:58,500 that the rules of arithmetic for complex numbers 354 00:16:58,500 --> 00:17:01,932 parallel the rules for real numbers. 355 00:17:01,932 --> 00:17:03,390 And by the way, it's very important 356 00:17:03,390 --> 00:17:07,530 here to make the aside that this is a crucial result. 357 00:17:07,530 --> 00:17:10,470 Notice that among the complex numbers, 358 00:17:10,470 --> 00:17:12,839 the real numbers are included, namely, 359 00:17:12,839 --> 00:17:16,319 thinking of it geometrically, if you look at the plane, 360 00:17:16,319 --> 00:17:17,819 the real numbers are those points 361 00:17:17,819 --> 00:17:20,790 in the plane which lie along the x-axis. 362 00:17:20,790 --> 00:17:23,520 It's the same as saying that when you extended the integers 363 00:17:23,520 --> 00:17:27,060 to form the rational numbers, the rational numbers included 364 00:17:27,060 --> 00:17:28,260 the integers. 365 00:17:28,260 --> 00:17:30,150 In other words, if I look at the number 3, 366 00:17:30,150 --> 00:17:32,160 I can think of it as the integer 3. 367 00:17:32,160 --> 00:17:36,570 I could also think of it as the ratio 3 divided by 1, you see? 368 00:17:36,570 --> 00:17:38,580 And therefore, I would like whatever 369 00:17:38,580 --> 00:17:41,070 rules I have for the complex numbers 370 00:17:41,070 --> 00:17:45,370 to remain valid if the complex numbers happen to be real. 371 00:17:45,370 --> 00:17:47,370 It would be terrible to have two different rules 372 00:17:47,370 --> 00:17:49,770 for multiplication, one for real numbers 373 00:17:49,770 --> 00:17:52,500 and one for complex numbers, and then 374 00:17:52,500 --> 00:17:54,900 given two real numbers, when I multiply them, 375 00:17:54,900 --> 00:17:57,610 I get one answer if I think of them as being real, 376 00:17:57,610 --> 00:18:01,080 and another answer if I think of them as being complex. 377 00:18:01,080 --> 00:18:04,230 At any rate, the important point is with this as motivation, 378 00:18:04,230 --> 00:18:08,730 I have a way of defining the product of two complex numbers 379 00:18:08,730 --> 00:18:10,208 to be a complex number. 380 00:18:10,208 --> 00:18:12,750 I don't mean that these things are equal to r, so they simply 381 00:18:12,750 --> 00:18:13,840 stand for Real. 382 00:18:13,840 --> 00:18:15,990 This is real plus i times a real, 383 00:18:15,990 --> 00:18:17,700 which is a complex number. 384 00:18:17,700 --> 00:18:21,800 A very special case occurs when you take the complex number 385 00:18:21,800 --> 00:18:27,210 a plus bi and multiply it by a minus bi, the same number only 386 00:18:27,210 --> 00:18:29,610 changing the plus to a minus. 387 00:18:29,610 --> 00:18:31,170 Notice that we would get what? 388 00:18:31,170 --> 00:18:33,605 The sum and difference here gives us the sum 389 00:18:33,605 --> 00:18:35,310 of difference of two squares. 390 00:18:35,310 --> 00:18:40,180 It's going to be what? a squared minus b squared i squared. 391 00:18:40,180 --> 00:18:44,520 But since i squared is minus 1, minus b squared i squared 392 00:18:44,520 --> 00:18:48,460 is minus b squared times minus 1, which is plus b squared. 393 00:18:48,460 --> 00:18:51,720 And notice, therefore, if you multiply a plus bi, which 394 00:18:51,720 --> 00:18:55,020 is a complex number, by this other complex number, a minus 395 00:18:55,020 --> 00:18:58,210 bi, you get a squared plus b squared, 396 00:18:58,210 --> 00:19:00,480 which is a non-negative real number. 397 00:19:00,480 --> 00:19:03,540 In particular, if you think back to the geometric interpretation 398 00:19:03,540 --> 00:19:09,930 of this, the point a plus bi in the plane is a comma b. 399 00:19:09,930 --> 00:19:12,630 And the distance of a comma b from the origin 400 00:19:12,630 --> 00:19:16,000 is the square root of a squared plus b squared. 401 00:19:16,000 --> 00:19:18,720 So this product is actually the square 402 00:19:18,720 --> 00:19:22,740 of the magnitude, the absolute value of a plus bi. 403 00:19:22,740 --> 00:19:25,620 This little gizmo is so important 404 00:19:25,620 --> 00:19:27,450 that it's given a special name. 405 00:19:27,450 --> 00:19:31,170 By definition, given any complex number z, 406 00:19:31,170 --> 00:19:35,640 written in the form x plus iy, the complex conjugate of z, 407 00:19:35,640 --> 00:19:37,080 called z bar-- 408 00:19:37,080 --> 00:19:40,350 not to be confused with a z with a bar underneath that we were 409 00:19:40,350 --> 00:19:43,110 using to denote vectors-- but the complex conjugate 410 00:19:43,110 --> 00:19:47,640 z bar is what you get by just changing the plus sign here 411 00:19:47,640 --> 00:19:48,720 to a minus. 412 00:19:48,720 --> 00:19:51,780 In other words, the complex conjugate of x plus iy 413 00:19:51,780 --> 00:19:53,490 is x minus iy. 414 00:19:53,490 --> 00:19:55,620 And geometrically, all this means 415 00:19:55,620 --> 00:20:00,090 is, you see, when you change the sign of the imaginary part, 416 00:20:00,090 --> 00:20:03,150 remember, the imaginary part is the y-axis. 417 00:20:03,150 --> 00:20:05,520 You're just reflecting the number symmetrically 418 00:20:05,520 --> 00:20:06,990 with respect to the x-axis. 419 00:20:06,990 --> 00:20:09,030 You're leaving the x-coordinate alone. 