[Script Info] Title: [Events] Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text Dialogue: 0,0:00:03.19,0:00:07.95,Default,,0000,0000,0000,,In this unit, we're going to\Nlook at how to divide 2 complex Dialogue: 0,0:00:07.95,0:00:11.24,Default,,0000,0000,0000,,numbers. Now, division of\Ncomplex numbers is rather more Dialogue: 0,0:00:11.24,0:00:13.44,Default,,0000,0000,0000,,complicated than addition,\NSubtraction, and multiplication. Dialogue: 0,0:00:14.14,0:00:17.40,Default,,0000,0000,0000,,And Division of complex numbers\Nrelies on two very important Dialogue: 0,0:00:17.40,0:00:20.99,Default,,0000,0000,0000,,principles. The first is that\Nwhen you take a complex number Dialogue: 0,0:00:20.99,0:00:24.25,Default,,0000,0000,0000,,and multiply by its complex\Nconjugate, you get a real Dialogue: 0,0:00:24.25,0:00:27.51,Default,,0000,0000,0000,,number. The second important\Nprinciple is that when you have Dialogue: 0,0:00:27.51,0:00:30.77,Default,,0000,0000,0000,,a fraction, you can multiply the\Nnumerator and the denominator. Dialogue: 0,0:00:30.77,0:00:35.33,Default,,0000,0000,0000,,That's the number on the top on\Nthe number on the bottom of the Dialogue: 0,0:00:35.33,0:00:39.24,Default,,0000,0000,0000,,fraction by the same value, and\Nnot change the value of a Dialogue: 0,0:00:39.24,0:00:43.37,Default,,0000,0000,0000,,fraction. So for example, if\Nyou start with a fraction of Dialogue: 0,0:00:43.37,0:00:47.17,Default,,0000,0000,0000,,half and you multiply the\Ntop and bottom by 5, you get Dialogue: 0,0:00:47.17,0:00:51.30,Default,,0000,0000,0000,,5/10 and the value of five\N10s is the same as the value Dialogue: 0,0:00:51.30,0:00:54.46,Default,,0000,0000,0000,,of 1/2. And that's really\Ngoing to be very important Dialogue: 0,0:00:54.46,0:00:58.27,Default,,0000,0000,0000,,when we come into being able\Nto workout. How to divide 1 Dialogue: 0,0:00:58.27,0:01:01.44,Default,,0000,0000,0000,,complex number by another.\NSo let's look at an example. Dialogue: 0,0:01:03.41,0:01:08.37,Default,,0000,0000,0000,,So we're going to take the\Ncomplex #4 + 7 I. I'm going to Dialogue: 0,0:01:08.37,0:01:12.97,Default,,0000,0000,0000,,divide it by the complex number\N1 - 3. I now remember the Dialogue: 0,0:01:12.97,0:01:16.86,Default,,0000,0000,0000,,division is the same thing. It's\Na fraction, so this complex Dialogue: 0,0:01:16.86,0:01:21.11,Default,,0000,0000,0000,,number divided by this one. We\Ncan just write a Swan complex Dialogue: 0,0:01:21.11,0:01:22.53,Default,,0000,0000,0000,,number over another complex Dialogue: 0,0:01:22.53,0:01:29.43,Default,,0000,0000,0000,,number. So now we have a\Nfraction we can say is that we Dialogue: 0,0:01:29.43,0:01:34.57,Default,,0000,0000,0000,,won't change the value of this\Nfraction if we multiply the Dialogue: 0,0:01:34.57,0:01:38.31,Default,,0000,0000,0000,,numerator and the denominator by\Nthe same value. Dialogue: 0,0:01:39.38,0:01:45.62,Default,,0000,0000,0000,,I'm going to choose to multiply\Nthe denominator by 1 + 3 I. Dialogue: 0,0:01:46.17,0:01:52.32,Default,,0000,0000,0000,,1 + 3 I is the complex conjugate\Nof 1 - 3 I and we choose this Dialogue: 0,0:01:52.32,0:01:55.94,Default,,0000,0000,0000,,complex conjugate so that when\Nwe do the multiplication, what's Dialogue: 0,0:01:55.94,0:01:59.93,Default,,0000,0000,0000,,in the denominator will turn out\Nto be a real number. Dialogue: 0,0:02:01.05,0:02:06.09,Default,,0000,0000,0000,,So