1 00:00:00,240 --> 00:00:07,080 Let us now go ahead and talk about more difficult operations, so here we will discuss the multiplication 2 00:00:07,080 --> 00:00:14,130 and division of complex numbers and how this works using the polar representation of a complex number. 3 00:00:15,820 --> 00:00:23,230 So the multiplication of a complex number is not so difficult if you just do it straightforwardly by 4 00:00:23,230 --> 00:00:24,650 writing down the two numbers. 5 00:00:25,150 --> 00:00:29,760 So you have the real part, imaginary part, real part, imaginary part of the two numbers. 6 00:00:30,340 --> 00:00:34,780 And so the result is that you have to calculate your four terms. 7 00:00:34,780 --> 00:00:45,410 You have to calculate X times, you plus X times V.I. plus Y times you, plus Y times the eye. 8 00:00:46,240 --> 00:00:54,490 So the only thing that may be a bit tricky here is the term Y times by eight times VI. 9 00:00:55,690 --> 00:01:01,520 So what we get here is why V Times Square and I square. 10 00:01:01,520 --> 00:01:02,780 It is of course minus one. 11 00:01:03,310 --> 00:01:12,520 So this means this product here, y Times VII becomes actually a real part because I square is minus 12 00:01:12,520 --> 00:01:13,840 one, which is a real number. 13 00:01:14,650 --> 00:01:16,810 So we get as the real part x2. 14 00:01:18,160 --> 00:01:20,270 Minus five. 15 00:01:21,190 --> 00:01:22,660 And then there's the imaginary part. 16 00:01:22,690 --> 00:01:28,580 We get the other two terms, we get X, times V plus you times Y. 17 00:01:29,680 --> 00:01:36,050 Now, the next thing we want to do is to calculate the quotient of two complex numbers. 18 00:01:36,820 --> 00:01:43,810 So what we have to do is we have to take the first numbers one and have to multiply by a reciprocal 19 00:01:43,810 --> 00:01:44,240 number. 20 00:01:45,040 --> 00:01:48,970 So here we first have to find out what is the reciprocal of a complex number. 21 00:01:49,720 --> 00:01:51,880 And it's not that straightforward. 22 00:01:51,970 --> 00:01:53,830 But instead, we have to use a trick. 23 00:01:54,430 --> 00:02:02,240 We have to multiply by one and instead of one, we write seastar, seastar divided by seastar. 24 00:02:02,620 --> 00:02:09,080 So this is here, the complex conjugate, where you change the sign of the imaginary part of the number. 25 00:02:10,060 --> 00:02:17,050 And so we have learned in the previous lecture that Z Times the star is equal to the absolute value 26 00:02:17,050 --> 00:02:18,640 of C squared. 27 00:02:19,420 --> 00:02:24,610 And the absolute value of C squared is, of course a real number because the absolute value is a real 28 00:02:24,610 --> 00:02:25,060 number. 29 00:02:25,870 --> 00:02:31,110 So this, in fact, here is the real part squared plus the imaginary part squared. 30 00:02:31,960 --> 00:02:34,880 And then on top here we have a complex number. 31 00:02:35,590 --> 00:02:38,350 So this is a perfectly fine, complex number. 32 00:02:38,740 --> 00:02:42,190 We can write down all of this mathematics here. 33 00:02:42,490 --> 00:02:51,130 And so we get X minus Y times I, which is seastar divided by X squared plus Y square, which is this 34 00:02:51,130 --> 00:02:53,190 one here, as I mentioned previously. 35 00:02:53,860 --> 00:02:58,540 And then if you really want to split it up into the real into the imaginary part, you write it down 36 00:02:58,540 --> 00:02:59,020 like this. 37 00:02:59,050 --> 00:03:03,250 So the real part is the real part of the first number. 38 00:03:04,510 --> 00:03:11,900 Yeah, part of the number divided by the absolute value squared minus the imaginary part times. 39 00:03:11,920 --> 00:03:16,150 I divide it by the absolute value squared of the number. 40 00:03:16,840 --> 00:03:23,720 OK, so now that we have the reciprocal of the number we can calculate Z two divided by zero one. 41 00:03:25,000 --> 00:03:34,150 So here I have used again U plus V for a real imaginary part of C two and X and Y for the real and imaginary 42 00:03:34,150 --> 00:03:35,500 parts obviously you want. 43 00:03:36,430 --> 00:03:41,560 And now we can just use this relation here for one over each one. 44 00:03:42,100 --> 00:03:48,270 So we have this one and we have to multiply with you plus V times I. 45 00:03:49,600 --> 00:03:57,130 So we will get in all terms the denominator X squared plus Y square, which is the absolute value square 46 00:03:57,130 --> 00:04:00,760 of C1 and you we will get four terms. 47 00:04:01,570 --> 00:04:10,210 So these are of course these terms here and the only difficult term is probably V times I times minus 48 00:04:10,210 --> 00:04:11,470 Y times I. 49 00:04:12,460 --> 00:04:20,110 And so this gives us V Times Y and I square and then another minus one minus sine. 50 00:04:20,740 --> 00:04:28,510 And so since I square is minus one and we have this minus sine here we get here the positive sign and 51 00:04:28,510 --> 00:04:30,850 we get this entry as a real part. 52 00:04:31,870 --> 00:04:34,480 So this is what the quotient looks like. 53 00:04:36,150 --> 00:04:41,520 So now we can go ahead and look at what this looks like in the complex plane. 