420 00:20:09,030 --> 00:20:12,750 And you're changing the y-coordinate to minus y. 421 00:20:12,750 --> 00:20:14,850 In polar coordinates, what you're saying 422 00:20:14,850 --> 00:20:17,490 is a complex number and its conjugate 423 00:20:17,490 --> 00:20:19,680 are the same distance from the origin. 424 00:20:19,680 --> 00:20:22,230 But in one case, the angle is theta. 425 00:20:22,230 --> 00:20:25,230 And in the other case, the angle is minus theta. 426 00:20:25,230 --> 00:20:28,860 One of the many applications of complex conjugates-- 427 00:20:28,860 --> 00:20:31,170 I'll give you some of these other applications 428 00:20:31,170 --> 00:20:32,100 in the exercises. 429 00:20:32,100 --> 00:20:34,200 But for now, what I think is important 430 00:20:34,200 --> 00:20:37,050 is a simple one that essentially shows us 431 00:20:37,050 --> 00:20:39,900 how we find the quotient of two complex numbers. 432 00:20:39,900 --> 00:20:42,180 Quite simply, sparing you the details, 433 00:20:42,180 --> 00:20:45,930 if I have c plus di divided by a plus bi, 434 00:20:45,930 --> 00:20:49,290 I simply multiply numerator and denominator 435 00:20:49,290 --> 00:20:53,370 by the complex conjugate of the denominator, OK? 436 00:20:53,370 --> 00:20:54,990 What will that do for me? 437 00:20:54,990 --> 00:20:59,220 Well, when I multiply the two factors in the numerator, 438 00:20:59,220 --> 00:21:04,070 I'm going to get a real part, namely, ac plus bd. 439 00:21:04,070 --> 00:21:06,420 In other words, it's minus bd i squared. 440 00:21:06,420 --> 00:21:07,880 But i squared is minus 1. 441 00:21:07,880 --> 00:21:10,130 So I'm going to get ac plus bd. 442 00:21:10,130 --> 00:21:15,140 I'm going to get an imaginary part, which is ad minus bc. 443 00:21:15,140 --> 00:21:19,110 But the denominator will just be a squared plus b squared, 444 00:21:19,110 --> 00:21:20,610 which is a real number. 445 00:21:20,610 --> 00:21:22,670 In other words, what this tells me is 446 00:21:22,670 --> 00:21:26,480 I can now write the quotient of two complex numbers in the form 447 00:21:26,480 --> 00:21:27,050 what-- 448 00:21:27,050 --> 00:21:30,740 a real number plus i times a real number. 449 00:21:30,740 --> 00:21:33,090 In fact, the only time I can't do this-- 450 00:21:33,090 --> 00:21:34,370 I have to be careful. 451 00:21:34,370 --> 00:21:37,790 a or b must be unequal to 0. 452 00:21:37,790 --> 00:21:40,460 Because if both and b are 0, this 453 00:21:40,460 --> 00:21:42,890 is a 0 denominator, which I don't allow myself 454 00:21:42,890 --> 00:21:43,970 to divide by. 455 00:21:43,970 --> 00:21:46,610 And the only way both a and b can be 0 456 00:21:46,610 --> 00:21:51,050 is if the complex number is 0 plus 0i, which is the number 0. 457 00:21:51,050 --> 00:21:53,250 So I'm saying, again, just like in the real case, 458 00:21:53,250 --> 00:21:55,280 I still can't divide by 0. 459 00:21:55,280 --> 00:21:57,740 Anyway, to give you a more practical illustration to see 460 00:21:57,740 --> 00:21:59,782 this with numbers-- not practical, but, at least, 461 00:21:59,782 --> 00:22:00,570 concrete-- 462 00:22:00,570 --> 00:22:05,720 3 plus 2i divided by 4 plus i, I simply multiply numerator 463 00:22:05,720 --> 00:22:08,450 and denominator by 4 minus i. 464 00:22:08,450 --> 00:22:10,730 You see, what that does is downstairs, 465 00:22:10,730 --> 00:22:15,200 I get 4 squared, which is 16, minus i squared, which 466 00:22:15,200 --> 00:22:18,830 is plus 1, 16 plus 1 is 17. 467 00:22:18,830 --> 00:22:20,480 The real part is going to be what? 468 00:22:20,480 --> 00:22:21,800 Here's 12. 469 00:22:21,800 --> 00:22:26,360 2i times minus i is plus 2 is 14. 470 00:22:26,360 --> 00:22:28,910 8i minus 3i is 5i. 471 00:22:28,910 --> 00:22:33,260 So this quotient, 3 plus 2i times 4 plus i, 472 00:22:33,260 --> 00:22:36,940 is 14/17 plus 5/17 i. 473 00:22:36,940 --> 00:22:39,590 And by the way, as a trivial check, and you 474 00:22:39,590 --> 00:22:41,630 can do this if you wish for an exercise, 475 00:22:41,630 --> 00:22:43,940 simply take the answer that we got here, 476 00:22:43,940 --> 00:22:46,940 multiply it by the denominator, 4 plus i 477 00:22:46,940 --> 00:22:49,640 here, and actually check that that product comes out 478 00:22:49,640 --> 00:22:51,590 to be 3 plus 2i. 479 00:22:51,590 --> 00:22:54,650 At any rate, this use of complex conjugates 480 00:22:54,650 --> 00:22:57,470 shows us that when you divide two complex numbers, 481 00:22:57,470 --> 00:22:59,630 the result will be a complex number, 482 00:22:59,630 --> 00:23:01,575 except for division by 0. 483 00:23:01,575 --> 00:23:03,950 You see, that's what went wrong with our integer problem, 484 00:23:03,950 --> 00:23:05,450 2x equals 3. 485 00:23:05,450 --> 00:23:08,450 The reason that we couldn't solve it is that to solve 2x 486 00:23:08,450 --> 00:23:12,350 equals 3 required that the process of taking 487 00:23:12,350 --> 00:23:15,650 the quotient of two integers, and the quotient of two 488 00:23:15,650 --> 00:23:17,758 integers did not have to be an integer. 489 00:23:17,758 --> 00:23:19,550 Notice on the other hand, that the quotient 490 00:23:19,550 --> 00:23:21,560 of two real numbers is a real number, 491 00:23:21,560 --> 00:23:23,510 as long as the denominator is not 0. 492 00:23:23,510 --> 00:23:25,280 And the quotient of two complex numbers 493 00:23:25,280 --> 00:23:29,360 is a complex number, except for division by 0. 