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Title: Complex numbers: Solving equations - with example
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YouTube-Title: Complex Numbers: Solving Equations (with Example)
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Subtitle in English
1 00:00:00.795 –> 00:00:02.705 Hello and welcome to a new video.
2 00:00:03.205 –> 00:00:06.225 And first let me thank all the very nice people
3 00:00:06.455 –> 00:00:08.625 that support this channel on Steady.
4 00:00:10.435 –> 00:00:14.405 Today’s topic is solving equations with complex numbers.
5 00:00:15.785 –> 00:00:17.965 In order to do this, you have to know
6 00:00:18.265 –> 00:00:22.045 how you can solve simple equations in the complex numbers.
7 00:00:23.185 –> 00:00:26.805 And when I say simple, I mean equations of the form,
8 00:00:27.025 –> 00:00:30.405 For example, z to the power of three is equal
9 00:00:30.625 –> 00:00:33.205 to a fixed complex number on the right hand side.
10 00:00:33.905 –> 00:00:36.965 For example, two plus two i
11 00:00:38.145 –> 00:00:39.405 In a descriptive way,
12 00:00:39.585 –> 00:00:42.845 you could say solving this equation means taking the third
13 00:00:42.995 –> 00:00:44.605 root on the right hand side.
14 00:00:45.315 –> 00:00:49.285 Therefore, often people talk about extracting roots when
15 00:00:49.285 –> 00:00:50.325 solving equations.
16 00:00:51.395 –> 00:00:54.485 Okay, so all the equations we consider here are of this form
17 00:00:54.665 –> 00:00:57.685 so z to given natural power n
18 00:00:58.065 –> 00:01:00.565 and on the right hand side a fixed complex number,
19 00:01:01.135 –> 00:01:05.085 which means a fixed real part and a fixed imaginary part.
20 00:01:05.995 –> 00:01:07.645 Obviously this is the first step.
21 00:01:07.865 –> 00:01:10.885 You first have to know how to solve such equations
22 00:01:10.985 –> 00:01:13.845 before you can solve more complicated equations.
23 00:01:15.435 –> 00:01:17.565 Okay, so this is the program for today.
24 00:01:17.835 –> 00:01:19.445 Give me all complex numbers.
25 00:01:19.745 –> 00:01:23.125 z that solve this equation here first.
26 00:01:23.225 –> 00:01:25.125 Now let’s go back to the real numbers.
27 00:01:25.665 –> 00:01:27.725 How can you solve an equation maybe
28 00:01:27.725 –> 00:01:31.805 of the form x squared equals to a fixed real number
29 00:01:32.265 –> 00:01:34.565 and maybe the easiest one would be four here.
30 00:01:35.585 –> 00:01:37.485 Now you learn, for example, in school
31 00:01:37.635 –> 00:01:40.085 that we have indeed two solutions here,
32 00:01:40.885 –> 00:01:44.845 a positive solution I could call x one, which is two,
33 00:01:45.305 –> 00:01:47.965 and a negative solution which we call x two,
34 00:01:48.455 –> 00:01:49.925 which is minus two.
35 00:01:51.265 –> 00:01:53.645 So exactly two solutions.
36 00:01:54.105 –> 00:01:55.205 And now let’s see
37 00:01:55.265 –> 00:01:57.885 how this looks like in the real number line.
38 00:01:58.545 –> 00:02:00.805 The line represents all real numbers
39 00:02:01.145 –> 00:02:03.245 and now I can put zero maybe here
40 00:02:04.425 –> 00:02:08.045 and now I see that my two solutions here x two
41 00:02:08.465 –> 00:02:12.565 and here x one have the same distance from zero.
42 00:02:13.955 –> 00:02:17.605 Also, you recognize the order we have a smaller solution
43 00:02:17.625 –> 00:02:18.765 and a bigger solution.
44 00:02:19.065 –> 00:02:21.165 In other words, we have a positive solution
45 00:02:21.265 –> 00:02:22.565 and a negative solution.
46 00:02:23.545 –> 00:02:24.965 And now you already know
47 00:02:25.195 –> 00:02:29.005 that we can’t have this nice order in the complex numbers
48 00:02:29.145 –> 00:02:34.045 as well because here now we have the whole complex plane.
