1 00:00:00,060 --> 00:00:06,870 All right, so now that we know what complex numbers are, let us continue and work on some simple operations 2 00:00:06,870 --> 00:00:12,830 like addition and subtraction, and I will also show you how you can do this in the complex plane. 3 00:00:14,280 --> 00:00:21,300 So previously we have introduced the complex number as some of a real part and the imaginary parts times 4 00:00:21,600 --> 00:00:22,650 the imaginary unit. 5 00:00:22,650 --> 00:00:25,190 I would use the square root of minus one. 6 00:00:25,710 --> 00:00:30,950 And we have also learned that we can represent a complex number as a vector. 7 00:00:31,920 --> 00:00:36,120 So adding to complex numbers is actually really simple. 8 00:00:36,690 --> 00:00:39,900 We just write them down and do it like with real numbers. 9 00:00:40,650 --> 00:00:45,540 So we have the first complex number that I have written down as X plus Y times. 10 00:00:45,540 --> 00:00:48,920 I read this is the real part and this is the imaginary part. 11 00:00:49,530 --> 00:00:55,650 And then we have another complex number which is determined by you and V as the real and imaginary parts. 12 00:00:56,550 --> 00:01:00,270 And now you can do it like with physical quantities that have a unit. 13 00:01:00,600 --> 00:01:04,550 You can only add up to the quantities that have the same unit. 14 00:01:05,130 --> 00:01:12,900 So you can add up X and you and you can add up Y and B because Y and we these parts here have the same 15 00:01:12,900 --> 00:01:21,450 unit, which is I so our sum is X plus you as the real part and Y plus B as the imaginary part. 16 00:01:21,960 --> 00:01:27,330 That's really nothing new, nothing special, just like for real numbers and for physical quantities. 17 00:01:28,500 --> 00:01:34,230 Now of course the subtraction is exactly the same because any subtraction is basically just an addition 18 00:01:35,160 --> 00:01:36,560 with a negative number here. 19 00:01:37,080 --> 00:01:40,550 So we get the same thing, but here have different signs. 20 00:01:40,560 --> 00:01:46,440 So we get X minus Uster apart and Y minus V as the imaginary part. 21 00:01:47,430 --> 00:01:49,290 So let's see how that works. 22 00:01:49,860 --> 00:01:54,380 So we take here two complex numbers and we want to add them up. 23 00:01:55,500 --> 00:02:05,630 So our complex numbers are C one is equal to two plus three times I. 24 00:02:06,360 --> 00:02:08,280 So that's that one here, C one. 25 00:02:08,880 --> 00:02:18,240 And then we have a Z two, which is just this one and the two is equal to minus three. 26 00:02:20,100 --> 00:02:29,070 Plus, one times I and of course, we do not really have to write down the one times, we can just write 27 00:02:29,280 --> 00:02:39,540 three plus I and now our our new number, zie one plus the two is, of course, the sum of the two real 28 00:02:39,540 --> 00:02:40,000 parts. 29 00:02:40,020 --> 00:02:54,330 So we have two minus three and then we have the sum of the imaginary parts which is three plus one times 30 00:02:54,350 --> 00:03:03,390 I so I was the one plus two is equal to minus one plus four times I. 31 00:03:05,250 --> 00:03:13,070 So what we have to do and what we can do is we can add these two numbers in terms of a vector addition. 32 00:03:13,650 --> 00:03:20,220 So you, you take this vector here and plotted like this and you take this vector plotted like this 33 00:03:20,550 --> 00:03:24,580 and where the two vectors meet, this is where our sum ends up being. 34 00:03:25,260 --> 00:03:32,220 So as you can see, we have the minus one as the real part and we have the four as the imaginary part, 35 00:03:32,220 --> 00:03:33,960 just like we calculated. 36 00:03:35,260 --> 00:03:35,730 All right. 37 00:03:35,730 --> 00:03:44,100 So let's do the same thing for the subtraction and take again our two numbers, C one and the number 38 00:03:44,100 --> 00:03:44,850 C two. 39 00:03:46,610 --> 00:03:52,940 Now, if we want to calculate the difference between these two numbers, see one minus C two, we must 40 00:03:52,940 --> 00:03:56,970 calculate the difference in the real part and in the imaginary part. 41 00:03:57,800 --> 00:04:03,660 So we have the difference here and the difference here. 42 00:04:04,760 --> 00:04:13,230 So we have in the rear part two minus minus three, and here we have three minus one. 43 00:04:14,120 --> 00:04:24,320 So the difference C one minus C two is equal to five plus two times by now. 44 00:04:24,320 --> 00:04:30,340 Again, we can do the same thing in terms of a vector addition or subtraction. 