494 00:23:29,360 --> 00:23:31,550 While we're talking about multiplication, 495 00:23:31,550 --> 00:23:33,620 a very enlightening thing happens 496 00:23:33,620 --> 00:23:37,160 if we think of multiplication in terms of polar coordinates, 497 00:23:37,160 --> 00:23:41,210 namely, take the two numbers, which in polar coordinates, 498 00:23:41,210 --> 00:23:45,350 are r1 comma theta 1 and r2 comma theta 2. 499 00:23:45,350 --> 00:23:48,480 Now again, remember from our Argand diagram 500 00:23:48,480 --> 00:23:50,900 what it means to say that the complex number has 501 00:23:50,900 --> 00:23:53,420 polar coordinates r1 comma theta 1. 502 00:23:53,420 --> 00:23:57,380 Remember, the complex number is written in the form x plus iy. 503 00:23:57,380 --> 00:24:00,470 The x-coordinate is r1 cosine theta 1. 504 00:24:00,470 --> 00:24:03,440 The y-coordinate is r1 sine theta 1. 505 00:24:03,440 --> 00:24:08,270 At any rate, this product, in terms of the standard notation, 506 00:24:08,270 --> 00:24:12,560 the x plus iy form, says we're multiplying these two numbers 507 00:24:12,560 --> 00:24:13,430 together. 508 00:24:13,430 --> 00:24:15,680 And using our rule for multiplication, 509 00:24:15,680 --> 00:24:18,470 multiplying term by term, and remembering 510 00:24:18,470 --> 00:24:21,890 that i squared is minus 1, observe that if we 511 00:24:21,890 --> 00:24:23,450 collect the terms here-- 512 00:24:23,450 --> 00:24:26,900 and I'll leave the details for you to verify at your leisure-- 513 00:24:26,900 --> 00:24:31,130 we wind up with r1 r2 being a common factor. 514 00:24:31,130 --> 00:24:34,940 The real part is r1 r2 times cosine theta 1 515 00:24:34,940 --> 00:24:39,140 cosine theta 2 minus sine theta 1 sine theta 2. 516 00:24:39,140 --> 00:24:41,600 The minus coming, because i squared is minus 1. 517 00:24:41,600 --> 00:24:47,840 The imaginary part is r1 r2 sine theta 1 cosine theta 2 sine 518 00:24:47,840 --> 00:24:50,890 theta 2 cosine theta 1. 519 00:24:50,890 --> 00:24:53,480 Remember, theta is a real number. 520 00:24:53,480 --> 00:24:55,760 If we recall our geometric-- 521 00:24:55,760 --> 00:24:58,610 trigonometric definitions, cosine theta 1 522 00:24:58,610 --> 00:25:01,880 cosine theta 2 minus sine theta 1 sine theta 2 523 00:25:01,880 --> 00:25:05,090 is cosine theta 1 plus theta 2. 524 00:25:05,090 --> 00:25:08,540 This expression this sine theta 1 plus theta 2. 525 00:25:08,540 --> 00:25:12,890 And remembering now that in polar form, what we're saying 526 00:25:12,890 --> 00:25:13,820 is what? 527 00:25:13,820 --> 00:25:17,060 This is the complex number whose magnitude 528 00:25:17,060 --> 00:25:23,210 is r1 r2 and whose argument is theta 1 plus theta 2. 529 00:25:23,210 --> 00:25:25,670 And putting these two steps together, what it says 530 00:25:25,670 --> 00:25:29,120 is that to multiply two complex numbers, the same definition 531 00:25:29,120 --> 00:25:30,620 of multiplication that we were using 532 00:25:30,620 --> 00:25:34,400 before, if we interpret this in terms of polar coordinate, 533 00:25:34,400 --> 00:25:39,230 it says, look at, to multiply two complex numbers [INAUDIBLE] 534 00:25:39,230 --> 00:25:42,230 lengths, you simply do what? 535 00:25:42,230 --> 00:25:45,110 Multiply the two lengths to get the resulting 536 00:25:45,110 --> 00:25:48,800 length of the product and add the two arguments, the two 537 00:25:48,800 --> 00:25:50,000 angles. 538 00:25:50,000 --> 00:25:51,530 That's a rather interesting result. 539 00:25:51,530 --> 00:25:53,720 You see, for example, if we now want 540 00:25:53,720 --> 00:25:56,940 to think of the complex numbers as being vectors, 541 00:25:56,940 --> 00:25:59,250 this gives us a third vector product 542 00:25:59,250 --> 00:26:01,530 that we had never talked about before 543 00:26:01,530 --> 00:26:05,010 and which I'll reinforce in the learning exercises. 544 00:26:05,010 --> 00:26:07,550 The idea is that using this as a model, 545 00:26:07,550 --> 00:26:09,330 and here's another real application, 546 00:26:09,330 --> 00:26:13,380 why not define a new product of two vectors obtained 547 00:26:13,380 --> 00:26:18,270 by multiplying the two lengths and adding the two angles? 548 00:26:18,270 --> 00:26:21,000 And the point was, that in our physical examples, 549 00:26:21,000 --> 00:26:23,340 there was really no motivation to invent this vector 550 00:26:23,340 --> 00:26:25,650 definition, in the same way that we 551 00:26:25,650 --> 00:26:28,620 could define the dot product and the cross product 552 00:26:28,620 --> 00:26:30,990 for complex numbers. 553 00:26:30,990 --> 00:26:34,380 Because after all, they are viewed as vectors in the plane. 554 00:26:34,380 --> 00:26:37,440 But again, we have no great practical application for this. 555 00:26:37,440 --> 00:26:39,300 So we don't bother doing it. 556 00:26:39,300 --> 00:26:40,830 By the way, just for kicks here, I 557 00:26:40,830 --> 00:26:42,660 thought you might enjoy the aside 558 00:26:42,660 --> 00:26:45,660 that this little interesting result explains 559 00:26:45,660 --> 00:26:50,890 why the product of two negative numbers happen to be positive. 