54 00:04:43,320 --> 00:04:48,420 So what we will we write down is, first of all, we write down to two complex numbers. 55 00:04:49,140 --> 00:04:49,860 Let's go ahead. 56 00:04:50,280 --> 00:04:51,170 We have Z. 57 00:04:51,180 --> 00:05:03,390 One is equal to one plus three times I and we have the two is equal to minus two plus PI. 58 00:05:04,680 --> 00:05:13,560 And then we take our equation for the multiplication so we can write down the one times C two is equal 59 00:05:13,560 --> 00:05:13,950 to. 60 00:05:14,550 --> 00:05:17,030 And no, I will not really rely on this equation. 61 00:05:17,040 --> 00:05:18,810 I will just show you how I would do it. 62 00:05:19,350 --> 00:05:21,330 First, let's write down these two numbers. 63 00:05:21,720 --> 00:05:28,140 Just copy them from here and then minus two plus one times. 64 00:05:28,140 --> 00:05:35,360 I know you calculate the four terms first to calculate one times, minus two, which is minus two, 65 00:05:35,850 --> 00:05:45,120 then you calculate one times I, which is I, then we have three times I times minus two, which is 66 00:05:45,120 --> 00:05:53,730 minus six I and then we have the last term, which is three times I, times I and I. 67 00:05:53,730 --> 00:05:54,960 Square is minus one. 68 00:05:54,960 --> 00:06:03,210 So we get minus three and now we can take the real parts which are minus two and minus three. 69 00:06:03,210 --> 00:06:14,720 So we get minus five and here we get one times I minus six times I which is minus five times I. 70 00:06:16,230 --> 00:06:19,230 And you would also get this if you would just try to use this formula. 71 00:06:19,350 --> 00:06:21,810 But I rather prefer do it like this. 72 00:06:22,980 --> 00:06:26,170 So now we can look at what this vector looks like. 73 00:06:26,190 --> 00:06:30,750 So here we have now our new vector Z one times Z two. 74 00:06:32,070 --> 00:06:36,240 So that is in fact really difficult to. 75 00:06:36,330 --> 00:06:36,680 Yeah. 76 00:06:36,690 --> 00:06:42,030 To think about what how we can get to this vector by by a simple vector operations. 77 00:06:42,420 --> 00:06:48,210 And in fact, it's not really possible because this is not a sum, not a subtraction, it's a product. 78 00:06:48,540 --> 00:06:52,710 And calculating the product of two vectors is not really that simple. 79 00:06:52,920 --> 00:06:56,370 We have to dot product, which is a scalar number, so not a vector. 80 00:06:56,730 --> 00:07:01,080 And we have the vector product which works just in three dimensional space. 81 00:07:01,080 --> 00:07:03,750 And what in this case point out of the plane. 82 00:07:04,650 --> 00:07:06,380 So how do we get this vector? 83 00:07:06,720 --> 00:07:10,410 Let's come back to this later when we have to pull our representation. 84 00:07:13,160 --> 00:07:20,750 But first, let us go ahead and calculate the quotient Ziyuan divided by Z2, so please don't be confused 85 00:07:20,750 --> 00:07:20,960 here. 86 00:07:20,960 --> 00:07:25,190 I used different nomenclature for Ziemann and Z2. 87 00:07:25,190 --> 00:07:29,740 So here's you want as you plus V.I., which was previously easy to. 88 00:07:30,890 --> 00:07:37,130 So what we need first is this, this absolute value square X squared plus Y squared. 89 00:07:38,390 --> 00:07:42,400 So this is the absolute value of C two square. 90 00:07:43,040 --> 00:07:50,300 And just for your information, we can also write down the real value of C two squared plus imaginary 91 00:07:51,050 --> 00:07:54,020 part of C two square. 92 00:07:55,010 --> 00:08:01,670 And so this is minus two square, which is four plus one square one. 93 00:08:01,680 --> 00:08:10,360 So we get five here and now we can write C one divided by C two is equal to one over five. 94 00:08:11,090 --> 00:08:16,450 Then we need these two terms, which are the products of the real parts and the imaginary parts. 95 00:08:16,450 --> 00:08:25,130 So we get one times minus two plus three times one, then we get plus I. 96 00:08:26,180 --> 00:08:36,860 And first we must calculate X times of V, which is the real part of C, two minus two times three. 97 00:08:37,500 --> 00:08:47,630 So this is minus six and then minus the other term, which is then just one times one. 98 00:08:49,880 --> 00:08:53,540 So we get one of a five 99 00:08:55,760 --> 00:09:05,750 um minus two, plus three is one and each here for the imaginary part we get minus six, minus one is 100 00:09:05,750 --> 00:09:07,430 minus seven. 101 00:09:09,590 --> 00:09:21,260 So we have C one divided by two is one over five as three apart and minus seven over five as the imaginary 102 00:09:21,260 --> 00:09:21,590 part. 103 00:09:22,910 --> 00:09:27,620 And this means in the complex plane, our vector is now here. 104 00:09:28,370 --> 00:09:34,340 And once again, it's really difficult to see how this vector is related to the two initial vectors. 105 00:09:34,820 --> 00:09:37,610 And we will find out now how they are related.