0:00:02.830,0:00:06.238 In this unit, went to look at[br]how to multiply 2 complex 0:00:06.238,0:00:09.240 numbers together. And[br]multiplying two complex numbers 0:00:09.240,0:00:13.344 simply requires us to be able to[br]multiply out brackets to collect 0:00:13.344,0:00:17.106 together like turns, and to[br]remember that I is the special 0:00:17.106,0:00:21.210 number whose property is that I[br]squared is equal to minus one. 0:00:21.860,0:00:24.280 So let's look at how that[br]all works in an example. 0:00:25.830,0:00:29.251 So here we have two complex[br]numbers 4 + 7 I. 0:00:29.760,0:00:34.110 And 2 + 3 I and we're going to[br]do is going to multiply these 0:00:34.110,0:00:35.270 two complex numbers together. 0:00:36.080,0:00:40.695 And the first thing we do is we[br]just multiply out the brackets 0:00:40.695,0:00:44.955 so we each term in the first[br]bracket must multiply each term 0:00:44.955,0:00:46.375 in the SEC bracket. 0:00:47.060,0:00:50.993 So we have 4 * 2 which is 8. 0:00:51.770,0:00:57.110 Four times plus three I, which[br]is plus 12 I. 0:00:58.100,0:01:02.042 Plus Seven I times two is 0:01:02.042,0:01:09.612 plus 49. And plus Seven[br]I times plus three. I is 0:01:09.612,0:01:11.406 21. I squared. 0:01:13.400,0:01:17.274 Now we see straight away that we[br]would be able to combine these 0:01:17.274,0:01:20.254 two terms because they're both[br]terms with eyes in them. 0:01:20.940,0:01:28.346 So the the front is going to[br]stay the same plus 12 I plus 0:01:28.346,0:01:32.049 14. I gives us plus 26 I. 0:01:33.380,0:01:38.770 Now over this term, on the end,[br]which is 21, I squared. Now we 0:01:38.770,0:01:44.160 have to remember what we know[br]about I. I is the square root of 0:01:44.160,0:01:49.165 minus one or said another way I[br]squared is equal to minus one. 0:01:49.165,0:01:54.170 So in this term 21 I squared we[br]can replace the isquared by 0:01:54.170,0:01:57.635 minus one. So that's plus 21[br]times minus one. 0:01:58.820,0:02:05.796 Now 21 times minus one is minus[br]21, so we have 8 - 21. You can 0:02:05.796,0:02:07.540 combine those two terms. 0:02:08.070,0:02:11.494 8 - 21 is 0:02:11.494,0:02:19.306 minus 13. Plus the[br]26 I was already there. 0:02:19.310,0:02:23.230 And so our answer, when we[br]multiply these two complex 0:02:23.230,0:02:27.934 numbers together, is this new[br]complex number minus 13 + 26? I 0:02:27.934,0:02:31.854 OK? We're going to look at[br]another example, two different 0:02:31.854,0:02:36.558 complex numbers. This time the[br]complex numbers minus 2 + 5. I 0:02:36.558,0:02:42.046 first one on 1 - 3 I is the[br]second one, exactly the same 0:02:42.046,0:02:46.750 principles before, but we have[br]to be a bit more careful 'cause 0:02:46.750,0:02:49.102 we got lots of minus signs 0:02:49.102,0:02:54.190 floating about. So we multiply[br]out the brackets minus 2 * 1. 0:02:54.770,0:02:56.018 Is minus 2. 0:02:56.780,0:03:02.360 Minus two times minus three I[br]gives us plus 6I. 0:03:03.030,0:03:06.612 Plus 5I Times one gives us 0:03:06.612,0:03:13.990 plus 5I. And plus 5I[br]Times minus three I years, minus 0:03:13.990,0:03:15.730 15 I squared. 0:03:17.600,0:03:23.827 Now we combine together our[br]items so we have minus 2 + 6 0:03:23.827,0:03:27.659 I plus 5I, giving us plus 11 I. 0:03:28.900,0:03:33.866 And in this term, remember that[br]I squared is minus one, so this 0:03:33.866,0:03:37.686 is minus 15 times minus one,[br]which is plus 15. 0:03:38.540,0:03:45.064 And so the final thing we do is[br]combine the minus two and the 0:03:45.064,0:03:50.190 plus 15 to get plus 13 and then[br]plus 11 I. 0:03:50.190,0:03:54.340 And so our answer, we multiply[br]these two complex numbers 0:03:54.340,0:03:58.075 together is the complex number[br]13 + 11 I. 0:03:59.930,0:04:02.810 So that's how we multiply[br]together to complex numbers 0:04:02.810,0:04:06.650 in the next unit. We're going[br]to look at a property that 0:04:06.650,0:04:08.890 complex numbers have called[br]the complex conjugate.