560 00:26:50,890 --> 00:26:53,740 Remember a negative number, if r has to-- see, 561 00:26:53,740 --> 00:26:56,050 a negative number lies on the real axis. 562 00:26:56,050 --> 00:26:58,210 And if r has to be positive, that 563 00:26:58,210 --> 00:26:59,980 means you're measuring in the direction 564 00:26:59,980 --> 00:27:03,700 of the negative x-axis, which means that the polar angle is 565 00:27:03,700 --> 00:27:05,410 180 degrees. 566 00:27:05,410 --> 00:27:08,200 If you multiply two numbers whose angles 567 00:27:08,200 --> 00:27:11,800 are 180 degrees, if you add the two angles, you get what? 568 00:27:11,800 --> 00:27:14,115 360 degrees. 569 00:27:14,115 --> 00:27:15,490 You see what I'm driving at here? 570 00:27:15,490 --> 00:27:18,960 In other words, if I multiply two numbers in polar form, each 571 00:27:18,960 --> 00:27:21,300 of whose angles is 180, the product 572 00:27:21,300 --> 00:27:25,320 will have the angle equal to 360, you see? 573 00:27:25,320 --> 00:27:28,800 And that puts you back on the positive real axis. 574 00:27:28,800 --> 00:27:31,020 And so you have such real interpretations 575 00:27:31,020 --> 00:27:33,390 as y in terms of complex numbers, 576 00:27:33,390 --> 00:27:36,240 we can explain very nicely what it means for the product of two 577 00:27:36,240 --> 00:27:38,250 negatives to be a positive. 578 00:27:38,250 --> 00:27:39,900 By the way, we can carry this result 579 00:27:39,900 --> 00:27:44,610 further if we had n factors written in polar form. 580 00:27:44,610 --> 00:27:46,770 Then to multiply these n factors, 581 00:27:46,770 --> 00:27:49,080 we would simply, by induction, so to speak, 582 00:27:49,080 --> 00:27:53,880 multiply the n magnitudes together and add the n angles. 583 00:27:53,880 --> 00:27:58,150 And a very interesting special case is if all of the factors 584 00:27:58,150 --> 00:27:59,800 happen to be equal-- in other words, 585 00:27:59,800 --> 00:28:03,480 if we want to raise the complex number written in polar form 586 00:28:03,480 --> 00:28:07,490 as r comma theta to the n-th power, a very interesting thing 587 00:28:07,490 --> 00:28:10,170 is that r comma theta to the n-th power is what? 588 00:28:10,170 --> 00:28:12,070 You multiply the magnitude. 589 00:28:12,070 --> 00:28:14,850 So the magnitude of the product will be r to the n. 590 00:28:14,850 --> 00:28:16,230 You add the angles. 591 00:28:16,230 --> 00:28:21,210 So the angle of the product, the argument will be n times theta. 592 00:28:21,210 --> 00:28:25,360 A special case of the special case is if r equals 1. 593 00:28:25,360 --> 00:28:29,160 And if r equals 1, that says that the complex number whose 594 00:28:29,160 --> 00:28:32,850 polar form is 1 comma theta, when raised to the n-th power, 595 00:28:32,850 --> 00:28:37,620 is that complex number in polar form 1 comma n theta. 596 00:28:37,620 --> 00:28:40,110 And by the way, again, remember what this thing means? 597 00:28:40,110 --> 00:28:44,070 To say that the complex number is 1 comma theta means what? 598 00:28:44,070 --> 00:28:48,130 That the distance from the origin is 1 and the angle 599 00:28:48,130 --> 00:28:49,510 is theta. 600 00:28:49,510 --> 00:28:54,250 See, that was what we mean by 1 comma theta over here. 601 00:28:54,250 --> 00:28:57,180 And notice that in Cartesian form, 602 00:28:57,180 --> 00:29:01,290 that makes this length cosine theta and this length sine 603 00:29:01,290 --> 00:29:02,020 theta. 604 00:29:02,020 --> 00:29:06,300 So i comma theta is cosine theta plus i sine theta. 605 00:29:06,300 --> 00:29:10,290 At any rate, translating both of these into Cartesian form, 606 00:29:10,290 --> 00:29:14,130 we wind up with a very famous result, 607 00:29:14,130 --> 00:29:17,130 called De Moivre's theorem. 608 00:29:17,130 --> 00:29:19,980 In my high school, it was called "De Moyvre's" theorem. 609 00:29:19,980 --> 00:29:21,780 But it's De Moivre's theorem. 610 00:29:21,780 --> 00:29:24,600 And it simply says that cosine theta plus i sine 611 00:29:24,600 --> 00:29:29,220 theta to the n is cosine n theta plus i sine n theta. 612 00:29:29,220 --> 00:29:32,430 That result may not seem that remarkable to you. 613 00:29:32,430 --> 00:29:36,090 But let me give you another example using this result that 614 00:29:36,090 --> 00:29:39,600 shows how we can get real results using 615 00:29:39,600 --> 00:29:41,010 imaginary numbers. 616 00:29:41,010 --> 00:29:43,890 Let me take the special case n equals 2, just 617 00:29:43,890 --> 00:29:45,120 for the sake of argument. 618 00:29:45,120 --> 00:29:47,820 If I take n equals 2, this gives me 619 00:29:47,820 --> 00:29:50,880 cosine theta plus i sine theta squared equals cosine 620 00:29:50,880 --> 00:29:54,600 2 theta plus i sine 2 theta. 621 00:29:54,600 --> 00:29:56,040 This says what? 622 00:29:56,040 --> 00:29:58,950 The real part of this is cosine squared theta 623 00:29:58,950 --> 00:30:00,450 minus sine squared theta. 624 00:30:00,450 --> 00:30:03,000 Again, it's i squared sine squared theta, which 625 00:30:03,000 --> 00:30:04,650 is minus sine squared theta. 626 00:30:04,650 --> 00:30:08,080 The imaginary part is 2 sine theta cosine theta. 627 00:30:08,080 --> 00:30:10,320 So if I square this, I get cosine 628 00:30:10,320 --> 00:30:14,040 squared theta minus sine squared theta plus i2 sine theta cosine 629 00:30:14,040 --> 00:30:14,730 theta. 630 00:30:14,730 --> 00:30:17,970 By De Moivre's theorem, that equals cosine 2 theta plus i 631 00:30:17,970 --> 00:30:19,350 sine 2 theta. 