1 00:00:00,300 --> 00:00:07,530 All right, so let us now come to the solutions to the exercises so he can see once again that three 2 00:00:07,530 --> 00:00:08,160 tasks. 3 00:00:08,910 --> 00:00:14,670 So if you have not done so, please stop the video and please take a sheet of paper and calculate these 4 00:00:14,670 --> 00:00:16,320 tasks one, two and three. 5 00:00:17,700 --> 00:00:19,380 So now let's come to the solutions. 6 00:00:20,760 --> 00:00:26,820 So our first task was to calculate to some difference and a product of two complex numbers. 7 00:00:27,420 --> 00:00:29,220 So that was pretty easy, I hope. 8 00:00:29,730 --> 00:00:31,690 So let's start with task number eight. 9 00:00:32,160 --> 00:00:34,040 So here we have to calculate the sum. 10 00:00:34,530 --> 00:00:45,540 So we have to calculate three plus two I plus minus one, plus four I and that is of course that you 11 00:00:45,540 --> 00:00:50,130 have to add the real parts first two and then the imaginary parts. 12 00:00:50,160 --> 00:01:00,720 This is six I then four B we have here the same thing and then we have to subtract minus one and four. 13 00:01:00,720 --> 00:01:08,910 I so we get for the real part four and for the imaginary part we get minus two I. 14 00:01:10,750 --> 00:01:18,190 Now for the product, you have to calculate three different terms, so you have to. 15 00:01:20,630 --> 00:01:27,590 Multiplied at two real paths to imaginary parts and to two mixed terms is what we get here is minus 16 00:01:27,710 --> 00:01:36,860 three, then we get three times for EI, which is 12 I and we have minus two EI for the other mixed 17 00:01:36,860 --> 00:01:44,540 term and for the imaginary product we have two times fours eight and items ie is minus one, so we get 18 00:01:44,540 --> 00:01:45,410 minus eight. 19 00:01:45,930 --> 00:01:48,350 So in total we get minus 11. 20 00:01:49,730 --> 00:01:51,940 Plus 10 I. 21 00:01:53,780 --> 00:01:58,230 OK, so while the first task was really easy, the second task is a bit more difficult. 22 00:01:58,700 --> 00:02:00,490 So we have again two numbers. 23 00:02:00,860 --> 00:02:06,550 First of all, we have here one number that is already separated into the real and the imaginary part. 24 00:02:08,000 --> 00:02:13,070 And then we have here a number that is represented in this oilor representation. 25 00:02:14,300 --> 00:02:17,090 So each of the power of AI and then something. 26 00:02:17,960 --> 00:02:22,850 So let's start and calculate, first of all, the complex conjugate. 27 00:02:24,500 --> 00:02:28,250 So this is C one star and this is just minus one. 28 00:02:28,640 --> 00:02:30,170 Minus two I. 29 00:02:31,460 --> 00:02:39,920 And for the other number, in the earlier representation, it is said to Star is equal to two times 30 00:02:40,520 --> 00:02:43,290 E to the minus five, PI over eight. 31 00:02:43,730 --> 00:02:46,250 This is because all of the numbers here are real numbers. 32 00:02:46,640 --> 00:02:51,800 And just this eye is a complex number and we have to calculate the complex conjugate. 33 00:02:52,700 --> 00:02:58,790 Also, another trick that I can show you here is you can write down that this is equal to two times 34 00:02:59,330 --> 00:03:06,620 cosine PI over eight plus I sign PI over eight. 35 00:03:10,190 --> 00:03:16,550 And so now you can see we have already separated this into the real and into the imaginary part, and 36 00:03:16,550 --> 00:03:26,870 this means we could write down that C two square, C two star is equal to two times cosine PI over eight 37 00:03:27,710 --> 00:03:28,460 minus. 38 00:03:28,460 --> 00:03:31,520 I sign PI over eight. 39 00:03:32,630 --> 00:03:39,290 And now, since cosine is an even function and sine is an odd function, we can also write that this 40 00:03:39,290 --> 00:03:49,020 is equal to cosine minus pi over eight plus I sign minus Pi over eight. 41 00:03:50,240 --> 00:03:54,230 And now this you can see is really what we have written down here with the minus sign. 42 00:03:55,430 --> 00:03:55,930 All right. 