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Title: Introduction
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Series: Start Learning Complex Numbers
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Parent Series: Start Learning Mathematics
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YouTube-Title: Start Learning Complex Numbers 1 | Introduction
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Bright video: https://youtu.be/bZSYcIJBMqE
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Dark video: https://youtu.be/bBkz_mNy6O4
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Quiz: Test your knowledge
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Subtitle on GitHub: slc01_sub_eng.srt
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Timestamps (n/a)
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Subtitle in English
1 00:00:00,409 –> 00:00:02,029 Hello and welcome to
2 00:00:02,039 –> 00:00:03,710 start learning complex
3 00:00:03,720 –> 00:00:05,489 numbers, a video course where
4 00:00:05,500 –> 00:00:07,190 I introduce you to the field
5 00:00:07,199 –> 00:00:08,550 of complex numbers.
6 00:00:09,039 –> 00:00:10,369 So I want to show you how
7 00:00:10,380 –> 00:00:12,180 we can construct these numbers
8 00:00:12,189 –> 00:00:13,270 and how we can calculate
9 00:00:13,279 –> 00:00:13,819 with them.
10 00:00:14,229 –> 00:00:15,670 However, before we do that,
11 00:00:15,680 –> 00:00:17,180 I first want to thank all
12 00:00:17,190 –> 00:00:18,569 the nice people that support
13 00:00:18,579 –> 00:00:19,979 this channel on Steady or
14 00:00:19,989 –> 00:00:20,620 paypal.
15 00:00:21,309 –> 00:00:22,510 Now, the starting point for
16 00:00:22,520 –> 00:00:24,069 the complex numbers is the
17 00:00:24,079 –> 00:00:25,760 real number line R
18 00:00:26,309 –> 00:00:26,610 there.
19 00:00:26,620 –> 00:00:28,239 You should know we find all
20 00:00:28,250 –> 00:00:29,809 integers on the number line,
21 00:00:29,819 –> 00:00:31,809 all fractions but also
22 00:00:31,819 –> 00:00:33,209 irrational numbers like the
23 00:00:33,220 –> 00:00:34,169 square root of two.
24 00:00:34,909 –> 00:00:36,639 Moreover, we also know that
25 00:00:36,650 –> 00:00:38,279 we have two operations, the
26 00:00:38,290 –> 00:00:40,180 addition and the multiplication
27 00:00:40,740 –> 00:00:41,939 and we know that they fulfill
28 00:00:41,950 –> 00:00:43,330 a lot of rules such that
29 00:00:43,340 –> 00:00:45,159 we call R a field.
30 00:00:45,590 –> 00:00:47,400 Then we also have an ordering
31 00:00:47,409 –> 00:00:49,259 so we can compare two numbers
32 00:00:49,840 –> 00:00:51,400 and this ordering also
33 00:00:51,409 –> 00:00:53,119 satisfies some rules.
34 00:00:53,169 –> 00:00:54,680 Finally, the last
35 00:00:54,689 –> 00:00:56,520 property is the completeness
36 00:00:56,529 –> 00:00:57,770 which tells us that there
37 00:00:57,779 –> 00:00:59,560 are no holes on this number
38 00:00:59,569 –> 00:00:59,919 line.
39 00:01:00,590 –> 00:01:01,049 OK.
40 00:01:01,060 –> 00:01:02,689 Then let’s discuss what one
41 00:01:02,700 –> 00:01:04,680 can do with these real numbers
42 00:01:04,690 –> 00:01:05,050 here.
43 00:01:05,680 –> 00:01:07,449 For example, we can say
44 00:01:07,480 –> 00:01:09,050 that we are able to solve
45 00:01:09,059 –> 00:01:10,279 a lot of equations
46 00:01:11,089 –> 00:01:12,959 solving X plus five
47 00:01:12,970 –> 00:01:14,860 is equal to one is no
48 00:01:14,870 –> 00:01:15,760 problem at all.
