1 00:00:00,166 --> 00:00:01,466 Hello everyone. 2 00:00:01,466 --> 00:00:05,300 In this video, we're going to 
talk about complex nth roots. 3 00:00:05,300 --> 00:00:08,166 In particular, complex third 
roots, or complex cube roots, 4 00:00:08,166 --> 00:00:10,800 because that's the problem that I picked. 5 00:00:10,800 --> 00:00:14,400 So find the cube roots of 1 minus 2i. 6 00:00:14,400 --> 00:00:17,433 You may leave part of your answer 
in terms of arctan. So this - 7 00:00:17,433 --> 00:00:21,266 the answer to this question 
is going to be ugly and use 8 00:00:21,266 --> 00:00:23,100 a lot of 9 00:00:23,100 --> 00:00:26,066 notation. It's just the way it's going to be. 10 00:00:26,066 --> 00:00:27,000 11 00:00:27,000 --> 00:00:28,766 Because I don’t… 12 00:00:28,766 --> 00:00:31,766 I didn't want to try to fudge the numbers so that 13 00:00:31,766 --> 00:00:34,700 they were correct. I just decided 
hey let's just leave it and 14 00:00:34,700 --> 00:00:38,466 let's just talk about the ideas because again 
the key ideas are what are really important 15 00:00:38,466 --> 00:00:42,866 in these examples, as opposed to actually getting a complete… 16 00:00:42,866 --> 00:00:45,466 well not a complete solution but 
you know what I mean, right, 17 00:00:45,466 --> 00:00:48,166 in terms of making the numbers perfect. 18 00:00:48,166 --> 00:00:49,866 So... 19 00:00:49,866 --> 00:00:52,033 what are we going to do in this 
question? So we're trying to find 20 00:00:52,033 --> 00:00:57,666 for the cube roots of 1 minus 2i, so we're 
trying to solve for z cubed equals 1 minus 2i, 21 00:00:57,666 --> 00:01:00,766 and when we're solving for z cubed 
equals 1 minus 2i, we usually like - 22 00:01:00,766 --> 00:01:04,533 well it's not usually, it's easier to deal with polar coordinates 23 00:01:04,533 --> 00:01:08,466 than it is to deal with standard form. 
So we're going to let z equal to r 24 00:01:08,466 --> 00:01:10,633 e to the i theta. 25 00:01:10,633 --> 00:01:12,333 26 00:01:12,333 --> 00:01:14,800 Then what do we have? Well if we 27 00:01:14,800 --> 00:01:18,400 plug in this z value into z 
cubed equals 1 minus 2i, 28 00:01:18,400 --> 00:01:24,266 we're going to get r cubed e to the i 3 theta is equal to 1 minus 2i, 29 00:01:24,266 --> 00:01:27,300 and from here we have 
options. What I'm going to do 30 00:01:27,300 --> 00:01:29,800 is I'm going to determine r first. 
r is always easy to determine 31 00:01:29,800 --> 00:01:32,900 because you can determine 
the lengths of both sides. 32 00:01:32,900 --> 00:01:35,800 The length of e to the i 3 
theta is always going to be - 33 00:01:35,800 --> 00:01:37,966 is always going to be 1, 34 00:01:37,966 --> 00:01:40,700 because this is just cosine plus i sine 35 00:01:40,700 --> 00:01:43,633 of our argument 3 theta, 36 00:01:43,633 --> 00:01:47,233 and cosine of 3 theta squared plus 
sine of 3 theta squared, that's always 1. 37 00:01:47,233 --> 00:01:52,000 So that's something you should double-check 
for yourselves, but taking lengths here is actually 38 00:01:52,000 --> 00:01:54,133 going to simplify this a lot. 39 00:01:54,133 --> 00:01:59,033 40 00:01:59,033 --> 00:02:01,433 So it's going to be r… 41 00:02:01,433 --> 00:02:02,666 42 00:02:02,666 --> 00:02:02,800 43 00:02:02,800 --> 00:02:06,800 okay so let's take lengths, what are 
