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