49 00:01:16,260 –> 00:01:17,800 In the same way, we can also
50 00:01:17,809 –> 00:01:19,760 solve X times five
51 00:01:19,769 –> 00:01:20,819 is equal to one.
52 00:01:21,330 –> 00:01:22,709 So here you can see we can
53 00:01:22,720 –> 00:01:24,080 use the property that we
54 00:01:24,089 –> 00:01:24,940 have a field.
55 00:01:25,650 –> 00:01:26,830 Then another equation we
56 00:01:26,839 –> 00:01:28,819 solved is X squared is
57 00:01:28,830 –> 00:01:29,860 equal to two.
58 00:01:30,680 –> 00:01:32,050 This one leads to the square
59 00:01:32,059 –> 00:01:34,050 of two which is in R
60 00:01:34,059 –> 00:01:35,370 by completeness.
61 00:01:35,910 –> 00:01:37,889 However, this does not mean
62 00:01:37,900 –> 00:01:39,209 that we are able to solve
63 00:01:39,220 –> 00:01:40,680 any quadratic equation.
64 00:01:41,099 –> 00:01:42,510 For example, the quadratic
65 00:01:42,519 –> 00:01:44,279 equation X squared is
66 00:01:44,290 –> 00:01:46,089 equal to minus one
67 00:01:46,190 –> 00:01:47,809 has no solutions.
68 00:01:48,500 –> 00:01:50,339 This is not hard to see because
69 00:01:50,349 –> 00:01:52,089 you can show that any
70 00:01:52,099 –> 00:01:53,839 square in the real numbers
71 00:01:53,849 –> 00:01:55,250 is greater or equal than
72 00:01:55,260 –> 00:01:55,800 zero.
73 00:01:56,230 –> 00:01:58,150 And we also need that minus
74 00:01:58,160 –> 00:01:59,830 one is less than
75 00:01:59,839 –> 00:02:00,410 zero.
76 00:02:00,980 –> 00:02:02,459 Here, I can tell you if you
77 00:02:02,470 –> 00:02:03,949 just work with the axioms
78 00:02:03,959 –> 00:02:05,370 of the real numbers, it’s
79 00:02:05,379 –> 00:02:06,989 a good exercise to show
80 00:02:07,000 –> 00:02:08,229 both of these things.
81 00:02:08,988 –> 00:02:10,967 Then this implies that this
82 00:02:10,979 –> 00:02:12,369 simple equation does not
83 00:02:12,378 –> 00:02:13,479 have any solution.
84 00:02:14,240 –> 00:02:15,740 However, for us, it makes
85 00:02:15,750 –> 00:02:17,490 sense that we extend our
86 00:02:17,500 –> 00:02:19,179 numbers such that this
87 00:02:19,190 –> 00:02:20,820 equation has solutions.
88 00:02:21,509 –> 00:02:22,429 The first thing you should
89 00:02:22,440 –> 00:02:24,380 note here is that X squared
90 00:02:24,389 –> 00:02:26,089 is nothing else than X
91 00:02:26,100 –> 00:02:26,889 times X.
92 00:02:27,610 –> 00:02:29,429 I say that because the same
93 00:02:29,440 –> 00:02:31,100 equation with the addition
94 00:02:31,169 –> 00:02:32,270 is solvable.
95 00:02:32,309 –> 00:02:34,070 So surprisingly, the
96 00:02:34,080 –> 00:02:35,869 addition and the multiplication
97 00:02:35,880 –> 00:02:37,570 act differently in R
98 00:02:37,639 –> 00:02:39,070 which also has something
99 00:02:39,080 –> 00:02:40,669 to do with the ordering.
100 00:02:41,190 –> 00:02:42,910 Hence, our idea is to
101 00:02:42,919 –> 00:02:44,880 expand the number set such
102 00:02:44,889 –> 00:02:46,669 that we can solve such equations.
103 00:02:46,679 –> 00:02:47,119 Here.