we going to get? We're going to get… 44 00:02:06,800 --> 00:02:12,000 45 00:02:12,000 --> 00:02:14,300 taking lengths gives 46 00:02:14,300 --> 00:02:17,666 this is equal to square root of… 47 00:02:17,666 --> 00:02:22,800 48 00:02:22,800 --> 00:02:25,633 49 00:02:25,633 --> 00:02:27,433 it's equal to the following. 50 00:02:27,433 --> 00:02:29,966 I guess I should say by… 51 00:02:29,966 --> 00:02:32,466 52 00:02:32,466 --> 00:02:35,466 okay, so let's take a look at 
everything that's just happened here. 53 00:02:35,466 --> 00:02:37,233 So we're going to take lengths. 54 00:02:37,233 --> 00:02:41,033 So by PM, what can we do? Well 
by PM, we can split this to a product, 55 00:02:41,033 --> 00:02:43,166 the other term drops out, 56 00:02:43,166 --> 00:02:46,433 so maybe I'll even include that step as well. 57 00:02:46,433 --> 00:02:49,433 This term is going to go away 58 00:02:49,433 --> 00:02:55,200 59 00:02:55,200 --> 00:02:58,133 So let me do this. 60 00:02:58,133 --> 00:03:01,866 61 00:03:01,866 --> 00:03:04,733 So r cubed e to the i 3 theta, that's the - 62 00:03:04,733 --> 00:03:07,566 if I split the modulus - or if I… 63 00:03:07,566 --> 00:03:11,333 the length of the product of two 
complex numbers is equal to the 64 00:03:11,333 --> 00:03:13,700 product of their lengths. 65 00:03:13,700 --> 00:03:16,033 The second term is 1, so 
that's going to go away, 66 00:03:16,033 --> 00:03:19,100 so then I'm left with - and I can move the cube outside 67 00:03:19,100 --> 00:03:22,166 because again it splits across products, 68 00:03:22,166 --> 00:03:25,233 and this value the length of this was 
equal to the length of 1 minus 2i, 69 00:03:25,233 --> 00:03:27,533 and that's going to be the square root of 5. 70 00:03:27,533 --> 00:03:28,666 71 00:03:28,666 --> 00:03:29,700 72 00:03:29,700 --> 00:03:31,800 So hence, 73 00:03:31,800 --> 00:03:34,300 74 00:03:34,300 --> 00:03:37,466 r is equal to the 75 00:03:37,466 --> 00:03:38,533 76 00:03:38,533 --> 00:03:40,933 sixth root of 5, 77 00:03:40,933 --> 00:03:43,300 78 00:03:43,300 --> 00:03:45,466 and the ugly numbers have already begun. 79 00:03:45,466 --> 00:03:46,233 80 00:03:46,233 --> 00:03:47,700 Okay? 81 00:03:47,700 --> 00:03:50,400 Again, because the modulus - 
so remember in this notation, 82 00:03:50,400 --> 00:03:55,000 r is equal to the modulus of z, which 
has to be a positive number, so 83 00:03:55,000 --> 00:03:59,666 r is positive and it's real so we - 
well it’s, okay, not negative. 84 00:03:59,666 --> 00:04:02,266 r isn’t always positive, it's always 
not negative, it could be 0. 85 00:04:02,266 --> 00:04:05,433 I mean clearly it's not in this case, but it could be 0, so it's non-negative 86 00:04:05,433 --> 00:04:09,433 and it's definitely real number, so 
the length is just going to be the 87 00:04:09,433 --> 00:04:11,733 positive cube root of 88 00:04:11,733 --> 00:04:14,800 the square root of 5, and
that's going to be this value. 89 00:04:14,800 --> 00:04:17,766 90 00:04:17,766 --> 00:04:22,600 Now we want to solve in terms of 3 theta. 91 00:04:22,600 --> 00:04:24,700 So next… 92 00:04:24,700 --> 00:04:27,200 93 00:04:27,200 --> 00:04:29,200 next we… 94 00:04:29,200 --> 00:04:30,733 95 00:04:30,733 --> 00:04:35,366 this back into - so we 