43 00:03:56,180 --> 00:04:02,270 So this little trick here I have only shown you to to prove that this is really correct, in case you 44 00:04:02,270 --> 00:04:03,080 didn't believe me. 45 00:04:03,350 --> 00:04:08,870 But also it helps us for the next task where we have to calculate the real part of the equation and 46 00:04:08,870 --> 00:04:10,880 see two and also the imaginary part. 47 00:04:11,600 --> 00:04:18,280 So here, obviously, it's very easy to have to write down these two numbers and please be careful. 48 00:04:18,290 --> 00:04:20,720 The imaginary part is a real number. 49 00:04:20,730 --> 00:04:27,380 So it's minus two and not sorry, it's of course, plus two because we want to calculate it of this 50 00:04:27,380 --> 00:04:27,740 number. 51 00:04:28,070 --> 00:04:31,430 That's plus two and not plus two eye. 52 00:04:31,970 --> 00:04:37,200 The eye is just the imaginary number, but the imaginary part is just a real number. 53 00:04:38,630 --> 00:04:39,020 All right. 54 00:04:39,020 --> 00:04:47,510 So here we have a real part of C two is equal to two times cosine of PI over eight. 55 00:04:48,050 --> 00:04:54,440 And the imaginary part of C two is two times sine PI over eight. 56 00:04:57,170 --> 00:04:57,710 All right. 57 00:04:57,710 --> 00:05:00,540 Now we want to calculate the absolute value. 58 00:05:01,970 --> 00:05:06,320 So this is the square root of the real part squared plus the imaginary part square. 59 00:05:06,360 --> 00:05:09,020 So this is one squared plus two square. 60 00:05:09,440 --> 00:05:11,690 Square root is a square root of five. 61 00:05:12,200 --> 00:05:12,890 Very easy. 62 00:05:13,280 --> 00:05:16,730 And here we have absolute value of C two. 63 00:05:17,300 --> 00:05:20,270 And here again, I want to show you two methods. 64 00:05:20,270 --> 00:05:23,840 First of all, the simple method which you should use. 65 00:05:24,590 --> 00:05:31,190 This is just absolute value of two times absolute value of E to the power of I pi over eight. 66 00:05:32,540 --> 00:05:36,360 And so the absolute value of each of the eyes, something is always one. 67 00:05:36,380 --> 00:05:38,300 So the absolute value is two. 68 00:05:39,060 --> 00:05:44,960 I mean, that's the whole trick and the whole purpose of writing it down in such an hourly the representation 69 00:05:44,990 --> 00:05:47,050 that you write, the absolute value in front of it. 70 00:05:47,990 --> 00:05:54,570 But if you didn't believe me, you could also write down that this is equal to the square root of. 71 00:05:55,220 --> 00:05:58,370 And now we have to use our real and imaginary parts. 72 00:05:58,820 --> 00:06:09,710 So we have two square times cosine PI over eight squared plus two square. 73 00:06:10,160 --> 00:06:13,520 Right down with these brackets sine square. 74 00:06:14,180 --> 00:06:15,080 Pi over eight. 75 00:06:17,980 --> 00:06:19,030 And, of course. 76 00:06:20,000 --> 00:06:25,900 If you have a cosine square plus assigned square, no matter which argument you have, it's always one. 77 00:06:26,300 --> 00:06:32,590 So you have that this is equal to two square square root is equal to two. 78 00:06:33,350 --> 00:06:34,310 So it's the same thing. 79 00:06:35,540 --> 00:06:38,330 OK, and then the only thing that's left is the reciprocal. 80 00:06:38,340 --> 00:06:40,280 So we have to calculate one of as you want. 81 00:06:41,540 --> 00:06:44,870 So here we have or we we can do multiple things. 82 00:06:44,870 --> 00:06:49,550 But what I like to do is multiply by one. 83 00:06:49,910 --> 00:06:52,370 So this is equal to one. 84 00:06:53,170 --> 00:07:00,620 And now what is written down here is the one star divided by absolute value of C one square. 