104 00:02:47,190 –> 00:02:48,759 But we already know that
105 00:02:48,770 –> 00:02:50,369 we can’t conserve all the
106 00:02:50,380 –> 00:02:52,350 properties in R in
107 00:02:52,360 –> 00:02:53,589 order to see what we can
108 00:02:53,600 –> 00:02:54,020 do.
109 00:02:54,039 –> 00:02:55,470 Let’s look at the number
110 00:02:55,479 –> 00:02:56,190 line again.
111 00:02:56,679 –> 00:02:57,860 First, let’s fill in the
112 00:02:57,869 –> 00:02:59,580 number 02
113 00:02:59,649 –> 00:03:00,419 and four.
114 00:03:00,949 –> 00:03:02,300 Now, you know, we can go
115 00:03:02,309 –> 00:03:04,229 from 2 to 4 just by
116 00:03:04,240 –> 00:03:05,880 using multiplication, we
117 00:03:05,889 –> 00:03:07,619 simply multiply by two.
118 00:03:08,139 –> 00:03:09,399 However, the interesting
119 00:03:09,410 –> 00:03:11,080 thing is now that we can
120 00:03:11,089 –> 00:03:12,779 split this big jump
121 00:03:12,789 –> 00:03:14,550 into small jumps,
122 00:03:15,089 –> 00:03:17,070 we just multiply by a number
123 00:03:17,080 –> 00:03:18,399 that is a little bit bigger
124 00:03:18,410 –> 00:03:19,119 than one.
125 00:03:19,630 –> 00:03:21,020 You see the smaller the one
126 00:03:21,029 –> 00:03:22,979 jump is the more jumps we
127 00:03:22,990 –> 00:03:23,339 need.
128 00:03:23,970 –> 00:03:24,460 OK.
129 00:03:24,470 –> 00:03:26,059 The splitting up does not
130 00:03:26,070 –> 00:03:27,630 work when we want to jump
131 00:03:27,639 –> 00:03:28,869 to minus two.
132 00:03:29,500 –> 00:03:31,080 For this big jump, we have
133 00:03:31,089 –> 00:03:32,880 to multiply by minus
134 00:03:32,889 –> 00:03:33,309 one.
135 00:03:34,000 –> 00:03:35,750 Also interesting here, if
136 00:03:35,759 –> 00:03:37,619 we multiply again by minus
137 00:03:37,630 –> 00:03:39,300 one, we get back to
138 00:03:39,309 –> 00:03:39,660 two.
139 00:03:40,770 –> 00:03:42,669 So here you see multiplying
140 00:03:42,679 –> 00:03:44,380 with a negative number is
141 00:03:44,389 –> 00:03:45,820 different than multiplying
142 00:03:45,830 –> 00:03:47,110 with a positive number
143 00:03:47,619 –> 00:03:48,990 because in the first case,
144 00:03:49,000 –> 00:03:50,690 we immediately change the
145 00:03:50,699 –> 00:03:52,059 side of the number line.
146 00:03:52,830 –> 00:03:54,509 Now looking back to the splitting
147 00:03:54,520 –> 00:03:55,710 of the jump for positive
148 00:03:55,720 –> 00:03:57,600 numbers here, we can see
149 00:03:57,610 –> 00:03:58,880 the equation from above
150 00:03:59,740 –> 00:04:01,490 because doing two jumps
151 00:04:01,500 –> 00:04:03,419 with the number X means
152 00:04:03,429 –> 00:04:05,240 we have X squared is
153 00:04:05,250 –> 00:04:06,199 equal to two.
154 00:04:06,970 –> 00:04:08,729 So exactly this equation
155 00:04:08,740 –> 00:04:10,389 where we know we have a solution.
156 00:04:11,470 –> 00:04:12,889 Now you might see the problem.
157 00:04:12,899 –> 00:04:14,500 How can we split up this
158 00:04:14,509 –> 00:04:16,238 jump into two parts?
159 00:04:17,040 –> 00:04:18,510 By looking at this picture,
160 00:04:18,608 –> 00:04:20,369 you might already see we
161 00:04:20,380 –> 00:04:21,839 could just add another
162 00:04:21,850 –> 00:04:23,220 direction for the numbers.
