0:00:02.990,0:00:06.721 In this unit, we're going to[br]look at how to add and subtract 0:00:06.721,0:00:11.425 complex numbers. Now when you're[br]at school, you first learn to 0:00:11.425,0:00:14.710 add and subtract using the[br]counting numbers. That's numbers 0:00:14.710,0:00:16.170 1234 and so on. 0:00:17.020,0:00:20.814 And every time you meet a new[br]set of numbers, you learn a new 0:00:20.814,0:00:24.337 rule or a new process for how[br]you can add and subtract them. 0:00:24.337,0:00:27.860 So for example, when you meet[br]fractions, you learn it to add 2 0:00:27.860,0:00:30.570 fractions, you must write them[br]both over a common denominator. 0:00:31.450,0:00:35.614 No, it's the same with complex[br]numbers. You just need to learn 0:00:35.614,0:00:39.778 the correct process to how to[br]add and subtract them, and with 0:00:39.778,0:00:42.554 complex numbers is quite[br]straightforward. It's based on 0:00:42.554,0:00:46.024 the principle you have seen[br]before in algebra where you 0:00:46.024,0:00:49.841 collect together some like[br]terms, so we're going to do is 0:00:49.841,0:00:53.311 going to start by adding two[br]algebraic expressions and then 0:00:53.311,0:00:56.781 see how that carries over into[br]adding together 2 complex 0:00:56.781,0:00:59.557 numbers. So let's take two[br]algebraic expressions. Let's 0:00:59.557,0:01:01.292 take the expressions for plus. 0:01:01.470,0:01:07.540 70 and 2[br]+ 3 two. 0:01:08.430,0:01:13.435 So we have two expressions are[br]going to do is going to add 0:01:13.435,0:01:18.055 these two things together, so[br]we're going to have 4 + 70. 0:01:19.110,0:01:22.630 Plus 2 + 3 T. 0:01:24.390,0:01:27.846 Now, the principle of collecting[br]together, like turns, simply 0:01:27.846,0:01:31.302 says looking your long[br]expression and look for terms 0:01:31.302,0:01:36.294 that are somehow the same. So in[br]this expression we have the four 0:01:36.294,0:01:41.286 on the two, which is just[br]numbers. So we can put those two 0:01:41.286,0:01:44.358 together. So 4 + 2 gives us 6. 0:01:45.340,0:01:49.960 And we have these two terms[br]which have both got Tees in 0:01:49.960,0:01:54.965 them, so we've got plus 70 + 3[br]more Tees so that gives 0:01:54.965,0:01:56.505 altogether plus 10 T. 0:01:58.200,0:02:02.640 So we started with two algebraic[br]expressions. We've added them 0:02:02.640,0:02:07.132 up. And we simplified by[br]collecting together terms that 0:02:07.132,0:02:12.254 are the same terms that are just[br]numbers and terms that have got 0:02:12.254,0:02:15.406 teasing them. Now adding[br]together complex numbers works 0:02:15.406,0:02:21.316 in exactly the same sort of way,[br]so we're going to do now is take 0:02:21.316,0:02:26.044 two complex numbers and add them[br]together. So I'm going to take 0:02:26.044,0:02:28.408 the complex #4 + 7 I. 0:02:29.360,0:02:33.248 And the complex number 2 + 3 I. 0:02:34.120,0:02:38.564 So this is first complex[br]number and this is my second 0:02:38.564,0:02:41.796 complex number. I'm going to[br]add them together. 0:02:42.870,0:02:44.854 Now what I do is I say well. 0:02:45.660,0:02:50.364 If if I take away the brackets,[br]I won't have changed anything. 0:02:56.300,0:02:59.600 Now I can look at this[br]expiration, which doesn't have 0:02:59.600,0:03:03.890 any brackets in and I can look[br]for terms that I can collect 0:03:03.890,0:03:08.180 together so I can collect the[br]four in the two together to get 0:03:08.180,0:03:12.800 6, and I can collect the plus[br]Seven I and the plus three I 0:03:12.800,0:03:14.450 together to get plus 10I. 0:03:17.240,0:03:21.368 And so that's the answer. When I[br]add these two complex numbers 0:03:21.368,0:03:26.184 together, I get this new complex[br]number 6 + 10 I and you'll see 0:03:26.184,0:03:30.312 that in fact all we've done is[br]we've added together the real 0:03:30.312,0:03:33.408 parts of the two complex[br]numbers, and we've added 0:03:33.408,0:03:37.192 together the imaginary parts of[br]the two complex numbers to get 0:03:37.192,0:03:41.258 the answer. So let's do[br]another example. OK, in this 0:03:41.258,0:03:44.057 example will just take two[br]different complex numbers and 0:03:44.057,0:03:44.990 add them together. 0:03:46.950,0:03:50.480 So we'll take the complex[br]numbers 5 + 6 I. 0:03:51.650,0:03:54.778 And the complex number 7 - 3 I. 0:03:56.340,0:03:58.475 Will add those two complex 0:03:58.475,0:04:04.016 numbers together. So just as[br]before will write everything out 0:04:04.016,0:04:05.432 without any brackets. 