85 00:07:03,210 --> 00:07:07,710 And we know, what, seven squares and we know what city one stars, so we can just write it down from 86 00:07:07,710 --> 00:07:14,510 above minus one, minus two, I divided by square root of five square, which is five. 87 00:07:15,120 --> 00:07:24,510 So this is equal to minus one five as the real part and minus two over five I or minus two or five as 88 00:07:24,510 --> 00:07:25,410 the imaginary part. 89 00:07:26,460 --> 00:07:29,430 And here we have one over Z. 90 00:07:29,700 --> 00:07:32,730 Two is equal to. 91 00:07:36,860 --> 00:07:45,970 Yeah, here we can do it really simple, we can just ride one of A to E to the I Pi over eight and so 92 00:07:45,980 --> 00:07:51,640 this is one over two times one over here to the I pi over eight. 93 00:07:52,310 --> 00:07:59,810 And if you know the rules for exponential functions, you know that this is one of a two E to the minus. 94 00:08:00,230 --> 00:08:02,540 I pi over it. 95 00:08:03,330 --> 00:08:09,170 So as you can see these, this is, are the representation has some advantages. 96 00:08:09,170 --> 00:08:15,590 For example, you can quite easily calculate the absolute value and the inverse, but it also has disadvantages. 97 00:08:15,620 --> 00:08:20,330 It was a bit more difficult to calculate the real and the imaginary parts compared to the left hand 98 00:08:20,330 --> 00:08:20,720 side. 99 00:08:21,980 --> 00:08:28,760 OK, and let's come to the last part where you should practice drawing these complex numbers as vectors 100 00:08:28,760 --> 00:08:29,840 in the complex plane. 101 00:08:30,800 --> 00:08:35,170 So first of all, let us draw the two numbers, C1 and Z two. 102 00:08:35,990 --> 00:08:39,170 And so let's start with the one in red. 103 00:08:39,860 --> 00:08:44,440 We have a real part of minus one and we have an imaginary part of two. 104 00:08:44,840 --> 00:08:54,510 So our vector points like this, sorry, it's a bit ugly, but hard to hard to draw on this tablet here. 105 00:08:55,100 --> 00:08:59,290 So then we can look at C to and draw in blue. 106 00:09:00,710 --> 00:09:07,250 So we know that the length of this vector is two and we know that the polar angle of this vector is 107 00:09:07,250 --> 00:09:08,530 PI over eight. 108 00:09:09,260 --> 00:09:15,620 So we know that here it is, imaginary axis is PI over two, which would be ninety degree, so PI over 109 00:09:15,620 --> 00:09:17,110 eight would be one quarter. 110 00:09:17,120 --> 00:09:20,120 So it would be approximately let's say. 111 00:09:21,300 --> 00:09:22,820 Would look like something like this. 112 00:09:25,440 --> 00:09:33,780 All right, so this is our too, and now we can draw all of these combinations, you had the some the 113 00:09:33,780 --> 00:09:36,470 difference, the product and the quotient. 114 00:09:37,410 --> 00:09:45,450 So first of all, let us calculate the sum of these two numbers so we have to draw a parallelogram so 115 00:09:45,450 --> 00:09:48,790 we could, for example, draw like this. 116 00:09:49,710 --> 00:09:53,370 So I think should it should look something like this. 117 00:09:56,620 --> 00:10:06,130 See one plus see two, then for C, one minor C to this is where we have to draw this arrow here into 118 00:10:06,130 --> 00:10:07,470 the opposite direction. 119 00:10:07,960 --> 00:10:11,350 So we have to draw the parallelogram along. 120 00:10:12,650 --> 00:10:13,280 Like this. 121 00:10:15,460 --> 00:10:23,110 So our Victor, I think, would point like this, this is the one liners minus to. 122 00:10:26,460 --> 00:10:32,970 All right, now it's a bit more difficult now we need to calculate the product and we have learned that 123 00:10:32,970 --> 00:10:39,750 the product of two complex numbers can be calculated by multiplying their lengths and by adding the 124 00:10:39,760 --> 00:10:40,810 polar angles. 125 00:10:42,250 --> 00:10:49,770 OK, so the length of these two vectors, so we know that the length of C2 is two and we know that the 126 00:10:49,770 --> 00:10:57,840 length of this vector here is a bit larger than two, but smaller than three. 127 00:10:58,830 --> 00:11:01,660 So that's estimated something like 2.5. 