49 00:02:35.585 –> 00:02:37.605 We have the real part on the x axis
50 00:02:37.865 –> 00:02:40.885 and the imaginary part on the Y axis.
51 00:02:41.625 –> 00:02:44.645 And the first part we immediately see is when we have zero
52 00:02:44.735 –> 00:02:48.485 again here in the middle that we have more numbers
53 00:02:48.635 –> 00:02:50.845 that have the same distance from zero.
54 00:02:51.425 –> 00:02:55.325 For example, for distance two we only have two numbers in
55 00:02:55.325 –> 00:02:58.805 the real number line, but immediately infinitely many
56 00:02:59.185 –> 00:03:00.285 In the complex plane
57 00:03:00.355 –> 00:03:03.405 because it describes a circle around zero
58 00:03:03.755 –> 00:03:08.325 With radius two also we don’t have an easy distinction
59 00:03:08.325 –> 00:03:10.965 between left and right as in a number line.
60 00:03:11.545 –> 00:03:13.565 So we can’t say we have a positive
61 00:03:13.565 –> 00:03:15.725 or negative solution here in a complex plane,
62 00:03:16.855 –> 00:03:19.605 hence we need another picture in the complex plane
63 00:03:19.665 –> 00:03:21.205 and that is what I will do later.
64 00:03:22.295 –> 00:03:24.205 First, I want some visualization
65 00:03:24.345 –> 00:03:25.805 for this complex number here.
66 00:03:25.905 –> 00:03:29.765 So I put this in the complex plane, so we have two here
67 00:03:30.625 –> 00:03:32.365 and the same way we have two here.
68 00:03:33.355 –> 00:03:35.965 This means the complex number with real part two
69 00:03:36.025 –> 00:03:37.925 and imaginary part two lies here.
70 00:03:38.105 –> 00:03:40.205 So we can visualize this with an error
71 00:03:40.315 –> 00:03:41.845 that goes to this point.
72 00:03:42.905 –> 00:03:46.885 One important thing I can immediately tell you now is if you
73 00:03:46.995 –> 00:03:49.885 have an exponent, so a power of z here,
74 00:03:50.115 –> 00:03:54.205 then it’s always helpful to rewrite this number on the right.
75 00:03:54.865 –> 00:03:57.245 You should rewrite it in the polar form
76 00:03:57.465 –> 00:03:59.725 or better in the exponential form.
77 00:04:00.545 –> 00:04:03.205 You will see soon that this is indeed very helpful.
78 00:04:04.115 –> 00:04:06.485 Okay, so we describe our complex number here
79 00:04:06.585 –> 00:04:07.765 now with two things.
80 00:04:08.815 –> 00:04:10.845 First, the distance from zero,
81 00:04:10.895 –> 00:04:12.565 which means the absolute value
82 00:04:13.185 –> 00:04:17.405 and second the angle from x axis
83 00:04:17.545 –> 00:04:18.805 to the vector itself.
84 00:04:19.705 –> 00:04:21.405 In order to calculate these two things,
85 00:04:21.465 –> 00:04:24.605 you can just draw the right angle triangle here
86 00:04:25.985 –> 00:04:28.645 and then you know you can calculate the length of this side
87 00:04:28.785 –> 00:04:30.765 by using Pythagoras theorem
88 00:04:31.665 –> 00:04:35.605 and to calculate the angle here, which I call phi now,
89 00:04:36.545 –> 00:04:38.685 you can use in trigonometric function.
90 00:04:39.475 –> 00:04:41.765 Most of the time one uses the tangent
91 00:04:42.715 –> 00:04:46.045 because then you can immediately use the Y coordinate
92 00:04:46.305 –> 00:04:47.805 and the X coordinate.
93 00:04:48.395 –> 00:04:51.565 However, in our example, we don’t have to calculate at all
94 00:04:51.635 –> 00:04:53.605 because we immediately see
95 00:04:53.875 –> 00:04:56.285 that we bisect the right angle here.
