1 00:00:02,830 --> 00:00:06,238 In this unit, went to look at how to multiply 2 complex 2 00:00:06,238 --> 00:00:09,240 numbers together. And multiplying two complex numbers 3 00:00:09,240 --> 00:00:13,344 simply requires us to be able to multiply out brackets to collect 4 00:00:13,344 --> 00:00:17,106 together like turns, and to remember that I is the special 5 00:00:17,106 --> 00:00:21,210 number whose property is that I squared is equal to minus one. 6 00:00:21,860 --> 00:00:24,280 So let's look at how that all works in an example. 7 00:00:25,830 --> 00:00:29,251 So here we have two complex numbers 4 + 7 I. 8 00:00:29,760 --> 00:00:34,110 And 2 + 3 I and we're going to do is going to multiply these 9 00:00:34,110 --> 00:00:35,270 two complex numbers together. 10 00:00:36,080 --> 00:00:40,695 And the first thing we do is we just multiply out the brackets 11 00:00:40,695 --> 00:00:44,955 so we each term in the first bracket must multiply each term 12 00:00:44,955 --> 00:00:46,375 in the SEC bracket. 13 00:00:47,060 --> 00:00:50,993 So we have 4 * 2 which is 8. 14 00:00:51,770 --> 00:00:57,110 Four times plus three I, which is plus 12 I. 15 00:00:58,100 --> 00:01:02,042 Plus Seven I times two is 16 00:01:02,042 --> 00:01:09,612 plus 49. And plus Seven I times plus three. I is 17 00:01:09,612 --> 00:01:11,406 21. I squared. 18 00:01:13,400 --> 00:01:17,274 Now we see straight away that we would be able to combine these 19 00:01:17,274 --> 00:01:20,254 two terms because they're both terms with eyes in them. 20 00:01:20,940 --> 00:01:28,346 So the the front is going to stay the same plus 12 I plus 21 00:01:28,346 --> 00:01:32,049 14. I gives us plus 26 I. 22 00:01:33,380 --> 00:01:38,770 Now over this term, on the end, which is 21, I squared. Now we 23 00:01:38,770 --> 00:01:44,160 have to remember what we know about I. I is the square root of 24 00:01:44,160 --> 00:01:49,165 minus one or said another way I squared is equal to minus one. 25 00:01:49,165 --> 00:01:54,170 So in this term 21 I squared we can replace the isquared by 26 00:01:54,170 --> 00:01:57,635 minus one. So that's plus 21 times minus one. 27 00:01:58,820 --> 00:02:05,796 Now 21 times minus one is minus 21, so we have 8 - 21. You can 28 00:02:05,796 --> 00:02:07,540 combine those two terms. 29 00:02:08,070 --> 00:02:11,494 8 - 21 is 30 00:02:11,494 --> 00:02:19,306 minus 13. Plus the 26 I was already there. 31 00:02:19,310 --> 00:02:23,230 And so our answer, when we multiply these two complex 32 00:02:23,230 --> 00:02:27,934 numbers together, is this new complex number minus 13 + 26? I 33 00:02:27,934 --> 00:02:31,854 OK? We're going to look at another example, two different 34 00:02:31,854 --> 00:02:36,558 complex numbers. This time the complex numbers minus 2 + 5. I 35 00:02:36,558 --> 00:02:42,046 first one on 1 - 3 I is the second one, exactly the same 36 00:02:42,046 --> 00:02:46,750 principles before, but we have to be a bit more careful 'cause 37 00:02:46,750 --> 00:02:49,102 we got lots of minus signs 38 00:02:49,102 --> 00:02:54,190 floating about. So we multiply out the brackets minus 2 * 1. 39 00:02:54,770 --> 00:02:56,018 Is minus 2. 40 00:02:56,780 --> 00:03:02,360 Minus two times minus three I gives us plus 6I. 41 00:03:03,030 --> 00:03:06,612 Plus 5I Times one gives us 42 00:03:06,612 --> 00:03:13,990 plus 5I. And plus 5I Times minus three I years, minus 43 00:03:13,990 --> 00:03:15,730 15 I squared. 44 00:03:17,600 --> 00:03:23,827 Now we combine together our items so we have minus 2 + 6 45 00:03:23,827 --> 00:03:27,659 I plus 5I, giving us plus 11 I. 46 00:03:28,900 --> 00:03:33,866 And in this term, remember that I squared is minus one, so this 47 00:03:33,866 --> 00:03:37,686 is minus 15 times minus one, which is plus 15. 48 00:03:38,540 --> 00:03:45,064 And so the final thing we do is combine the minus two and the 49 00:03:45,064 --> 00:03:50,190 plus 15 to get plus 13 and then plus 11 I. 50 00:03:50,190 --> 00:03:54,340 And so our answer, we multiply these two complex numbers 51 00:03:54,340 --> 00:03:58,075 together is the complex number 13 + 11 I. 52 00:03:59,930 --> 00:04:02,810 So that's how we multiply together to complex numbers 53 00:04:02,810 --> 00:04:06,650 in the next unit. We're going to look at a property that 54 00:04:06,650 --> 00:04:08,890 complex numbers have called the complex conjugate.