for multiplying the\Ndenominator by 1 + 3 I we've got Dialogue: 0,0:02:06.09,0:02:09.03,Default,,0000,0000,0000,,to multiply the numerator by 1 + Dialogue: 0,0:02:09.03,0:02:13.78,Default,,0000,0000,0000,,3 I. So that way we have\Nmultiplied the numerator and Dialogue: 0,0:02:13.78,0:02:17.60,Default,,0000,0000,0000,,denominator by the same value,\Nso we haven't changed the value Dialogue: 0,0:02:17.60,0:02:18.64,Default,,0000,0000,0000,,of the answer. Dialogue: 0,0:02:19.56,0:02:23.92,Default,,0000,0000,0000,,So let's now multiply these two\Nfractions together. We multiply Dialogue: 0,0:02:23.92,0:02:28.72,Default,,0000,0000,0000,,out the two terms in the\Nnumerator. We multiply out the Dialogue: 0,0:02:28.72,0:02:34.38,Default,,0000,0000,0000,,two terms in the denominator, so\Nwe get 4 * 1 is 4. Dialogue: 0,0:02:34.99,0:02:38.60,Default,,0000,0000,0000,,4 * 3 I is 12 I. Dialogue: 0,0:02:39.69,0:02:43.22,Default,,0000,0000,0000,,Seven 8 * 1 is 7 I. Dialogue: 0,0:02:44.10,0:02:49.27,Default,,0000,0000,0000,,+78 times plus three. I is plus\N21 by squares. Dialogue: 0,0:02:50.16,0:02:54.35,Default,,0000,0000,0000,,So that's multiplied. The two\Nterms in the numerator. Now we Dialogue: 0,0:02:54.35,0:02:59.30,Default,,0000,0000,0000,,multiply the two terms in the\Ndenominator to get 1 * 1 one Dialogue: 0,0:02:59.30,0:03:00.45,Default,,0000,0000,0000,,times plus 3I. Dialogue: 0,0:03:01.85,0:03:03.50,Default,,0000,0000,0000,,Minus three items one. Dialogue: 0,0:03:04.60,0:03:08.12,Default,,0000,0000,0000,,And minus three I times\Nplus three. I give this Dialogue: 0,0:03:08.12,0:03:09.53,Default,,0000,0000,0000,,minus nine I squared. Dialogue: 0,0:03:13.59,0:03:15.78,Default,,0000,0000,0000,,What time do this up? Dialogue: 0,0:03:16.38,0:03:22.27,Default,,0000,0000,0000,,21 I squared is 21 times minus\None, so that's minus 21, so Dialogue: 0,0:03:22.27,0:03:25.89,Default,,0000,0000,0000,,we've got 4 - 21 is minus 17. Dialogue: 0,0:03:26.96,0:03:32.04,Default,,0000,0000,0000,,12 + 7 I is 99, so we've\Ngot plus 99. Dialogue: 0,0:03:34.06,0:03:35.57,Default,,0000,0000,0000,,And then in the denominator. Dialogue: 0,0:03:36.51,0:03:40.23,Default,,0000,0000,0000,,I squared is minus one, so we've\Ngot minus nine times minus one Dialogue: 0,0:03:40.23,0:03:42.52,Default,,0000,0000,0000,,is plus nine, 1 + 9 is 10. Dialogue: 0,0:03:43.67,0:03:49.24,Default,,0000,0000,0000,,And three I minus three. I is\Nnothing. So the management turns Dialogue: 0,0:03:49.24,0:03:54.34,Default,,0000,0000,0000,,disappear. So we've ended up\Nwith a real denominator so we Dialogue: 0,0:03:54.34,0:04:00.37,Default,,0000,0000,0000,,could leave our answer like\Nthis. Or we could split it up as Dialogue: 0,0:04:00.37,0:04:04.09,Default,,0000,0000,0000,,minus 17 over 10 + 19 over 10 Dialogue: 0,0:04:04.09,0:04:10.71,Default,,0000,0000,0000,,I. And if we want we\Ncould write as minus one point 7 Dialogue: 0,0:04:10.71,0:04:12.02,Default,,0000,0000,0000,,+ 10.9 I. Dialogue: 0,0:04:14.58,0:04:20.70,Default,,0000,0000,0000,,So that's our answer. When we\Ndivide 4 + 7, I buy 1 - 3. Dialogue: 0,0:04:20.70,0:04:23.96,Default,,0000,0000,0000,,I get minus one point 7 + 1.9. Dialogue: 0,0:04:24.97,0:04:28.18,Default,,0000,0000,0000,,Now let's do another example to\Nillustrate the principals again. Dialogue: 0,0:04:29.28,0:04:34.66,Default,,0000,0000,0000,,Here are two more complex\Nnumbers 2 - 5 I and minus 4 + 3 