632 00:30:19,350 --> 00:30:21,990 We saw that the only way that two complex numbers can 633 00:30:21,990 --> 00:30:24,690 be equal is if the real parts are equal 634 00:30:24,690 --> 00:30:27,000 and if the imaginary parts are equal. 635 00:30:27,000 --> 00:30:30,930 Comparing the real parts and comparing the imaginary parts, 636 00:30:30,930 --> 00:30:34,710 we get that sine 2 theta equals 2 sine theta cosine theta. 637 00:30:34,710 --> 00:30:38,760 Cosine 2 theta is cosine squared theta minus sine squared theta. 638 00:30:38,760 --> 00:30:43,020 Notice, by the way, that these are real results. 639 00:30:43,020 --> 00:30:45,450 And also, notice, by the way, that even though these 640 00:30:45,450 --> 00:30:47,550 may look like old hat to you, I could 641 00:30:47,550 --> 00:30:50,970 just as easily, for example, I could have picked 642 00:30:50,970 --> 00:30:54,210 n 5 here or 6. 643 00:30:54,210 --> 00:30:57,840 And in fact, I will do that in the learning exercises. 644 00:30:57,840 --> 00:31:00,060 The point being that, what, I can raise this 645 00:31:00,060 --> 00:31:04,370 to the fifth power, compare this with cosine 5 theta plus i 646 00:31:04,370 --> 00:31:07,350 sine 5 theta, and wind up with real identities, 647 00:31:07,350 --> 00:31:08,460 in fact, for what? 648 00:31:08,460 --> 00:31:12,360 Sine n theta and cosine n theta for any whole number 649 00:31:12,360 --> 00:31:13,915 value of n. 650 00:31:13,915 --> 00:31:15,540 See, I'm trying to hammer home the fact 651 00:31:15,540 --> 00:31:18,090 that as we're doing the complex number arithmetic, 652 00:31:18,090 --> 00:31:21,798 don't forget that this stuff does have real applications, 653 00:31:21,798 --> 00:31:24,090 and we haven't even started to scratch the surface yet. 654 00:31:24,090 --> 00:31:27,420 This is just our baby lecture introducing the arithmetic 655 00:31:27,420 --> 00:31:28,440 of complex numbers. 656 00:31:28,440 --> 00:31:30,060 We haven't even gotten to anything 657 00:31:30,060 --> 00:31:31,752 like algebra or calculus yet. 658 00:31:31,752 --> 00:31:33,710 Wait till that happens, and you're really going 659 00:31:33,710 --> 00:31:35,850 to see some nice applications. 660 00:31:35,850 --> 00:31:38,745 At any rate, by the way, the polar form of multiplication 661 00:31:38,745 --> 00:31:42,390 that leads to a very interesting way of extracting 662 00:31:42,390 --> 00:31:44,760 roots of complex numbers. 663 00:31:44,760 --> 00:31:48,260 For example, suppose I want to find the sixth root of i. 664 00:31:48,260 --> 00:31:49,930 In other words, what complex number, 665 00:31:49,930 --> 00:31:52,200 raised to the sixth power, gives i? 666 00:31:52,200 --> 00:31:55,020 In fact, is there such a complex number? 667 00:31:55,020 --> 00:31:58,013 After all, we could raise real numbers to powers. 668 00:31:58,013 --> 00:31:59,430 But one of the reasons that we had 669 00:31:59,430 --> 00:32:02,550 to invent the complex numbers is that we couldn't extract 670 00:32:02,550 --> 00:32:04,380 the square root of minus 1. 671 00:32:04,380 --> 00:32:07,590 There was no real number whose square was minus 1. 672 00:32:07,590 --> 00:32:11,320 The question now is there a complex number, 673 00:32:11,320 --> 00:32:14,460 which when raised to the sixth power, equals i? 674 00:32:14,460 --> 00:32:16,650 One way of doing this is to say, OK, 675 00:32:16,650 --> 00:32:18,600 let's assume there is a complex number. 676 00:32:18,600 --> 00:32:22,020 We'll call it x plus iy, which when raised to the sixth power, 677 00:32:22,020 --> 00:32:22,680 equals i. 678 00:32:22,680 --> 00:32:25,920 In other words, the sixth root the i is x plus iy. 679 00:32:25,920 --> 00:32:28,770 And let's see if we can solve for x and y. 680 00:32:28,770 --> 00:32:31,320 One way of doing this is to raise both sides here 681 00:32:31,320 --> 00:32:33,150 to the sixth power, in which case 682 00:32:33,150 --> 00:32:37,560 we see that i has to be x plus iy to the sixth power. 683 00:32:37,560 --> 00:32:41,790 On the other hand, i is written as 0 plus 1i. 684 00:32:41,790 --> 00:32:43,510 If I raise this to the sixth power, 685 00:32:43,510 --> 00:32:45,060 I don't know if you've noticed this, 686 00:32:45,060 --> 00:32:48,870 every time I raise i to an even power, I get a real number. 687 00:32:48,870 --> 00:32:49,410 Why? 688 00:32:49,410 --> 00:32:51,880 Because i squared is minus 1. 689 00:32:51,880 --> 00:32:55,170 Therefore, i to the fourth is i squared squared, which 690 00:32:55,170 --> 00:32:57,330 is minus 1 squared, which is 1. 691 00:32:57,330 --> 00:32:59,790 i to the sixth is i to the fourth times 692 00:32:59,790 --> 00:33:04,500 i squared, which is 1 times i squared, which is minus 1. 693 00:33:04,500 --> 00:33:08,460 And in the same way, if I take i cubed, that's what? 694 00:33:08,460 --> 00:33:11,040 i squared times i, which is minus i. 695 00:33:11,040 --> 00:33:14,610 In other words, the even powers of i are real, 696 00:33:14,610 --> 00:33:18,690 the odd powers of i give me back plus or minus i. 697 00:33:18,690 --> 00:33:21,600 So if I raise this to the sixth power and collect terms, 698 00:33:21,600 --> 00:33:24,450 I'll get a certain number of real terms 699 00:33:24,450 --> 00:33:27,370 and a certain number of purely imaginary terms. 