96 00:04:57.105 –> 00:05:01.565 So we cut pi over two in half, which means we get phi
97 00:05:01.705 –> 00:05:05.125 as pi over four, which means
98 00:05:05.655 –> 00:05:08.445 45 degree. just as well,
99 00:05:08.705 –> 00:05:11.845 We also get our absolute value, which I called r
100 00:05:12.945 –> 00:05:16.885 by using Pythagoras we know now r squared is equal
101 00:05:16.885 –> 00:05:21.445 to two squared plus two squared, which is eight.
102 00:05:22.825 –> 00:05:26.725 Now r is of course the positive solution here, so r is equal
103 00:05:27.105 –> 00:05:29.125 to the square root of eight.
104 00:05:30.385 –> 00:05:31.925 Now to summarize this,
105 00:05:32.305 –> 00:05:36.085 we can rewrite our complex number two plus two i in
106 00:05:36.085 –> 00:05:37.125 an exponential form.
107 00:05:38.065 –> 00:05:42.205 We have now number equal to first the absolute value r
108 00:05:42.705 –> 00:05:45.405 and then times e to the power I,
109 00:05:45.785 –> 00:05:47.525 and then comes the angle phi,
110 00:05:47.655 –> 00:05:50.445 which is now a case PI over four.
111 00:05:52.505 –> 00:05:54.795 Okay, so this is the exponential form
112 00:05:55.135 –> 00:05:58.525 and this is always helpful when you are consider
113 00:05:59.145 –> 00:06:00.325 powers in an equation
114 00:06:01.075 –> 00:06:02.405 because calculating
115 00:06:02.405 –> 00:06:06.245 with exponents is much easier when you have the exponential
116 00:06:06.525 –> 00:06:08.165 function here in this case.
117 00:06:08.235 –> 00:06:11.125 Then you can just multiply the exponents
118 00:06:11.425 –> 00:06:14.685 and then it does not matter how big this power is indeed
119 00:06:14.685 –> 00:06:18.405 because you just have to multiply them. well
120 00:06:18.405 –> 00:06:21.365 and now we can solve our equation from the beginning.
121 00:06:22.095 –> 00:06:23.685 Still we have z cubed,
122 00:06:23.745 –> 00:06:26.565 but the right hand side is now much easier.
123 00:06:27.745 –> 00:06:29.365 Now my first advice here is
124 00:06:29.395 –> 00:06:31.725 that you should rephrase that a little bit.
125 00:06:32.785 –> 00:06:34.525 Namely you used the fact
126 00:06:34.795 –> 00:06:37.085 that we don’t change our complex number
127 00:06:37.625 –> 00:06:39.325 if we add a full turn,
128 00:06:41.295 –> 00:06:44.525 which means we add two pi to our angle here,
129 00:06:44.905 –> 00:06:46.245 two pi is the full circle
130 00:06:46.705 –> 00:06:50.765 so we end up at the same point in the complex plane.
131 00:06:51.065 –> 00:06:52.925 Now we still get to this point
132 00:06:53.305 –> 00:06:55.205 by writing this complex number
133 00:06:56.105 –> 00:06:59.005 or in other words this is just another represenation
134 00:06:59.145 –> 00:07:01.605 of the same complex number from the beginning.
135 00:07:02.075 –> 00:07:05.165 However, at this point we can do even more
136 00:07:05.315 –> 00:07:09.965 because if we can add one turn we also can add two turns
137 00:07:10.345 –> 00:07:12.645 and even three and so on.
138 00:07:12.785 –> 00:07:15.125 We still end at the same complex number
139 00:07:15.225 –> 00:07:16.325 in the complex plane.
140 00:07:17.145 –> 00:07:21.885 In short, we can add K turns, so K times two pi,
141 00:07:22.675 –> 00:07:25.485 therefore we can choose as many turns as we want.
142 00:07:25.585 –> 00:07:28.765 So we can choose our K maybe first as zero,
143 00:07:28.895 –> 00:07:32.885 which is this representation from before or one turn
144 00:07:33.065 –> 00:07:36.125 or two turns or three turns and so on.
145 00:07:37.555 –> 00:07:39.205 Okay, now before we go further,
146 00:07:39.465 –> 00:07:41.325 we should first answer the question,
147 00:07:41.905 –> 00:07:46.365 how many solutions can this equation have there is helpful
148 00:07:46.505 –> 00:07:49.765 to first look back at our real example from
149 00:07:49.785 –> 00:07:53.525 before there we had two solutions
150 00:07:53.625 –> 00:07:54.925 for a quadratic equation.
