0:00:02.800,0:00:05.880 In this unit, we're going to[br]look at the complex conjugate. 0:00:06.690,0:00:09.834 Every complex number as[br]associated with it, another 0:00:09.834,0:00:12.978 complex number, which is called[br]its complex conjugate. 0:00:13.510,0:00:17.130 And you find the complex[br]conjugate of a complex number 0:00:17.130,0:00:20.388 simply by changing the imaginary[br]part of that number. 0:00:20.940,0:00:23.937 This is best illustrated by[br]looking at some examples. 0:00:25.210,0:00:28.862 So here in this table we've got[br]three different complex numbers, 0:00:28.862,0:00:33.178 and we're going to do is going[br]to find the complex conjugate of 0:00:33.178,0:00:34.838 each of these three numbers. 0:00:35.420,0:00:40.280 So we start by looking at the[br]complex #4 + 7 I. 0:00:40.890,0:00:45.661 On the way to find the complex[br]conjugate is to change the sign 0:00:45.661,0:00:50.065 of the imaginary part. So that[br]means that the plus sign changes 0:00:50.065,0:00:54.102 to a minus sign, so the complex[br]conjugate is 4 minus. 0:00:54.790,0:00:55.660 Seven I. 0:00:57.130,0:01:01.629 Here's another complex number 1[br]- 3. I defined its complex 0:01:01.629,0:01:06.128 number. We change the sign of[br]the imaginary part. In other 0:01:06.128,0:01:11.854 words, we change this minus sign[br]to a plus. So we get the complex 0:01:11.854,0:01:13.899 number 1 + 3 I. 0:01:15.680,0:01:19.523 As another complex number minus[br]4 - 3 I. 0:01:20.310,0:01:23.514 And defined its complex[br]conjugate. Again we change the 0:01:23.514,0:01:27.786 sign of the imaginary part. We[br]don't need to be worried about 0:01:27.786,0:01:32.770 what the sign of the real part[br]is. We just changing the sign of 0:01:32.770,0:01:37.042 the imaginary part and so we get[br]minus 4 + 3 I. 0:01:38.380,0:01:42.450 So whenever we start with any[br]complex number, we can find 0:01:42.450,0:01:45.780 its complex conjugate very[br]easily. We just change the 0:01:45.780,0:01:47.630 sign of the imaginary[br]partners. 0:01:48.940,0:01:52.746 Now the complex conjugate has a[br]very special property and we'll 0:01:52.746,0:01:55.168 see what that is by doing an 0:01:55.168,0:02:00.286 example. OK, what we're going to[br]do is we're going to take a 0:02:00.286,0:02:04.738 complex #4 + 7 I I'm going to[br]multiply it by its own complex 0:02:04.738,0:02:09.508 conjugate, which is 4 - 7 I, and[br]we're going to see what we get. 0:02:10.340,0:02:17.774 So we do. 4 * 4 is[br]16 four times minus Seven. I is 0:02:17.774,0:02:19.367 minus 28 I. 0:02:20.610,0:02:24.180 Plus Seven I times four is 0:02:24.180,0:02:31.425 plus 28I. And plus Seven[br]I minus Seven I is minus 0:02:31.425,0:02:33.180 49 I squared. 0:02:34.530,0:02:36.252 Now when we come to tidy this 0:02:36.252,0:02:39.288 up. The 16 stays there. 0:02:40.000,0:02:45.213 We have minus 28I Plus 28I, so[br]they cancel each other out, so 0:02:45.213,0:02:47.218 we're left with no eyes. 0:02:48.080,0:02:51.985 So there's nothing coming from[br]those two terms, and from this 0:02:51.985,0:02:56.600 term on the end, we've got minus[br]49. I squared. We remember that 0:02:56.600,0:03:01.570 I squared is minus one, so we[br]got minus 49 times minus one, so 0:03:01.570,0:03:02.635 that's plus 49. 0:03:03.430,0:03:07.690 And 16 +[br]49 is 65. 0:03:08.850,0:03:13.556 So when we multiply the two[br]complex numbers together 4 + 7 I 0:03:13.556,0:03:18.986 and its complex conjugate 4 - 7[br]I we find that the answer we get 0:03:18.986,0:03:24.054 is 65. There was the answer is a[br]purely real number, it has no 0:03:24.054,0:03:26.950 imaginary part or an imaginary[br]part of 0. 0:03:28.230,0:03:32.118 That is quite important. So two[br]complex numbers multiplying 0:03:32.118,0:03:34.710 together to give a real number. 0:03:35.400,0:03:38.610 Let's see if it's always[br]happens. Let's try another pair 0:03:38.610,0:03:41.820 and complex number and its[br]complex conjugate and see what 0:03:41.820,0:03:45.796 happens then. OK, in this[br]example we're just going to take 0:03:45.796,0:03:48.532 another complex number and its[br]complex conjugate and multiply 0:03:48.532,0:03:54.685 them together. So what we've got[br]is 1 - 3 I. Its complex 0:03:54.685,0:04:01.055 conjugate is 1 + 3 I let's[br]multiply them together. 1 * 1 is 0:04:01.055,0:04:06.164 one. One times plus three. I[br]is plus 3I. 0:04:06.680,0:04:09.338 Minus three items, one is minus 0:04:09.338,0:04:15.971 three I. And minus three I times[br]plus three I is minus 9. 0:04:16.480,0:04:17.790 I squat. 0:04:19.010,0:04:23.654 Always do now is tidy this up.[br]That means we combined together 0:04:23.654,0:04:29.459 are terms in I and we use the[br]fact that I squared is equal to 0:04:29.459,0:04:34.877 minus one. So we get one start[br]plus three. I minus three I, so 0:04:34.877,0:04:38.747 that's no eyes and then minus[br]nine isquared. Remembering that 0:04:38.747,0:04:43.391 I squared is minus one, we've[br]got minus nine times minus one, 0:04:43.391,0:04:47.261 giving is plus 9, which is an[br]answer of text. 0:04:47.990,0:04:51.270 So once again we've[br]multiplied complex number by 0:04:51.270,0:04:54.960 its complex conjugate and[br]we've got a real number. 0:04:56.310,0:04:59.632 Now this is a very important[br]property and it doesn't just 0:04:59.632,0:05:02.954 happen in the two examples that[br]I've picked, it happens that 0:05:02.954,0:05:06.276 every complex number. If you[br]pick any complex, then be like 0:05:06.276,0:05:09.900 and multiply it by its complex[br]conjugate, you will get a real 0:05:09.900,0:05:13.826 number and that turns out to be[br]very important when we come to 0:05:13.826,0:05:17.148 learn how to divide complex[br]numbers, which is what will be 0:05:17.148,0:05:18.658 doing in the next unit.