128 00:11:01,830 --> 00:11:04,410 So we get as a product the length of five. 129 00:11:05,580 --> 00:11:11,250 OK, so our vector must be approximately the length of five and now we just have to add up these two 130 00:11:11,250 --> 00:11:12,150 polar angles. 131 00:11:12,610 --> 00:11:14,400 So this angle here is a bit small. 132 00:11:14,410 --> 00:11:17,990 It's it's 90 degree over force or twenty two point five. 133 00:11:18,270 --> 00:11:21,410 So we have to add twenty two point five to this angle here. 134 00:11:21,720 --> 00:11:27,030 So I would say it's pointing along this direction and it has a length of five. 135 00:11:27,660 --> 00:11:30,720 So, yeah, it's hard to say maybe. 136 00:11:31,230 --> 00:11:31,800 Maybe here. 137 00:11:34,240 --> 00:11:37,440 OK, I hope that this is correct. 138 00:11:37,590 --> 00:11:44,750 Let's see later see time two and then we calculate the quotient. 139 00:11:45,350 --> 00:11:50,980 So here we have to divide this length here, which is we said approximately 2.5. 140 00:11:51,500 --> 00:11:53,780 We have to divide by this length, which is two. 141 00:11:53,780 --> 00:11:54,700 So we get our length. 142 00:11:54,710 --> 00:11:57,050 That's a bit longer than one. 143 00:11:58,210 --> 00:12:05,270 And for the polar angles, we have to take this polar angle and we have to subtract this polar angle. 144 00:12:05,740 --> 00:12:10,510 So I think this fact is pointing something like this. 145 00:12:10,840 --> 00:12:13,860 So this is sea one divided by Sea two. 146 00:12:14,950 --> 00:12:19,060 And I told you, you should compare these vectors with the actual numbers. 147 00:12:19,570 --> 00:12:21,660 And I have calculated the actual numbers. 148 00:12:21,700 --> 00:12:22,360 So let's see. 149 00:12:24,240 --> 00:12:34,220 All right, so let's check we have the one plus two is on zero point eight plus two point seven nine, 150 00:12:34,620 --> 00:12:39,090 OK, your report was maybe a bit smaller, but two point seven is OK. 151 00:12:39,120 --> 00:12:40,320 So I think it's quite good. 152 00:12:41,130 --> 00:12:45,930 Then the one minus the two, we have minus two point eight plus one point two. 153 00:12:45,930 --> 00:12:47,970 I OK here. 154 00:12:47,970 --> 00:12:48,310 Yeah. 155 00:12:48,330 --> 00:12:53,550 The the rear part is a bit too bit of should have been actually a bit longer this whole thing. 156 00:12:54,360 --> 00:12:54,860 Yeah. 157 00:12:54,900 --> 00:12:55,380 OK. 158 00:12:57,710 --> 00:13:05,300 But I think it's OK and it's just a practice for two to get the feeling it's much more important that, 159 00:13:05,300 --> 00:13:08,110 you know, it points along this direction roughly. 160 00:13:08,990 --> 00:13:16,580 And then for the product, we have minus three point four and two point nine K minus three point four 161 00:13:16,580 --> 00:13:18,110 is actually really good. 162 00:13:18,110 --> 00:13:24,020 But two point nine here would be you would be rather here, but yeah, it's fine I guess. 163 00:13:24,410 --> 00:13:28,510 And Fawzy one divided by two, we have almost no real part. 164 00:13:28,520 --> 00:13:29,180 Yeah that's good. 165 00:13:29,450 --> 00:13:32,180 And we have one point one as the imaginary part. 166 00:13:32,180 --> 00:13:33,090 That's also great. 167 00:13:33,860 --> 00:13:39,950 OK, so as you have seen this, this practice was not about getting the perfect estimation, but to 168 00:13:39,950 --> 00:13:45,740 get a feeling when someone gives you two numbers, especially if these numbers are in different representation, 169 00:13:45,740 --> 00:13:52,820 where it's quite difficult to multiply them to or to calculate the quotient, then you can just draw 170 00:13:52,820 --> 00:13:58,730 them and come up with the idea where this product or this quotient points along. 171 00:13:59,570 --> 00:14:05,810 OK, so this was the exercise number three and we have completed the whole three exercises and I hope 172 00:14:05,810 --> 00:14:10,100 now you know how to calculate with complex numbers.