700 00:33:27,370 --> 00:33:29,460 In fact, using the binomial theorem 701 00:33:29,460 --> 00:33:31,380 and raising this to the sixth power 702 00:33:31,380 --> 00:33:33,780 and separating the terms for you in advance, 703 00:33:33,780 --> 00:33:34,990 I wind up with what? 704 00:33:34,990 --> 00:33:39,450 x to the sixth plus 6x to the fifth iy plus 15x 705 00:33:39,450 --> 00:33:45,120 to the fourth iy squared plus 20x cubed iy cubed plus 15x 706 00:33:45,120 --> 00:33:48,270 squared iy to the fourth plus 6x iy 707 00:33:48,270 --> 00:33:50,580 to the fifth plus iy to the sixth. 708 00:33:50,580 --> 00:33:51,930 I went through that rapidly. 709 00:33:51,930 --> 00:33:53,970 It's just using the binomial theorem, 710 00:33:53,970 --> 00:33:57,300 noticing that all of these terms will turn out to be real. 711 00:33:57,300 --> 00:33:59,970 All of these terms will be purely imaginary. 712 00:33:59,970 --> 00:34:02,010 In other words, getting rid of the i's 713 00:34:02,010 --> 00:34:03,480 to the best of my ability. 714 00:34:03,480 --> 00:34:06,720 See, squaring over here, this is a minus y squared term. 715 00:34:06,720 --> 00:34:08,520 This is just y to the fourth. 716 00:34:08,520 --> 00:34:11,610 This is i to the sixth, which is the same as i squared. 717 00:34:11,610 --> 00:34:13,210 Because i to the fourth is 1. 718 00:34:13,210 --> 00:34:16,989 i to the sixth is i fourth times i squared, which is i squared. 719 00:34:16,989 --> 00:34:19,710 So this just comes out as minus 1, et cetera. 720 00:34:19,710 --> 00:34:23,130 And making these translations, we wind up 721 00:34:23,130 --> 00:34:28,050 with the complicated algebraic system that to find x and y, 722 00:34:28,050 --> 00:34:31,150 we must be able to solve this system of equations. 723 00:34:31,150 --> 00:34:33,210 In other words, the root part must be 0, 724 00:34:33,210 --> 00:34:36,810 the imaginary part must be 1, all right? 725 00:34:36,810 --> 00:34:38,940 Now at this stage of the game, not only 726 00:34:38,940 --> 00:34:41,130 may it seem difficult to solve this, 727 00:34:41,130 --> 00:34:44,130 but it may be that there are no real values of x and y 728 00:34:44,130 --> 00:34:44,880 which solve this. 729 00:34:44,880 --> 00:34:47,070 And if I can't find x and y, if there 730 00:34:47,070 --> 00:34:50,429 are no values for x and y, it means that x plus iy 731 00:34:50,429 --> 00:34:52,489 doesn't exist. 732 00:34:52,489 --> 00:34:55,340 Well, here's where I wanted to show you the tremendous power 733 00:34:55,340 --> 00:34:57,680 of polar coordinates. 734 00:34:57,680 --> 00:35:00,860 You see, in polar coordinates, how would I write i? 735 00:35:00,860 --> 00:35:02,060 I is what? 736 00:35:02,060 --> 00:35:03,880 It's magnitude is 1. 737 00:35:03,880 --> 00:35:07,010 See, it's the 0.0 comma 1 in the Argand diagram. 738 00:35:07,010 --> 00:35:08,540 It's magnitude is 1. 739 00:35:08,540 --> 00:35:11,330 And it's argument is pi over 2. 740 00:35:11,330 --> 00:35:15,650 By the way, again, I'm in this trouble with multiple angles. 741 00:35:15,650 --> 00:35:17,510 You see, it's only pi over 2. 742 00:35:17,510 --> 00:35:21,290 It could be 5 pi over 2, 9 pi over 2, et cetera. 743 00:35:21,290 --> 00:35:24,480 Every time I go through 2 pi, I come back to the same point. 744 00:35:24,480 --> 00:35:26,460 I'm going to mention that in a moment. 745 00:35:26,460 --> 00:35:30,300 But the idea is, look at, this is what I want. 746 00:35:30,300 --> 00:35:31,110 This is i. 747 00:35:31,110 --> 00:35:32,672 And I want the [INAUDIBLE] divide. 748 00:35:32,672 --> 00:35:34,130 Let's assume that the answer can be 749 00:35:34,130 --> 00:35:35,882 written in polar coordinates. 750 00:35:35,882 --> 00:35:37,340 If I write it in polar coordinates, 751 00:35:37,340 --> 00:35:39,190 the answer is r comma theta. 752 00:35:39,190 --> 00:35:41,720 All I've got to do now is solve for r and theta. 753 00:35:41,720 --> 00:35:44,270 And I claim, believe it or not, that that's trivial, 754 00:35:44,270 --> 00:35:48,650 namely, given this, I raise both sides to the sixth power. 755 00:35:48,650 --> 00:35:52,350 If I do that, I wind up here with i, which just, 756 00:35:52,350 --> 00:35:54,350 to get this all on one line, I just repeat this. 757 00:35:54,350 --> 00:35:57,470 This is 1 comma pi over 2 in polar coordinates. 758 00:35:57,470 --> 00:36:00,050 And this is r comma theta to the sixth power. 759 00:36:00,050 --> 00:36:03,290 But the beauty of multiplication in polar coordinates 760 00:36:03,290 --> 00:36:05,570 is that r comma theta to the sixth power 761 00:36:05,570 --> 00:36:08,720 is r to the sixth comma 6 theta. 762 00:36:08,720 --> 00:36:10,630 The magnitude is this-- 763 00:36:10,630 --> 00:36:12,680 so in other words, you multiply the magnitudes, 764 00:36:12,680 --> 00:36:15,140 and you add the angles. 765 00:36:15,140 --> 00:36:18,170 Therefore, what this tells me is, remember, r must be real. 766 00:36:18,170 --> 00:36:20,510 There is only one real number, which 767 00:36:20,510 --> 00:36:23,640 when raised to the sixth power, is 1. 768 00:36:23,640 --> 00:36:25,980 And that's 1 itself. 769 00:36:25,980 --> 00:36:33,080 See, the n-th root of any positive number 770 00:36:33,080 --> 00:36:34,910 has exactly one real solution. 771 00:36:34,910 --> 00:36:36,410 That's why we can always find the r. 772 00:36:36,410 --> 00:36:38,118 In this case, I picked the simple example 773 00:36:38,118 --> 00:36:39,980 where r turns out to be 1. 