163 00:04:24,059 –> 00:04:25,600 Therefore, the idea is that
164 00:04:25,609 –> 00:04:27,429 we also include a Y
165 00:04:27,440 –> 00:04:27,880 axis.
166 00:04:27,890 –> 00:04:29,760 Now here
167 00:04:29,769 –> 00:04:31,459 jumping from one to
168 00:04:31,470 –> 00:04:33,220 minus one is still
169 00:04:33,230 –> 00:04:35,040 given by multiplying with
170 00:04:35,049 –> 00:04:36,019 minus one.
171 00:04:36,480 –> 00:04:38,000 However, now we are able
172 00:04:38,010 –> 00:04:39,850 to split it up into smaller
173 00:04:39,859 –> 00:04:41,320 parts, for
174 00:04:41,329 –> 00:04:43,239 example, into two parts where
175 00:04:43,250 –> 00:04:45,119 we multiply with the number
176 00:04:45,130 –> 00:04:45,720 X.
177 00:04:46,440 –> 00:04:47,859 And there you see this is
178 00:04:47,869 –> 00:04:49,299 our equation from before
179 00:04:49,350 –> 00:04:51,279 X squared is equal to
180 00:04:51,290 –> 00:04:52,140 minus one.
181 00:04:52,940 –> 00:04:54,339 Hence you see the number
182 00:04:54,350 –> 00:04:56,200 we search for is this
183 00:04:56,209 –> 00:04:57,720 point in the plane.
184 00:04:58,720 –> 00:05:00,100 So we need a new name for
185 00:05:00,109 –> 00:05:00,890 this number.
186 00:05:00,899 –> 00:05:02,529 We call it I.
187 00:05:03,799 –> 00:05:05,500 Now with our picture in mind
188 00:05:05,510 –> 00:05:07,279 this I is now a special
189 00:05:07,290 –> 00:05:08,739 number because when we
190 00:05:08,750 –> 00:05:10,170 multiply I with
191 00:05:10,179 –> 00:05:11,820 itself, we get out the
192 00:05:11,829 –> 00:05:13,809 jump minus one or
193 00:05:13,820 –> 00:05:14,290 to put it.
194 00:05:14,299 –> 00:05:15,640 In other words, I
195 00:05:15,649 –> 00:05:17,519 squared is equal to
196 00:05:17,529 –> 00:05:18,420 minus one.
197 00:05:19,329 –> 00:05:20,910 And this is now the important
198 00:05:20,920 –> 00:05:22,670 formula we should remember
199 00:05:22,679 –> 00:05:23,630 all the time.
200 00:05:24,600 –> 00:05:24,869 OK.
201 00:05:24,880 –> 00:05:26,760 With all this, we have some
202 00:05:26,769 –> 00:05:28,510 idea what our goal is to
203 00:05:28,519 –> 00:05:29,970 expand the number set.
204 00:05:30,100 –> 00:05:31,529 And how we can reach that
205 00:05:31,540 –> 00:05:33,489 goal extending the
206 00:05:33,500 –> 00:05:35,109 number line into a second
207 00:05:35,119 –> 00:05:36,690 direction is of course a
208 00:05:36,700 –> 00:05:38,049 completely new idea.
209 00:05:38,910 –> 00:05:40,149 Therefore, we really have
210 00:05:40,160 –> 00:05:41,670 to do this in a formal way,
211 00:05:41,700 –> 00:05:43,329 what we can do in the next
212 00:05:43,339 –> 00:05:45,239 video in this way.
213 00:05:45,250 –> 00:05:46,649 I hope I see you there and
214 00:05:46,660 –> 00:05:47,549 have a nice day.
215 00:05:47,700 –> 00:05:48,420 Bye.
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Quiz Content
Q1: This is how the quizzes for the topics look like. You should start with the next video :)
A1: Yes!
A2: No!
A3: Never!
A4: Why???
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Last update: 2024-10
