﻿[Script Info] Title: [Events] Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text Dialogue: 0,0:00:02.80,0:00:05.88,Default,,0000,0000,0000,,In this unit, we're going to\Nlook at the complex conjugate. Dialogue: 0,0:00:06.69,0:00:09.83,Default,,0000,0000,0000,,Every complex number as\Nassociated with it, another Dialogue: 0,0:00:09.83,0:00:12.98,Default,,0000,0000,0000,,complex number, which is called\Nits complex conjugate. Dialogue: 0,0:00:13.51,0:00:17.13,Default,,0000,0000,0000,,And you find the complex\Nconjugate of a complex number Dialogue: 0,0:00:17.13,0:00:20.39,Default,,0000,0000,0000,,simply by changing the imaginary\Npart of that number. Dialogue: 0,0:00:20.94,0:00:23.94,Default,,0000,0000,0000,,This is best illustrated by\Nlooking at some examples. Dialogue: 0,0:00:25.21,0:00:28.86,Default,,0000,0000,0000,,So here in this table we've got\Nthree different complex numbers, Dialogue: 0,0:00:28.86,0:00:33.18,Default,,0000,0000,0000,,and we're going to do is going\Nto find the complex conjugate of Dialogue: 0,0:00:33.18,0:00:34.84,Default,,0000,0000,0000,,each of these three numbers. Dialogue: 0,0:00:35.42,0:00:40.28,Default,,0000,0000,0000,,So we start by looking at the\Ncomplex #4 + 7 I. Dialogue: 0,0:00:40.89,0:00:45.66,Default,,0000,0000,0000,,On the way to find the complex\Nconjugate is to change the sign Dialogue: 0,0:00:45.66,0:00:50.06,Default,,0000,0000,0000,,of the imaginary part. So that\Nmeans that the plus sign changes Dialogue: 0,0:00:50.06,0:00:54.10,Default,,0000,0000,0000,,to a minus sign, so the complex\Nconjugate is 4 minus. Dialogue: 0,0:00:54.79,0:00:55.66,Default,,0000,0000,0000,,Seven I. Dialogue: 0,0:00:57.13,0:01:01.63,Default,,0000,0000,0000,,Here's another complex number 1\N- 3. I defined its complex Dialogue: 0,0:01:01.63,0:01:06.13,Default,,0000,0000,0000,,number. We change the sign of\Nthe imaginary part. In other Dialogue: 0,0:01:06.13,0:01:11.85,Default,,0000,0000,0000,,words, we change this minus sign\Nto a plus. So we get the complex Dialogue: 0,0:01:11.85,0:01:13.90,Default,,0000,0000,0000,,number 1 + 3 I. Dialogue: 0,0:01:15.68,0:01:19.52,Default,,0000,0000,0000,,As another complex number minus\N4 - 3 I. Dialogue: 0,0:01:20.31,0:01:23.51,Default,,0000,0000,0000,,And defined its complex\Nconjugate. Again we change the Dialogue: 0,0:01:23.51,0:01:27.79,Default,,0000,0000,0000,,sign of the imaginary part. We\Ndon't need to be worried about Dialogue: 0,0:01:27.79,0:01:32.77,Default,,0000,0000,0000,,what the sign of the real part\Nis. We just changing the sign of Dialogue: 0,0:01:32.77,0:01:37.04,Default,,0000,0000,0000,,the imaginary part and so we get\Nminus 4 + 3 I. Dialogue: 0,0:01:38.38,0:01:42.45,Default,,0000,0000,0000,,So whenever we start with any\Ncomplex number, we can find Dialogue: 0,0:01:42.45,0:01:45.78,Default,,0000,0000,0000,,its complex conjugate very\Neasily. We just change the Dialogue: 0,0:01:45.78,0:01:47.63,Default,,0000,0000,0000,,sign of the imaginary\Npartners. Dialogue: 0,0:01:48.94,0:01:52.75,Default,,0000,0000,0000,,Now the complex conjugate has a\Nvery special property and we'll Dialogue: 0,0:01:52.75,0:01:55.17,Default,,0000,0000,0000,,see what that is by doing an Dialogue: 0,0:01:55.17,0:02:00.29,Default,,0000,0000,0000,,example. OK, what we're going to\Ndo is we're going to take a Dialogue: 0,0:02:00.29,0:02:04.74,Default,,0000,0000,0000,,complex #4 + 7 I I'm going to\Nmultiply it by its own complex Dialogue: 0,0:02:04.74,0:02:09.51,Default,,0000,0000,0000,,conjugate, which is 4 - 7 I, and\Nwe're going to see what we get. Dialogue: 0,0:02:10.34,0:02:17.77,Default,,0000,0000,0000,,So we do. 4 * 4 is\N16 four times minus Seven. I is Dialogue: 0,0:02:17.77,0:02:19.37,Default,,0000,0000,0000,,minus 28 I. Dialogue: 0,0:02:20.61,0:02:24.18,Default,,0000,0000,0000,,Plus Seven I times four is Dialogue: 0,0:02:24.18,0:02:31.42,Default,,0000,0000,0000,,plus 28I. And plus Seven\NI minus Seven I is minus Dialogue: 0,0:02:31.42,0:02:33.18,Default,,0000,0000,0000,,49 I squared. Dialogue: 0,0:02:34.53,0:02:36.25,Default,,0000,0000,0000,,Now when we come to tidy this Dialogue: 0,0:02:36.25,0:02:39.29,Default,,0000,0000,0000,,up. The 16 stays there. Dialogue: 