774 00:36:39,980 --> 00:36:43,670 See, minus 1 also raised to the sixth power is 1. 775 00:36:43,670 --> 00:36:44,660 But that doesn't count. 776 00:36:44,660 --> 00:36:47,480 Because we agreed that r had to be positive. 777 00:36:47,480 --> 00:36:50,730 r was measuring the magnitude of the complex number, 778 00:36:50,730 --> 00:36:53,150 so r must be 1 by virtue of the fact 779 00:36:53,150 --> 00:36:55,100 that r is greater than or equal to 0. 780 00:36:55,100 --> 00:36:56,630 That eliminates minus 1. 781 00:36:56,630 --> 00:37:01,190 Now 6 theta can either be pi over 2 or 5 pi 782 00:37:01,190 --> 00:37:02,840 over 2, et cetera. 783 00:37:02,840 --> 00:37:06,470 The important point being that we now tack on the 2 pi k. 784 00:37:06,470 --> 00:37:10,610 Because notice that as this changes by 360 degrees, 785 00:37:10,610 --> 00:37:13,370 theta only changes by 60 degrees. 786 00:37:13,370 --> 00:37:16,430 Because 6 theta is changing by 360. 787 00:37:16,430 --> 00:37:19,250 Therefore, you see, we're going to get a whole bunch of theta 788 00:37:19,250 --> 00:37:20,700 values that work this way. 789 00:37:20,700 --> 00:37:23,390 And again, I'll explain this in more detail 790 00:37:23,390 --> 00:37:26,720 as we go through the exercises on this unit. 791 00:37:26,720 --> 00:37:28,340 But the idea is what? 792 00:37:28,340 --> 00:37:32,330 If I keep tacking on multiples of 2 pi, what I'm saying 793 00:37:32,330 --> 00:37:35,420 is r must be 1, and theta is what? 794 00:37:35,420 --> 00:37:39,990 It's pi over 12 plus what? 795 00:37:39,990 --> 00:37:45,870 2 pi k over 6 pi over 3k, 60 [INAUDIBLE].. 796 00:37:45,870 --> 00:37:48,170 I essentially, in terms of angles, 797 00:37:48,170 --> 00:37:50,510 tack on 60 degree increments here. 798 00:37:50,510 --> 00:37:53,390 To make a long story short, r must be 1. 799 00:37:53,390 --> 00:37:57,380 But theta could either be pi over 12, 5 pi over 12, 800 00:37:57,380 --> 00:38:00,195 9 pi over 12, 13 pi over 12. 801 00:38:00,195 --> 00:38:03,680 See, I'm adding on 4 pi over 12 each time, pi over 3, 802 00:38:03,680 --> 00:38:05,960 60 degrees, in degree measure. 803 00:38:05,960 --> 00:38:09,660 17 pi over 12, 21 pi over 12. 804 00:38:09,660 --> 00:38:12,410 The next one would be 25 pi over 12. 805 00:38:12,410 --> 00:38:14,180 But I hope you can see that that's 806 00:38:14,180 --> 00:38:18,830 the same as, position-wise, 1 comma pi over 12 807 00:38:18,830 --> 00:38:22,370 gives me the same thing as 1 comma 25 pi over 12. 808 00:38:22,370 --> 00:38:27,230 But all six of these angles give me different positions. 809 00:38:27,230 --> 00:38:30,080 Just by way of illustration, pi over 12 810 00:38:30,080 --> 00:38:32,180 turns out to be 15 degrees. 811 00:38:32,180 --> 00:38:35,090 What this says is that one of the six roots of i, 812 00:38:35,090 --> 00:38:36,800 geometrically, is what? 813 00:38:36,800 --> 00:38:41,720 It's that complex number, which is 1 unit from the origin. 814 00:38:41,720 --> 00:38:44,000 That means it's on the circle of radius 1, 815 00:38:44,000 --> 00:38:46,070 centered at the origin. 816 00:38:46,070 --> 00:38:48,620 And the angle must be 15 degrees. 817 00:38:48,620 --> 00:38:50,760 The argument is 15 degrees. 818 00:38:50,760 --> 00:38:52,550 And by the way, just to check this 819 00:38:52,550 --> 00:38:55,880 out that this really is at 1 comma pi over 12 820 00:38:55,880 --> 00:38:58,130 really is an answer here. 821 00:38:58,130 --> 00:39:01,080 How do you raise a complex number to a power 822 00:39:01,080 --> 00:39:03,580 if we view it as a length? 823 00:39:03,580 --> 00:39:05,690 To raise this to the sixth power, 824 00:39:05,690 --> 00:39:08,150 we must raise 1 to the sixth power. 825 00:39:08,150 --> 00:39:09,590 That will still be 1. 826 00:39:09,590 --> 00:39:11,660 So I'm still going to be on the circle. 827 00:39:11,660 --> 00:39:15,390 When I multiply, I add angles. 828 00:39:15,390 --> 00:39:17,990 So when I raise this to the sixth power, 829 00:39:17,990 --> 00:39:21,710 I'm taking, what, 15 degrees six times is 90 degrees. 830 00:39:21,710 --> 00:39:24,540 And that puts me right up where i is supposed to be. 831 00:39:24,540 --> 00:39:27,380 In other words, to this thing backwards, so to speak, 832 00:39:27,380 --> 00:39:32,390 I know that 6 times the angle I'm looking for must be 90. 833 00:39:32,390 --> 00:39:35,160 So the angle itself must be 15. 834 00:39:35,160 --> 00:39:38,735 For example, if I was looking for the eighth root of i, 835 00:39:38,735 --> 00:39:40,550 I would do what? 836 00:39:40,550 --> 00:39:43,850 I would know that when I add the angle to itself 8 times, 837 00:39:43,850 --> 00:39:45,020 I want 90. 838 00:39:45,020 --> 00:39:48,230 So the angle would have had to been 90 over 8. 839 00:39:48,230 --> 00:39:50,990 And again, I'll leave this for you as exercises. 840 00:39:50,990 --> 00:39:54,890 The point is that geometrically, the sixth roots of i 841 00:39:54,890 --> 00:39:57,990 are all equally spaced points. 842 00:39:57,990 --> 00:40:01,280 The first one is the point 1 comma pi over 12. 843 00:40:01,280 --> 00:40:05,220 And the rest are spaced equally along the circle at 60 degree 844 00:40:05,220 --> 00:40:05,720 intervals. 845 00:40:05,720 --> 00:40:09,680 You see, it breaks the circle up into six equal parts. 