151 00:07:55.865 –> 00:07:59.325 Of course, we also can consider the equation in the complex
152 00:07:59.325 –> 00:08:03.525 numbers, which means we have to add the imaginary axis
153 00:08:03.825 –> 00:08:05.325 to our picture here.
154 00:08:06.465 –> 00:08:09.755 However, we still stay at two solutions
155 00:08:10.335 –> 00:08:13.755 and what we also find is that all solutions have
156 00:08:13.775 –> 00:08:16.395 to lie on the circle around zero,
157 00:08:16.685 –> 00:08:18.075 which I told you at the beginning.
158 00:08:19.215 –> 00:08:21.595 So this is now the circle with radius two
159 00:08:21.895 –> 00:08:24.555 around zero in the complex plane.
160 00:08:25.205 –> 00:08:27.435 Hence if you search for real solutions,
161 00:08:27.695 –> 00:08:30.955 you only have two possible points on the real line,
162 00:08:31.175 –> 00:08:33.355 namely the intersections with the circle
163 00:08:33.545 –> 00:08:35.475 with the line, okay?
164 00:08:35.575 –> 00:08:38.395 One result is all complex solutions have
165 00:08:38.395 –> 00:08:42.035 to lie on the circle with the given fixed radius.
166 00:08:42.935 –> 00:08:44.595 And the other result is
167 00:08:44.785 –> 00:08:49.355 that the solutions divide this circle in same size pieces.
168 00:08:50.495 –> 00:08:52.355 In this example you see this very easily.
169 00:08:52.735 –> 00:08:54.795 So this is the first solution, so the arrow
170 00:08:55.185 –> 00:08:58.845 to the right. Then comes one half turn
171 00:08:59.105 –> 00:09:01.685 and then we find the second arrow to the left.
172 00:09:02.225 –> 00:09:05.125 So half a circle here and then half a circle here.
173 00:09:05.975 –> 00:09:09.005 Hence this is what you can remember if you have the quadratic
174 00:09:09.325 –> 00:09:13.845 equation here to the circle is divided in two half
175 00:09:13.875 –> 00:09:15.885 circles, so it’s cut in half.
176 00:09:16.665 –> 00:09:18.725 Now on the our case with power three,
177 00:09:19.105 –> 00:09:22.405 we know the circle is cut in three equal parts
178 00:09:23.465 –> 00:09:25.805 in the same way, if you have a power four,
179 00:09:26.105 –> 00:09:29.445 you cut the circle in four equal parts and so on.
180 00:09:30.295 –> 00:09:34.045 Hence, please remember the exponent here gives you the
181 00:09:34.045 –> 00:09:35.925 number of solutions on this circle.
182 00:09:36.225 –> 00:09:38.365 And so when we’re here, it’s not so hard to see
183 00:09:38.825 –> 00:09:41.365 and let’s draw the circle in this case again.
184 00:09:42.305 –> 00:09:44.165 So maybe let’s put in the assumption
185 00:09:44.195 –> 00:09:46.165 that we have found one solution.
186 00:09:46.345 –> 00:09:49.165 So maybe this point here, this means
187 00:09:49.165 –> 00:09:52.445 that this point solves this equation here
188 00:09:53.505 –> 00:09:56.605 and now we find also all the other solutions,
189 00:09:57.025 –> 00:10:01.405 namely there are two, because we know the circle is divided
190 00:10:01.595 –> 00:10:06.525 into three equal parts, which means if we go 120 degree
191 00:10:07.115 –> 00:10:10.005 here to the left, we find this arrow
192 00:10:12.755 –> 00:10:16.735 and then we go 120 degree in the positive direction here
193 00:10:17.315 –> 00:10:20.775 and we find this arrow, hence this point.
194 00:10:20.875 –> 00:10:24.535 And this point is also a solution if we now go further.
195 00:10:24.635 –> 00:10:27.375 So if you add again 120 degree,
196 00:10:27.835 –> 00:10:29.895 you end up at the first solution.
