• Title: Introduction

  • Series: Start Learning Complex Numbers

  • Parent Series: Start Learning Mathematics

  • YouTube-Title: Start Learning Complex Numbers - Part 1 - Introduction

  • Bright video: https://youtu.be/bZSYcIJBMqE

  • Dark video: https://youtu.be/bBkz_mNy6O4

  • Quiz: Test your knowledge

  • PDF: Download PDF version of the bright video

  • Print-PDF: Download printable PDF version

  • Thumbnail (bright): Download PNG

  • Thumbnail (dark): Download PNG

  • Subtitle on GitHub: slc01_sub_eng.srt

  • Timestamps (n/a)
  • 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.

  • 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???

  • Back to overview page