846 00:40:09,680 --> 00:40:10,910 I come back to here. 847 00:40:10,910 --> 00:40:12,710 Notice that, for example, if I take 848 00:40:12,710 --> 00:40:19,190 this one, which is 75 degrees, if I take 75 degrees 6 times, 849 00:40:19,190 --> 00:40:23,270 notice that I come back, what, 75 times 6 is 450. 850 00:40:23,270 --> 00:40:27,560 It means I go all the way around and come back to i when I 851 00:40:27,560 --> 00:40:30,560 raise this to the sixth power. 852 00:40:30,560 --> 00:40:33,170 To help you see this geometrically, I'll pick-- 853 00:40:33,170 --> 00:40:35,390 of these six, one of these happens 854 00:40:35,390 --> 00:40:37,400 to be very easy, at least, to me. 855 00:40:37,400 --> 00:40:41,480 9 pi over 12 is 3 pi over 4, which 856 00:40:41,480 --> 00:40:44,240 happens to be 135 degrees. 857 00:40:44,240 --> 00:40:49,970 And i-- 1 comma 3 pi over 4 is cosine 3 pi over 4 plus i 858 00:40:49,970 --> 00:40:52,010 sine 3 pi over 4. 859 00:40:52,010 --> 00:40:56,920 The cosine of 3 pi over 4 is minus 1 over square root of 2. 860 00:40:56,920 --> 00:40:59,450 The sine is plus 1 over square root of 2. 861 00:40:59,450 --> 00:41:04,880 So in typical x plus iy form, one of the roots 862 00:41:04,880 --> 00:41:08,960 is 1 over the square root of 2 times minus 1 plus i. 863 00:41:08,960 --> 00:41:12,320 In other words, the real part of this complex number is minus 1 864 00:41:12,320 --> 00:41:13,640 over the square root of 2. 865 00:41:13,640 --> 00:41:17,090 The imaginary part is 1 over the square root of 2. 866 00:41:17,090 --> 00:41:20,180 And I leave it, again, as a voluntary exercise for you 867 00:41:20,180 --> 00:41:23,420 to do to actually raise this to the sixth power 868 00:41:23,420 --> 00:41:27,190 and find, amazingly enough, that you do get 1 for an answer-- 869 00:41:27,190 --> 00:41:28,880 i for an answer. 870 00:41:28,880 --> 00:41:32,570 Now you see, I just picked one particular example. 871 00:41:32,570 --> 00:41:34,430 But this would have worked for any roots 872 00:41:34,430 --> 00:41:35,510 that I wanted to extract. 873 00:41:35,510 --> 00:41:38,930 And this is very important from a mathematical point of view. 874 00:41:38,930 --> 00:41:41,630 The complex numbers are closed, with respect 875 00:41:41,630 --> 00:41:43,130 to extracting roots. 876 00:41:43,130 --> 00:41:47,210 And let me summarize that for a part of today's lesson 877 00:41:47,210 --> 00:41:48,050 over here. 878 00:41:48,050 --> 00:41:50,960 The idea is that one of the reasons 879 00:41:50,960 --> 00:41:53,990 that we had to invent the fractions after we knew 880 00:41:53,990 --> 00:41:57,170 the integers was the fact that the integers were not closed 881 00:41:57,170 --> 00:41:59,990 with respect to division, that the quotient of two integers 882 00:41:59,990 --> 00:42:01,730 didn't have to be an integer. 883 00:42:01,730 --> 00:42:06,260 One of the reasons that we had to invent the complex numbers 884 00:42:06,260 --> 00:42:08,120 after we had the real numbers was 885 00:42:08,120 --> 00:42:10,550 that the real numbers were not closed with respect 886 00:42:10,550 --> 00:42:12,150 to extracting roots. 887 00:42:12,150 --> 00:42:14,900 What I've just shown you in terms of a particular example 888 00:42:14,900 --> 00:42:17,630 is that the complex numbers are closed with respect 889 00:42:17,630 --> 00:42:18,980 to extracting roots. 890 00:42:18,980 --> 00:42:22,310 This means that, in particular, the basic operations that 891 00:42:22,310 --> 00:42:24,227 are involved in solving polynomial equations-- 892 00:42:24,227 --> 00:42:25,685 in other words, what you have to do 893 00:42:25,685 --> 00:42:27,680 is solve a polynomial equation, nothing 894 00:42:27,680 --> 00:42:31,050 more than the basic operations of adding, subtracting, 895 00:42:31,050 --> 00:42:33,860 multiplying, dividing, raising the powers, 896 00:42:33,860 --> 00:42:35,300 and extracting roots. 897 00:42:35,300 --> 00:42:37,790 All of these operations are closed with respect 898 00:42:37,790 --> 00:42:39,320 to the complex numbers. 899 00:42:39,320 --> 00:42:41,420 And what this means is if you wanted 900 00:42:41,420 --> 00:42:44,480 to write a polynomial equation which 901 00:42:44,480 --> 00:42:47,630 had complex numbers as coefficients, 902 00:42:47,630 --> 00:42:51,050 you would not have to invent any more complex numbers. 903 00:42:51,050 --> 00:42:53,030 You would not have to invent a new number 904 00:42:53,030 --> 00:42:55,580 system to solve this equation, namely, 905 00:42:55,580 --> 00:43:00,230 any polynomial with complex coefficients has complex roots. 906 00:43:00,230 --> 00:43:02,270 And I think that's enough for today. 907 00:43:02,270 --> 00:43:04,880 I want you to drill now on the exercises. 908 00:43:04,880 --> 00:43:07,640 Next time, we will talk about functions 909 00:43:07,640 --> 00:43:09,140 using complex numbers. 910 00:43:09,140 --> 00:43:11,210 At any rate, until next time, goodbye. 911 00:43:15,730 --> 00:43:18,130 Funding for the publication of this video 912 00:43:18,130 --> 00:43:23,020 was provided by the Gabriella and Paul Rosenbaum Foundation. 913 00:43:23,020 --> 00:43:27,160 Help OCW continue to provide free and open access to MIT 914 00:43:27,160 --> 00:43:32,595 courses by making a donation at ocw.mit.edu/donate.