0,0:02:40.00,0:02:45.21,Default,,0000,0000,0000,,We have minus 28I Plus 28I, so\Nthey cancel each other out, so Dialogue: 0,0:02:45.21,0:02:47.22,Default,,0000,0000,0000,,we're left with no eyes. Dialogue: 0,0:02:48.08,0:02:51.98,Default,,0000,0000,0000,,So there's nothing coming from\Nthose two terms, and from this Dialogue: 0,0:02:51.98,0:02:56.60,Default,,0000,0000,0000,,term on the end, we've got minus\N49. I squared. We remember that Dialogue: 0,0:02:56.60,0:03:01.57,Default,,0000,0000,0000,,I squared is minus one, so we\Ngot minus 49 times minus one, so Dialogue: 0,0:03:01.57,0:03:02.64,Default,,0000,0000,0000,,that's plus 49. Dialogue: 0,0:03:03.43,0:03:07.69,Default,,0000,0000,0000,,And 16 +\N49 is 65. Dialogue: 0,0:03:08.85,0:03:13.56,Default,,0000,0000,0000,,So when we multiply the two\Ncomplex numbers together 4 + 7 I Dialogue: 0,0:03:13.56,0:03:18.99,Default,,0000,0000,0000,,and its complex conjugate 4 - 7\NI we find that the answer we get Dialogue: 0,0:03:18.99,0:03:24.05,Default,,0000,0000,0000,,is 65. There was the answer is a\Npurely real number, it has no Dialogue: 0,0:03:24.05,0:03:26.95,Default,,0000,0000,0000,,imaginary part or an imaginary\Npart of 0. Dialogue: 0,0:03:28.23,0:03:32.12,Default,,0000,0000,0000,,That is quite important. So two\Ncomplex numbers multiplying Dialogue: 0,0:03:32.12,0:03:34.71,Default,,0000,0000,0000,,together to give a real number. Dialogue: 0,0:03:35.40,0:03:38.61,Default,,0000,0000,0000,,Let's see if it's always\Nhappens. Let's try another pair Dialogue: 0,0:03:38.61,0:03:41.82,Default,,0000,0000,0000,,and complex number and its\Ncomplex conjugate and see what Dialogue: 0,0:03:41.82,0:03:45.80,Default,,0000,0000,0000,,happens then. OK, in this\Nexample we're just going to take Dialogue: 0,0:03:45.80,0:03:48.53,Default,,0000,0000,0000,,another complex number and its\Ncomplex conjugate and multiply Dialogue: 0,0:03:48.53,0:03:54.68,Default,,0000,0000,0000,,them together. So what we've got\Nis 1 - 3 I. Its complex Dialogue: 0,0:03:54.68,0:04:01.06,Default,,0000,0000,0000,,conjugate is 1 + 3 I let's\Nmultiply them together. 1 * 1 is Dialogue: 0,0:04:01.06,0:04:06.16,Default,,0000,0000,0000,,one. One times plus three. I\Nis plus 3I. Dialogue: 0,0:04:06.68,0:04:09.34,Default,,0000,0000,0000,,Minus three items, one is minus Dialogue: 0,0:04:09.34,0:04:15.97,Default,,0000,0000,0000,,three I. And minus three I times\Nplus three I is minus 9. Dialogue: 0,0:04:16.48,0:04:17.79,Default,,0000,0000,0000,,I squat. Dialogue: 0,0:04:19.01,0:04:23.65,Default,,0000,0000,0000,,Always do now is tidy this up.\NThat means we combined together Dialogue: 0,0:04:23.65,0:04:29.46,Default,,0000,0000,0000,,are terms in I and we use the\Nfact that I squared is equal to Dialogue: 0,0:04:29.46,0:04:34.88,Default,,0000,0000,0000,,minus one. So we get one start\Nplus three. I minus three I, so Dialogue: 0,0:04:34.88,0:04:38.75,Default,,0000,0000,0000,,that's no eyes and then minus\Nnine isquared. Remembering that Dialogue: 0,0:04:38.75,0:04:43.39,Default,,0000,0000,0000,,I squared is minus one, we've\Ngot minus nine times minus one, Dialogue: 0,0:04:43.39,0:04:47.26,Default,,0000,0000,0000,,giving is plus 9, which is an\Nanswer of text. Dialogue: 0,0:04:47.99,0:04:51.27,Default,,0000,0000,0000,,So once again we've\Nmultiplied complex number by Dialogue: 0,0:04:51.27,0:04:54.96,Default,,0000,0000,0000,,its complex conjugate and\Nwe've got a real number. Dialogue: 0,0:04:56.31,0:04:59.63,Default,,0000,0000,0000,,Now this is a very important\Nproperty and it doesn't just Dialogue: 0,0:04:59.63,0:05:02.95,Default,,0000,0000,0000,,happen in the two examples that\NI've picked, it happens that Dialogue: 0,0:05:02.95,0:05:06.28,Default,,0000,0000,0000,,every complex number. If you\Npick any complex, then be like Dialogue: 0,0:05:06.28,0:05:09.90,Default,,0000,0000,0000,,and multiply it by its complex\Nconjugate, you will get a real Dialogue: 0,0:05:09.90,0:05:13.83,Default,,0000,0000,0000,,number and that turns out to be\Nvery important when we come to Dialogue: 0,0:05:13.83,0:05:17.15,Default,,0000,0000,0000,,learn how to divide complex\Nnumbers, which is what will be Dialogue: 0,0:05:17.15,0:05:18.66,Default,,0000,0000,0000,,doing in the next unit.