197 00:10:30.565 –> 00:10:34.615 Here you see the uniform distribution on the circle is a
198 00:10:34.615 –> 00:10:38.135 general result that you can always use when you want
199 00:10:38.135 –> 00:10:41.815 to generate all the solutions from one known solution.
200 00:10:42.635 –> 00:10:45.575 In short, if you have one, you get all the others back then
201 00:10:46.835 –> 00:10:47.975 and you will see that.
202 00:10:48.075 –> 00:10:52.095 Now in our calculation here on the left, in a formal way,
203 00:10:52.355 –> 00:10:56.695 we now use the exponent one over three on both sides.
204 00:10:57.125 –> 00:10:59.855 Yeah. And then we find indeed all the solutions
205 00:11:00.595 –> 00:11:03.895 and I will count the solutions with an index K.
206 00:11:04.315 –> 00:11:08.615 So z index K is equal to: first,
207 00:11:08.675 –> 00:11:12.415 we have the square root of eight to the power one over three,
208 00:11:13.155 –> 00:11:15.455 and then the exponential function
209 00:11:15.595 –> 00:11:16.935 to the power one over three.
210 00:11:17.485 –> 00:11:22.295 However, this simply means that we multiply this exponent
211 00:11:22.295 –> 00:11:23.735 with one over three.
212 00:11:24.925 –> 00:11:27.935 Okay, so let’s rephrase that a little bit even more
213 00:11:28.315 –> 00:11:31.735 so we can write the eight to the power of one over six.
214 00:11:31.955 –> 00:11:34.575 So combine the square word with the one over three.
215 00:11:35.435 –> 00:11:38.215 And the same way here we do the multiplication.
216 00:11:38.215 –> 00:11:41.295 So we have PI over 12 plus,
217 00:11:41.915 –> 00:11:46.215 and there we have two over three pi times K
218 00:11:48.005 –> 00:11:51.495 well, and here you should immediately recognize our two over
219 00:11:51.505 –> 00:11:55.605 three pi is exactly our 120 degree.
220 00:11:55.945 –> 00:11:58.485 well, we saw earlier in our circle here,
221 00:11:59.945 –> 00:12:02.885 and maybe you also recognize our first solution here.
222 00:12:02.995 –> 00:12:06.045 This is pi over 12, so 15 degree.
223 00:12:06.385 –> 00:12:10.595 Uh, this is the angle here and this makes totally sense
224 00:12:10.745 –> 00:12:15.075 because now if you go to the power three, you have
225 00:12:15.075 –> 00:12:17.275 to add up the angle three times.
226 00:12:18.015 –> 00:12:22.595 So three times this angle you end up by 45 degree,
227 00:12:22.645 –> 00:12:24.835 which is our original angle from
228 00:12:24.835 –> 00:12:25.875 the number at the beginning.
229 00:12:27.045 –> 00:12:29.745 So this is indeed our first solution
230 00:12:30.765 –> 00:12:34.585 and we find that if we put K equal to zero,
231 00:12:35.535 –> 00:12:39.345 then we find the second solution if we add 120 degree,
232 00:12:39.605 –> 00:12:42.145 so this would be K equal to one,
233 00:12:42.805 –> 00:12:44.865 and if we put K equal to two,
234 00:12:45.405 –> 00:12:48.625 we find the third solution we could go further.
235 00:12:48.885 –> 00:12:51.825 But then you see, we don’t find any new numbers and
236 00:12:51.825 –> 00:12:54.705 therefore we can stop here with K equals to two.
237 00:12:55.695 –> 00:12:58.585 Okay, so all this we should put in a
238 00:12:58.745 –> 00:12:59.985 tidy solution in the end.
239 00:13:00.605 –> 00:13:03.865 So the first solution z zero is eight
240 00:13:03.865 –> 00:13:08.105 to the power one over six, which is just the square root of two,
241 00:13:08.395 –> 00:13:09.985 right, is what you can calculate.
242 00:13:10.325 –> 00:13:12.305 But remember this is just the radius
243 00:13:12.485 –> 00:13:14.025 of the whole circle here.