substitute this back into the 96 00:04:35,366 --> 00:04:37,600 first displayed equation 97 00:04:37,600 --> 00:04:40,133 98 00:04:40,133 --> 00:04:42,633 to get that 99 00:04:42,633 --> 00:04:46,933 100 00:04:46,933 --> 00:04:50,500 e i to the 3 theta is equal to 101 00:04:50,500 --> 00:04:52,133 102 00:04:52,133 --> 00:04:55,566 r inverse of 1 minus 2i. 103 00:04:55,566 --> 00:04:56,500 104 00:04:56,500 --> 00:04:56,866 105 00:04:56,866 --> 00:05:01,766 So e to the i 3 theta is equal 
to r inverse of 1 minus 2i. 106 00:05:01,766 --> 00:05:03,500 107 00:05:03,500 --> 00:05:05,633 So again, I could… 108 00:05:05,633 --> 00:05:10,200 I mean, I can - I want to solve for 3 theta. 109 00:05:10,200 --> 00:05:11,866 110 00:05:11,866 --> 00:05:15,900 You pick your way that you'd 
like to solve this, I'm going to do - 111 00:05:15,900 --> 00:05:18,766 I don't even know what am I going 
to do. What am I going to do? 112 00:05:18,766 --> 00:05:22,366 I'm going to solve for the argument by taking - 113 00:05:22,366 --> 00:05:26,200 what can I do? I could try to convert 
this to polar coordinates, I guess. 114 00:05:26,200 --> 00:05:27,100 115 00:05:27,100 --> 00:05:29,433 Is that the best option? Probably. 116 00:05:29,433 --> 00:05:30,600 117 00:05:30,600 --> 00:05:34,300 So to convert this into polar coordinates, it's going to be… 118 00:05:34,300 --> 00:05:35,100 119 00:05:35,100 --> 00:05:37,233 what do I want to say? 120 00:05:37,233 --> 00:05:39,500 121 00:05:39,500 --> 00:05:42,633 How do I want to do this? 122 00:05:42,633 --> 00:05:44,566 123 00:05:44,566 --> 00:05:47,666 I can write this as e to the power of 124 00:05:47,666 --> 00:05:48,700 125 00:05:48,700 --> 00:05:51,033 pick your favorite Greek letter, 
I'm going to pick alpha. 126 00:05:51,033 --> 00:05:51,966 127 00:05:51,966 --> 00:05:57,033 I'm going to say where alpha is equal to arctan 128 00:05:57,033 --> 00:05:58,633 of... 129 00:05:58,633 --> 00:06:01,166 130 00:06:01,166 --> 00:06:03,466 what am I going to say? 2r… 131 00:06:03,466 --> 00:06:05,033 132 00:06:05,033 --> 00:06:07,900 2r negative 1, 133 00:06:07,900 --> 00:06:09,866 134 00:06:09,866 --> 00:06:13,166 divided by r negative 1. 135 00:06:13,166 --> 00:06:14,966 136 00:06:14,966 --> 00:06:16,900 Let's see. 137 00:06:16,900 --> 00:06:21,933 138 00:06:21,933 --> 00:06:24,233 Oh negative 2, oops. 139 00:06:24,233 --> 00:06:25,133 140 00:06:25,133 --> 00:06:30,400 Negative 2, which is equal to arctan of minus 2. 141 00:06:30,400 --> 00:06:33,733 142 00:06:33,733 --> 00:06:38,166 Now something to keep in 
mind, is this the actual correct 143 00:06:38,166 --> 00:06:40,533 one? Do we want 144 00:06:40,533 --> 00:06:44,566 arctan of negative 2 or do we 
want arctan of negative 2 plus pi? 145 00:06:44,566 --> 00:06:46,466 And again we can answer this, 146 00:06:46,466 --> 00:06:48,833 arctan of negative 2, I don't 
know the exact number of this, 147 00:06:48,833 --> 00:06:52,266 but that doesn't matter. I 
know that arctan of minus 2 148 00:06:52,266 --> 00:06:56,366 is going to be a negative radian value. 149 00:06:56,366 --> 00:06:57,433 150 00:06:57,433 --> 00:07:00,800 So that's going to put us in the 
fourth quadrant, and 1 minus 2i 151 00:07:00,800 --> 00:07:04,800 scaled, so 1 minus 2i lives 152 00:07:04,800 --> 00:07:08,766 in the fourth quadrant, so life is okay. 
So I don't need to add pi here, I can just 153 00:07:08,766 --> 00:07:11,166 take alpha to be arctan of minus 2. 154 00:07:11,166 --> 00:07:14,633 155 00:07:14,633 --> 00:07:17,100 So then what does this give? 