Dialogue: 0,0:04:34.66,0:04:38.61,Default,,0000,0000,0000,,i's going to divide the first\None by the second one. Dialogue: 0,0:04:39.43,0:04:46.30,Default,,0000,0000,0000,,And we write those as a fraction\N2 - 5 I over minus 4 + Dialogue: 0,0:04:46.30,0:04:52.71,Default,,0000,0000,0000,,3. I now the way to do it\Nis to multiply. Want to multiply Dialogue: 0,0:04:52.71,0:04:58.21,Default,,0000,0000,0000,,the denominator by its complex\Nconjugate, which is minus 4 - 3 Dialogue: 0,0:04:58.21,0:05:02.79,Default,,0000,0000,0000,,I. And because we're multiplying\Nthe denominator by this value, Dialogue: 0,0:05:02.79,0:05:05.08,Default,,0000,0000,0000,,we must multiply the numerator. Dialogue: 0,0:05:06.89,0:05:08.02,Default,,0000,0000,0000,,By this value as well. Dialogue: 0,0:05:10.28,0:05:14.70,Default,,0000,0000,0000,,Now we multiply out the\Nnumerator and denominator. Dialogue: 0,0:05:15.41,0:05:20.83,Default,,0000,0000,0000,,So we have two times minus four\Nis minus 8 two times minus Dialogue: 0,0:05:20.83,0:05:23.33,Default,,0000,0000,0000,,three. I is minus six I. Dialogue: 0,0:05:24.22,0:05:31.10,Default,,0000,0000,0000,,Minus 5I Times minus four is\Nplus 20I and minus 5I times Dialogue: 0,0:05:31.10,0:05:35.68,Default,,0000,0000,0000,,minus three. I is plus 15\NI squared. Dialogue: 0,0:05:36.87,0:05:41.63,Default,,0000,0000,0000,,And then in the dominator we\Nhave minus four times minus 4 Dialogue: 0,0:05:41.63,0:05:45.71,Default,,0000,0000,0000,,inches 16. Minus four\Ntimes minus three I, Dialogue: 0,0:05:45.71,0:05:47.68,Default,,0000,0000,0000,,which is plus 12 I. Dialogue: 0,0:05:48.77,0:05:52.70,Default,,0000,0000,0000,,Plus three I times minus 4\Ninches minus 12 I. Dialogue: 0,0:05:53.40,0:05:57.62,Default,,0000,0000,0000,,I'm plus three I times\Nminus three I, which is Dialogue: 0,0:05:57.62,0:05:59.31,Default,,0000,0000,0000,,minus nine. I squared. Dialogue: 0,0:06:02.11,0:06:03.71,Default,,0000,0000,0000,,And now he tidies up. Dialogue: 0,0:06:04.30,0:06:11.68,Default,,0000,0000,0000,,59 squared is minus 15, so we've\Ngot minus 8 - 15 is minus 23. Dialogue: 0,0:06:12.54,0:06:16.80,Default,,0000,0000,0000,,Minus six I plus\N20I is plus 49. Dialogue: 0,0:06:18.04,0:06:22.26,Default,,0000,0000,0000,,So that's the numerator\Nsimplified, and then the Dialogue: 0,0:06:22.26,0:06:28.13,Default,,0000,0000,0000,,denominator. We've got minus\Nnine. I squared, so that's plus Dialogue: 0,0:06:28.13,0:06:35.32,Default,,0000,0000,0000,,nine. We got 16 + 9 is\N25 and 12. I minus 12. I Dialogue: 0,0:06:35.32,0:06:40.46,Default,,0000,0000,0000,,that disappears, leaving us with\Na real denominator, which is Dialogue: 0,0:06:40.46,0:06:47.66,Default,,0000,0000,0000,,what we wanted. So we can write\Nthat as minus 23 over 25 + Dialogue: 0,0:06:47.66,0:06:49.71,Default,,0000,0000,0000,,14 over 25 I. Dialogue: 0,0:06:51.17,0:06:58.53,Default,,0000,0000,0000,,Which we could also write\Nus minus N .92 + Dialogue: 0,0:06:58.53,0:07:00.00,Default,,0000,0000,0000,,.56 high. Dialogue: 0,0:07:03.92,0:07:06.84,Default,,0000,0000,0000,,And so that's the result\Nof doing this division. Dialogue: 0,0:07:09.08,0:07:12.96,Default,,0000,0000,0000,,Now in the next unit, we'll\Nlook at something called the Dialogue: 0,0:07:12.96,0:07:16.14,Default,,0000,0000,0000,,organ diagram, which is a way\Nof graphically representing Dialogue: 0,0:07:16.14,0:07:16.85,Default,,0000,0000,0000,,complex numbers.