244 00:13:15.365 –> 00:13:18.785 And as the first solution, we have the angle PI over 12
245 00:13:19.375 –> 00:13:22.865 plus zero for the second solution z one,
246 00:13:23.645 –> 00:13:25.905 we have still the same absolute value,
247 00:13:26.365 –> 00:13:28.745 but then e to power i
248 00:13:29.565 –> 00:13:33.145 and now PI over 12 plus two over three pi,
249 00:13:33.925 –> 00:13:36.865 but this one is just eight over 12 pi.
250 00:13:37.085 –> 00:13:40.625 So in sum we have nine over 12 pi.
251 00:13:42.215 –> 00:13:44.545 Then the same holds for the third solution,
252 00:13:45.035 –> 00:13:46.625 still the same absolute value,
253 00:13:46.925 –> 00:13:51.505 but now we have again to add eight over 12 pi,
254 00:13:52.035 –> 00:13:54.545 which means we have 17 over 12.
255 00:13:57.215 –> 00:13:58.315 Now using some colors,
256 00:13:58.655 –> 00:14:01.275 you now see the solutions in the complex plane
257 00:14:01.495 –> 00:14:03.755 and first solution with the small angle.
258 00:14:04.065 –> 00:14:07.995 Then we add up the 120 degree, find the second solution,
259 00:14:08.335 –> 00:14:12.755 and the same way the third solution, I strongly advise you
260 00:14:13.335 –> 00:14:15.875 to do that after your calculation.
261 00:14:15.975 –> 00:14:19.035 So draw a short sketch with your solutions
262 00:14:19.035 –> 00:14:23.835 to see if the solutions divide the circle indeed into equal
263 00:14:23.835 –> 00:14:25.795 parts, okay?
264 00:14:25.855 –> 00:14:28.235 And the same way, the same thing works
265 00:14:28.255 –> 00:14:30.115 for higher powers, for example.
266 00:14:30.415 –> 00:14:33.835 Now you should know how to solve z to the power four
267 00:14:34.495 –> 00:14:36.995 equals to a complex number on the right.
268 00:14:37.175 –> 00:14:40.315 So for example, three plus two i,
269 00:14:41.525 –> 00:14:44.675 first rewrite this number in the exponential form
270 00:14:44.935 –> 00:14:46.955 and then do this scheme here,
271 00:14:48.135 –> 00:14:50.635 and then you find the radius and
272 00:14:50.855 –> 00:14:53.485 The first angle for the first solution.
273 00:14:54.225 –> 00:14:58.645 Now, because you know the power, you already know the angle
274 00:14:58.835 –> 00:15:00.205 between two solutions.
275 00:15:00.825 –> 00:15:03.365 You know you need four equal parts in a circle.
276 00:15:03.705 –> 00:15:06.165 So the angle is 90 degree.
277 00:15:07.395 –> 00:15:09.405 This means with the one exception
278 00:15:09.515 –> 00:15:12.165 that the right hand side is equal to zero.
279 00:15:12.785 –> 00:15:15.845 You always find exactly four solutions
280 00:15:15.985 –> 00:15:18.045 for your equation on this circle.
281 00:15:18.985 –> 00:15:21.605 Or in other words, if the right hand side is not zero,
282 00:15:22.225 –> 00:15:26.125 the power here gives you exactly the number of solutions.
283 00:15:27.065 –> 00:15:29.605 And please note you don’t have such a nice
284 00:15:29.625 –> 00:15:30.965 result for the real numbers.
285 00:15:31.795 –> 00:15:35.365 However, the good thing is you also see the real solutions
286 00:15:35.395 –> 00:15:37.645 then in your picture if there are any.
287 00:15:38.715 –> 00:15:42.845 Okay? Now I strongly advise you to do some more calculations
288 00:15:43.155 –> 00:15:44.325 with the complex numbers
289 00:15:44.745 –> 00:15:49.645 and then you really know how to solve such easy equations
290 00:15:49.715 –> 00:15:50.725 with complex numbers.
291 00:15:51.905 –> 00:15:55.645 And after this, you can also solve more complicated
292 00:15:56.005 –> 00:15:57.565 equations in the complex plane.
293 00:15:58.315 –> 00:16:02.245 Then Thank you for listening and see you next time. Bye.
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Last update: 2025-10