0:04:08.960,0:04:13.003 And then we look for the terms[br]that we can collect together so 0:04:13.003,0:04:16.424 we can collect together the real[br]parts. That's just the numbers. 0:04:16.424,0:04:18.912 The real numbers 5 + 7 to give 0:04:18.912,0:04:24.350 us 12. And then we can collect[br]together the two imaginary parts 0:04:24.350,0:04:28.450 plus six I minus three I giving[br]us plus 3I. 0:04:31.220,0:04:34.828 And so the answer when we had[br]these two complex numbers 0:04:34.828,0:04:39.748 together is 12 + 3 I and once[br]again we see that we've done is. 0:04:39.748,0:04:43.028 We've just added together the[br]real parts and we've added 0:04:43.028,0:04:44.340 together the imaginary parts. 0:04:45.980,0:04:48.950 Now subtraction works in[br]exactly the same way, we just 0:04:48.950,0:04:52.811 have to be careful that we[br]make sure that we do take away 0:04:52.811,0:04:55.187 the whole of the second[br]complex number. 'cause 0:04:55.187,0:04:58.157 sometimes people might just[br]take away the real part and 0:04:58.157,0:05:01.127 forget to take away the[br]imaginary part as well. So 0:05:01.127,0:05:03.206 let's see how we can do that. 0:05:04.370,0:05:10.430 So if we go back to our first[br]pair of numbers 4 + 7 I. 0:05:11.220,0:05:15.730 We're going to take away from[br]that 2 + 3 I. 0:05:18.310,0:05:22.236 So when we remove the brackets,[br]this time we have to be careful 0:05:22.236,0:05:25.860 to make the minus operate on the[br]whole of this complex number. 0:05:26.620,0:05:30.700 So from the first complex number[br]we have 4 + 7 I. 0:05:32.110,0:05:35.818 But then when we take these[br]brackets out, we get minus 2. 0:05:37.990,0:05:42.202 Then we get minus plus three[br]eyes are getting minus three I. 0:05:43.590,0:05:46.505 And now we're in a position to[br]collect together like terms. 0:05:46.505,0:05:49.420 Just as we've been doing in all[br]the examples so far. 0:05:50.950,0:05:54.734 So our numbers are 4 - 2 which 0:05:54.734,0:06:00.370 is 2. And then our terms[br]with I in them are plus 0:06:00.370,0:06:04.660 Seven. I minus three I,[br]so that gives us plus 4I. 0:06:07.590,0:06:12.336 And so the answer when we do[br]this subtraction 4 + 7 I take 0:06:12.336,0:06:17.421 away 2 + 3. I gives us a[br]complex number 2 + 4 I and 0:06:17.421,0:06:21.489 again we can see that what[br]we've done is. We know this, 0:06:21.489,0:06:25.557 I'm subtracting the real parts[br]4 takeaway two to give us two 0:06:25.557,0:06:28.947 and subtracted the imaginary[br]part 7 takeaway three to give 0:06:28.947,0:06:32.676 us four and that goes with the[br]eye because that's the 0:06:32.676,0:06:33.354 imaginary part. 0:06:35.400,0:06:39.820 The one last example now just to[br]make sure we've got this well 0:06:39.820,0:06:43.220 and truly understood, we're[br]going to take start with a 0:06:43.220,0:06:45.260 complex number 5 + 6 I. 0:06:46.960,0:06:53.106 I'm going to take away from that[br]the complex number 7 - 3 I. 0:06:54.580,0:06:58.228 Let me just repeat the process.[br]We've seen several Times Now we 0:06:58.228,0:07:00.052 remove the brackets to start off 0:07:00.052,0:07:04.270 with. So nothing happens to[br]the first 2 terms, but the 0:07:04.270,0:07:07.370 minus operates on everything[br]that's in this next set of 0:07:07.370,0:07:09.230 brackets. So we get minus 7. 0:07:10.390,0:07:14.926 Then we get minus minus three.[br]I2 minus is making a plus. 0:07:16.260,0:07:21.221 Three, I and now we collect[br]together the real numbers we 0:07:21.221,0:07:23.025 collect together the imaginary 0:07:23.025,0:07:30.453 numbers. We have 5 - 7 which[br]is minus two. We have plus 6I 0:07:30.453,0:07:33.890 plus three I which is +9 I. 0:07:36.220,0:07:40.731 And so that's our answer. This[br]time and again we see that what 0:07:40.731,0:07:45.242 we've done is, we've subtracted[br]the real parts 5 - 7 gives us 0:07:45.242,0:07:48.712 minus two, and then we've[br]subtracted the imaginary parts 6 0:07:48.712,0:07:53.223 minus minus three is 6 + 3,[br]which is 9 because there's the 0:07:53.223,0:07:55.999 imaginary parts. That's nine I[br]in the answer. 0:07:57.090,0:08:02.173 So the way we add or subtract[br]2 complex numbers is simply to 0:08:02.173,0:08:05.692 use algebraic manipulation and[br]collect together like terms in 0:08:05.692,0:08:10.384 the next unit. We look at how[br]to multiply 2 complex numbers 0:08:10.384,0:08:10.775 together.