So if we want to solve for 156 00:07:17,100 --> 00:07:19,100 theta, so hence, 157 00:07:19,100 --> 00:07:21,366 158 00:07:21,366 --> 00:07:24,366 3 theta is equal to 159 00:07:24,366 --> 00:07:25,333 160 00:07:25,333 --> 00:07:28,666 alpha plus 2 pi k 161 00:07:28,666 --> 00:07:29,600 162 00:07:29,600 --> 00:07:30,300 163 00:07:30,300 --> 00:07:32,400 for any integer k. 164 00:07:32,400 --> 00:07:35,000 165 00:07:35,000 --> 00:07:37,300 So if I compare these two things - 
oh I should have mentioned. 166 00:07:37,300 --> 00:07:39,633 This should be e to the i alpha, let me 167 00:07:39,633 --> 00:07:41,766 typo that in there. 168 00:07:41,766 --> 00:07:43,233 169 00:07:43,233 --> 00:07:47,533 I want e to the i alpha, where 
alpha is equal to the angle, right? 170 00:07:47,533 --> 00:07:48,700 171 00:07:48,700 --> 00:07:51,666 So this is going to give us 
3 theta is equal to alpha plus 172 00:07:51,666 --> 00:07:54,766 2 pi k for any integer k, right, because again, up to… 173 00:07:54,766 --> 00:07:56,333 174 00:07:56,333 --> 00:08:00,066 these values, e to the i... 175 00:08:00,066 --> 00:08:02,633 e to the i 2 pi is always 
going to be equal to 1. 176 00:08:02,633 --> 00:08:07,933 So if I take any multiple here, it's going to 
give me the same value for e to the i alpha. 177 00:08:07,933 --> 00:08:10,466 178 00:08:10,466 --> 00:08:13,300 I mean these two angles must be equal up some multiples of 2 pi, right, 179 00:08:13,300 --> 00:08:18,066 because I can keep rotating around the complex 
plane, it's going to give me the same angle. 180 00:08:18,066 --> 00:08:18,800 181 00:08:18,800 --> 00:08:21,700 So doing this, and then just solving for theta, 182 00:08:21,700 --> 00:08:26,900 so solving for theta gives 183 00:08:26,900 --> 00:08:29,033 184 00:08:29,033 --> 00:08:31,600 theta is equal to alpha over 3 185 00:08:31,600 --> 00:08:35,933 plus 2 pi k divided by 3. 186 00:08:35,933 --> 00:08:40,266 187 00:08:40,266 --> 00:08:43,566 So these are our possible solutions. 188 00:08:43,566 --> 00:08:46,366 Again, as in class, 189 00:08:46,366 --> 00:08:51,166 190 00:08:51,166 --> 00:08:54,966 right, we've already seen that 
two of these angles are equal if 191 00:08:54,966 --> 00:08:55,566 192 00:08:55,566 --> 00:08:58,800 the k values are congruent mod 3. 193 00:08:58,800 --> 00:09:02,833 We don't want to keep doing that. We 
only want a unique set because we want - 194 00:09:02,833 --> 00:09:05,366 because they correspond 
to complex numbers and 195 00:09:05,366 --> 00:09:09,800 for complex numbers, eventually you 
just start to cycle on to yourself. 196 00:09:09,800 --> 00:09:10,866 197 00:09:10,866 --> 00:09:12,933 And when does that 
happen? Every 2 pi multiple, 198 00:09:12,933 --> 00:09:15,533 right? So when I hit 3, I hit a 2 pi multiple, 199 00:09:15,533 --> 00:09:19,633 and so I’m going to go back to when 
k was 0, and so on and so forth. 200 00:09:19,633 --> 00:09:20,533 201 00:09:20,533 --> 00:09:22,566 So, great. So solving for theta gives us this, 202 00:09:22,566 --> 00:09:25,933 and as in class, we have that 
k must be either 0, 1, 2, or 3… 203 00:09:25,933 --> 00:09:29,600 or 0, 1, or 2 to give us the 3 possible solutions. 204 00:09:29,600 --> 00:09:30,966 205 00:09:30,966 --> 00:09:33,200 So hence, 206 00:09:33,200 --> 00:09:35,533 207 00:09:35,533 --> 00:09:38,733 how do I want to say this? So hence, 208 00:09:38,733 --> 00:09:43,733 209 00:09:43,733 --> 00:09:48,233 z must be an element of… 210 00:09:48,233 --> 00:09:49,966 211 00:09:49,966 --> 00:09:52,466 which values? 212 00:09:52,466 --> 00:09:53,533 213 00:09:53,533 --> 00:09:55,533 The sixth… 214 00:09:55,533 --> 00:09:56,233 215 00:09:56,233 --> 00:09:59,933 the sixth root of 5 times e to the power of 216 00:09:59,933 --> 00:10:00,666 217 00:10:00,666 --> 00:10:02,600 i arc- 218 00:10:02,600 --> 00:10:03,433 219 00:10:03,433 --> 00:10:06,700 -tan of minus 2 over 3, 220 00:10:06,700 --> 00:10:08,300 221 00:10:08,300 --> 00:10:10,300 comma, 222 00:10:10,300 --> 00:10:11,233 223 00:10:11,233 --> 00:10:14,000 this is going to look nasty but that's okay. 224 00:10:14,000 --> 00:10:14,966 225 00:10:14,966 --> 00:10:15,166 226 00:10:15,166 --> 00:10:19,500 227 00:10:19,500 --> 00:10:20,866 228 00:10:20,866 --> 00:10:22,900 Alright. 229 00:10:22,900 --> 00:10:27,566 Let's see if I did this correctly. So hence 
z is an element of - oh my God. Okay. 230 00:10:27,566 --> 00:10:28,400 231 00:10:28,400 --> 00:10:33,700 The sixth root of 5 times e to the i arctan of 232 00:10:33,700 --> 00:10:36,066 negative 2 divided by 3, 233 00:10:36,066 --> 00:10:38,266 234 00:10:38,266 --> 00:10:42,900 and then if I add a 2 pi over 3 multiple, 
and if I add a 4 pi over 3 multiple. 235 00:10:42,900 --> 00:10:44,566 236 00:10:44,566 --> 00:10:46,800 Again if you wanted to, if you're a lot 237 00:10:46,800 --> 00:10:49,966 more patient than I'm going to 
be, you could convert these back 238 00:10:49,966 --> 00:10:52,066 to standard form. 239 00:10:52,066 --> 00:10:54,866 It's going to take a little bit of effort since 
we don't know what arctan of minus 2 is. 240 00:10:54,866 --> 00:10:58,500 If you have a calculator, this is very easy. 