45 00:04:30,710 --> 00:04:37,190 So we have to do is we have to take to C to Vector here and plotted along the opposite direction. 46 00:04:37,340 --> 00:04:38,620 So basically mirror it. 47 00:04:39,020 --> 00:04:42,940 And then again, we have to draw our parallelogram and what the two vectors meet. 48 00:04:43,370 --> 00:04:47,330 This is where our difference C one minus C2 is. 49 00:04:48,350 --> 00:04:55,460 So we get five as the real part and we get to see the imaginary part, just like we have here, five 50 00:04:55,460 --> 00:04:57,020 plus two I. 51 00:05:00,000 --> 00:05:06,870 Now, we can also do other things with complex numbers, so calculating products is possible, also 52 00:05:07,710 --> 00:05:10,970 calculating quotients is possible, but that's a bit difficult. 53 00:05:11,010 --> 00:05:13,740 We will do this in the next lecture and this lecture. 54 00:05:13,740 --> 00:05:20,040 We will also discuss operations that are only possible for complex numbers and not for real numbers. 55 00:05:20,520 --> 00:05:24,480 However, they are also only based on addition and subtraction. 56 00:05:25,470 --> 00:05:31,910 So first of all, what we can do is we can calculate to so-called complex conjugate of a complex number 57 00:05:32,640 --> 00:05:35,310 and that is indicated by the star up here. 58 00:05:36,540 --> 00:05:43,200 So the complex conjugate, it means that you take your complex numbers, Z, and you just reverse the 59 00:05:43,200 --> 00:05:45,210 sign of the imaginary part. 60 00:05:45,960 --> 00:05:53,240 So instead of writing down X plus items, I, we get now X minus Y plus Y times I. 61 00:05:54,480 --> 00:06:02,010 So just switch around this sign here and the real part of this number is invariant. 62 00:06:02,130 --> 00:06:03,270 That remains the same. 63 00:06:04,740 --> 00:06:12,750 Now we can use this complex conjugate of a number to to do special operations and two of them are to 64 00:06:12,750 --> 00:06:19,890 calculate the real and the imaginary part of a number quite easily, because not all of the time you 65 00:06:19,890 --> 00:06:22,850 will find a complex number that has such a simple shape. 66 00:06:23,310 --> 00:06:28,620 So you ideally want to have to shape real part plus imaginary part. 67 00:06:29,190 --> 00:06:35,580 But when someone gives you a complex number, it could also be that you have some products and some 68 00:06:35,580 --> 00:06:43,770 exponents and you first have to really simplify and an easy way to find out what the real and the imaginary 69 00:06:43,770 --> 00:06:49,400 parts of a number are, is to just calculate the sum of the number. 70 00:06:49,410 --> 00:06:53,490 Plus it's complex conjugate and then divide by two. 71 00:06:54,390 --> 00:07:01,830 Because as you can see, what happens then is you add up the real parts, but the imaginary parts they 72 00:07:01,830 --> 00:07:04,400 cancel out because here you have this negative sign. 73 00:07:04,950 --> 00:07:11,430 So you will get here two times X divided by two is equal to X and X is our real part. 74 00:07:12,870 --> 00:07:15,350 For the imaginary part, you take the difference. 75 00:07:15,960 --> 00:07:22,100 This is because here you will then get this minus sign here, and this means the real parts will cancel 76 00:07:22,260 --> 00:07:25,250 and this time the imaginary parts will add up. 77 00:07:25,590 --> 00:07:33,640 So because you have here y time as I minus minus Y times AI, which gives us two times Y times. 78 00:07:34,740 --> 00:07:39,300 So when we take this and divide by two AI, we get Y, which is the imaginary part. 79 00:07:41,740 --> 00:07:49,020 Now, another thing we can do is we can calculate the absolute value, and this is exactly as in, you 80 00:07:49,030 --> 00:07:50,220 know, in vector algebra. 81 00:07:51,010 --> 00:08:00,520 So you take the absolute value of this number X Y times eye by squaring the individual components, 82 00:08:00,550 --> 00:08:03,640 the real part and the imaginary part and adding them up. 83 00:08:04,730 --> 00:08:05,500 So please note this. 84 00:08:05,500 --> 00:08:07,010 Here is not the absolute value yet. 85 00:08:07,030 --> 00:08:12,260 This is the absolute value square and this is equal to X squared plus Y square. 86 00:08:12,670 --> 00:08:21,040 And then if you know about the complex conjugate, you can also just take the number itself and multiply 87 00:08:21,040 --> 00:08:22,810 it with its complex conjugate. 88 00:08:22,810 --> 00:08:31,460 And you can also do this in the opposite order because then you would get by using the nominal formulae 89 00:08:31,510 --> 00:08:40,660 formula, you get X squared minus Y squared, Times Square and I square is minus one, so you get X 90 00:08:40,660 --> 00:08:41,780 squared, plus Y square. 