You can convert back and forth no problem. 241 00:10:58,500 --> 00:10:59,633 242 00:10:59,633 --> 00:11:02,233 As it is though, I'll leave the answer like this. 243 00:11:02,233 --> 00:11:07,900 244 00:11:07,900 --> 00:11:11,133 Okay, so what else do I want to say? Is 
there anything else I need to say about this? 245 00:11:11,133 --> 00:11:15,300 Notice that the angles do differ by 2 pi 
over 3, and that's what's given by the 246 00:11:15,300 --> 00:11:20,100 Complex nth Roots Theorem, it says once you 
have a root, all the other ones you just need to 247 00:11:20,100 --> 00:11:24,800 rotate by 2 pi divided by n where n was the z to the n part, 248 00:11:24,800 --> 00:11:27,666 and here n is 3. 249 00:11:27,666 --> 00:11:30,333 So that's fantastic. Oh I forgot a little 250 00:11:30,333 --> 00:11:31,133 251 00:11:31,133 --> 00:11:33,466 bracket over here, that's okay. 252 00:11:33,466 --> 00:11:36,500 253 00:11:36,500 --> 00:11:39,733 Yeah that's basically it. I mean, that's how you solve something like this. 254 00:11:39,733 --> 00:11:42,466 You need to be a little bit careful 
when you're doing these, but 255 00:11:42,466 --> 00:11:47,500 again, the only real part where you 
could get a little bit tripped up is this 256 00:11:47,500 --> 00:11:51,933 e to the i alpha part, right, just the 
same thing as in the previous video 257 00:11:51,933 --> 00:11:56,600 about changing to polar coordinates, you might 
have to worry about adding pi or not adding pi. 258 00:11:56,600 --> 00:11:58,500 259 00:11:58,500 --> 00:12:01,133 Here again, we know that two of these 260 00:12:01,133 --> 00:12:04,266 theta values are equal if they 
differ by a multiple of 2 pi, 261 00:12:04,266 --> 00:12:07,766 and so we only need to consider 
three values for k because 262 00:12:07,766 --> 00:12:12,100 after three values of k, so after 0, 1, and 
2, they start to repeat themselves up to 263 00:12:12,100 --> 00:12:16,200 2 pi multiples. You could think 
of this as like modulo 2 pi, 264 00:12:16,200 --> 00:12:19,400 but that would need a lot of explanation 
that we haven't talked about, 265 00:12:19,400 --> 00:12:22,666 but that's the way you can sort 
of think of it in your head. 266 00:12:22,666 --> 00:12:27,133 And otherwise, I think that's 267 00:12:27,133 --> 00:12:29,600 all I want to say. So 268 00:12:29,600 --> 00:12:33,633 like cube roots of 1 minus 2i, not pretty. 269 00:12:33,633 --> 00:12:36,400 If you plug these values in they will work, 270 00:12:36,400 --> 00:12:40,066 ideally, if everything - if I didn't make any typos or anything. 271 00:12:40,066 --> 00:12:41,333 272 00:12:41,333 --> 00:12:43,966 But yeah, hopefully this gives you an idea 
of how to solve these types of problems, 273 00:12:43,966 --> 00:12:46,866 how to set them up and how to attack them. 274 00:12:46,866 --> 00:12:50,066 So that's great. Thank you very 
much for your time and good luck.