91 00:08:42,790 --> 00:08:48,180 And now when you have this, it's very easy to get the absolute value you just take to square root. 92 00:08:48,490 --> 00:08:54,100 So you have you to square root of X squared plus by square or you have the square root of this product 93 00:08:54,100 --> 00:08:58,360 of the complex conjugate and the complex numbers. 94 00:08:59,020 --> 00:09:02,370 So let us now go ahead and see how this works in practice. 95 00:09:03,220 --> 00:09:10,060 So let's take our number Z, which is equal to two plus three times I. 96 00:09:10,750 --> 00:09:13,840 And now we want to calculate the complex conjugate. 97 00:09:14,740 --> 00:09:22,180 So we write seastar is equal to two minus three times. 98 00:09:23,290 --> 00:09:28,600 So we have here there's minus sine where we had here the positive side. 99 00:09:29,980 --> 00:09:37,780 And now what this means in terms of writing down complex numbers as vectors is you take to the vector 100 00:09:37,780 --> 00:09:43,190 after complex numbers and you move right at the real axis. 101 00:09:43,480 --> 00:09:47,300 This is because the sign of the imaginary part has changed. 102 00:09:47,530 --> 00:09:50,680 So now our complex numbers seastar is here. 103 00:09:52,240 --> 00:09:58,450 So now that we have the numbers and it's complex conjugate, we can calculate the real. 104 00:09:59,580 --> 00:10:02,450 So let's do this according to this formula here. 105 00:10:03,640 --> 00:10:14,210 So our real part of C is equal to one half times and then we add up the real parts. 106 00:10:14,620 --> 00:10:16,360 So this is then. 107 00:10:18,760 --> 00:10:26,650 Yeah, two times or let's let's write it down like this, two plus two and the imaginary part, it's 108 00:10:26,650 --> 00:10:32,590 of course three plus minus three I. 109 00:10:36,740 --> 00:10:47,840 And this is equal to one 1/2 times four plus zero, and this is equal to two. 110 00:10:48,410 --> 00:10:52,180 And I mean, of course, in our case, we already knew before that this was true. 111 00:10:52,580 --> 00:10:55,790 So we have now that the real part is in fact, too. 112 00:10:56,390 --> 00:10:58,570 But here it's a bit useless, to be honest. 113 00:10:58,580 --> 00:11:04,370 But in general, you can have very difficult looking expressions here where it's not really separated 114 00:11:04,370 --> 00:11:05,870 in real and imaginary part. 115 00:11:06,080 --> 00:11:08,360 And you first have to calculate this. 116 00:11:09,640 --> 00:11:16,310 So now we can go ahead and look at what this means for this vector, for these vectors here. 117 00:11:17,460 --> 00:11:23,070 So what we have to do is we have to take the numbers and it's complex conjugate, we have to add them 118 00:11:23,070 --> 00:11:26,910 up, which brings us here and we have to multiply it by one half. 119 00:11:27,480 --> 00:11:30,180 So this means the real part is exactly here. 120 00:11:30,330 --> 00:11:33,240 And of course, that's really the real part. 121 00:11:33,250 --> 00:11:33,840 It's to. 122 00:11:36,320 --> 00:11:41,090 Now, let's do the same thing for the imaginary part here, we calculate the difference. 123 00:11:41,750 --> 00:11:48,140 So this means the imaginary part of Z is equal to one over two AI. 124 00:11:48,890 --> 00:11:53,940 And then we have the difference of the, you know, the real parts of the axis. 125 00:11:54,260 --> 00:12:03,920 So we have two minus two plus three minus minus three times AI. 126 00:12:05,180 --> 00:12:10,070 And this is of course one or two AI zero. 127 00:12:11,160 --> 00:12:23,370 Plus six times I and this is equal to three I divided by I is one, so we get three as our imaginary 128 00:12:23,370 --> 00:12:23,650 part. 129 00:12:24,030 --> 00:12:25,050 So let's that one here. 130 00:12:25,740 --> 00:12:30,420 And again, we can practice by subtracting these two vectors, C and seastar. 131 00:12:30,900 --> 00:12:34,320 We mirror this vector here and draw the parallelogram. 132 00:12:34,680 --> 00:12:44,700 So we end up being here at the top and now we just have to multiply by one half or we could also multiply 133 00:12:44,880 --> 00:12:49,680 by one over Twohy, which would rotate this vector here in the real plane. 134 00:12:49,680 --> 00:12:54,520 But we will talk about how to multiply in the complex plane later. 135 00:12:56,220 --> 00:12:56,730 All right. 136 00:12:56,880 --> 00:13:01,230 So now we know how we can determine the real part and the imaginary part. 137 00:13:01,740 --> 00:13:07,290 In the next lecture, we will talk about multiplication, division and DiPaola representation.