0:00:00.740,0:00:05.390 >> Let's take a look at how we do[br]adding, subtracting, multiplying, 0:00:05.390,0:00:07.650 and dividing complex numbers and see 0:00:07.650,0:00:10.260 how different forms of[br]the complex number lends 0:00:10.260,0:00:15.600 themselves to easier algebraic[br]or arithmetic computations. 0:00:15.600,0:00:17.820 We have two different[br]complex numbers here Z1, 0:00:17.820,0:00:21.270 which is equal to four plus J3[br]in rectangular coordinates. 0:00:21.270,0:00:22.785 In its polar form, 0:00:22.785,0:00:26.780 Z1 is also equal to[br]five angle 37.87 degrees. 0:00:26.780,0:00:31.580 Of course, in it's complex exponential[br]form which is still polar, 0:00:31.580,0:00:34.460 it still has the polar values magnitude 0:00:34.460,0:00:40.820 five angle e to the j angle 37.87 degrees. 0:00:40.820,0:00:46.350 Z2 equal to 2 minus J5, 0:00:46.350,0:00:50.895 in polar form it's that and it's[br]complex exponential form is that. 0:00:50.895,0:00:53.580 Let's just plot these out right here, 0:00:53.580,0:00:56.555 tag it in just so that we[br]can see what we have here. 0:00:56.555,0:01:00.110 In the complex plane we're[br]going over 4 and up 3. 0:01:00.110,0:01:02.370 So it's about like that. 0:01:02.750,0:01:09.275 On this one, we're coming[br]over two and going down five. 0:01:09.275,0:01:17.700 So maybe it's down here something[br]like that for Z2 and Z1. All right. 0:01:17.700,0:01:20.670 Let's start this off by[br]just adding Z1 plus Z2. 0:01:20.670,0:01:25.550 Z1 plus Z2, addition is most easily[br]done in rectangular coordinates. 0:01:25.550,0:01:29.225 Because in that you simply add[br]the real parts and the imaginary parts. 0:01:29.225,0:01:34.710 So Z1 plus Z2 will be 4 plus J3 plus Z2, 0:01:34.710,0:01:37.860 which is 2 minus J5. 0:01:37.860,0:01:41.280 When you do that, you get, let's see, 0:01:41.280,0:01:49.605 4 plus 2 is 6 and J3[br]minus J5 is a minus J2. 0:01:49.605,0:01:52.110 Pretty easy. All right. 0:01:52.110,0:01:53.325 How about multiplying. 0:01:53.325,0:01:57.165 Let's do Z1 times Z2. 0:01:57.165,0:01:59.935 Well, before we go on, let me[br]just say that multiplying or 0:01:59.935,0:02:04.545 adding in polar form really isn't very, 0:02:04.545,0:02:09.245 polar form isn't conducive to adding[br]unless you want to do it graphically. 0:02:09.245,0:02:15.600 Of course, adding graphically you take[br]Z1 and Z2 and you tip to tail them. 0:02:15.600,0:02:17.520 So if that's Z2, 0:02:17.520,0:02:25.165 then you'd add in Z1 like that and the[br]resulting complex number would be that. 0:02:25.165,0:02:28.145 Generally speaking, we[br]don't add in polar form 0:02:28.145,0:02:31.580 unless you're using your calculator[br]and your calculator can do all things. 0:02:31.580,0:02:36.125 So now, let's look at[br]multiplying Z1 times Z2. 0:02:36.125,0:02:39.450 We get then Z1 is 0:02:39.450,0:02:48.885 4 plus J3 times Z2 which is 2 minus J5. 0:02:48.885,0:02:52.670 We see here that we have[br]two binomials multiplied together. 0:02:52.670,0:02:54.335 That means we need to foil them. 0:02:54.335,0:02:56.900 Distribute both of these terms[br]to both of those terms. 0:02:56.900,0:03:02.275 When you do that you get 4 times 2 is 8, 0:03:02.275,0:03:07.840 4 times a minus J5 is a minus J20. 0:03:08.270,0:03:13.590 J3 times 2 is a plus J6. 0:03:13.590,0:03:18.420 J3 times minus J5, 0:03:18.420,0:03:19.560 we've go to be careful here. 0:03:19.560,0:03:22.035 3 times 5 is 15. 0:03:22.035,0:03:26.840 But we've go to be careful with[br]the sign plus times a minus is minus. 0:03:26.840,0:03:30.725 But J times J is a negative one. 0:03:30.725,0:03:34.775 So we have a minus times[br]minus becomes plus. 0:03:34.775,0:03:36.920 So we have two imaginary terms, 0:03:36.920,0:03:42.090 two real terms.8 plus 15 is 23. 0:03:42.560,0:03:50.830 The negative J20 plus J6 is a minus J14. 0:03:52.390,0:03:57.560 So multiplying two complex numbers[br]in rectangular form requires 0:03:57.560,0:04:02.930 us to foil them. 0:04:02.930,0:04:05.420 I'll leave it to you and you stop 0:04:05.420,0:04:07.715 the video right here and[br]go ahead and convert 0:04:07.715,0:04:13.115 this rectangular complex number[br]into its polar form. 0:04:13.115,0:04:23.370 Let me just give the answer. The answer[br]is 26.93 angle negative 31.33. 0:04:24.220,0:04:30.680 All right. Now, let's multiply Z1 times Z2. 0:04:30.680,0:04:35.830 Only doing it in polar coordinates[br]using our polar form of these. 0:04:35.830,0:04:40.925 In polar we have 0:04:40.925,0:04:47.675 Z1 times Z2 that's equal to 5. 0:04:47.675,0:04:51.130 I'm going to write it using[br]its complex exponential form. 0:04:51.130,0:04:57.840 5e to the J37.87 times 0:04:57.840,0:05:05.470 Z2 which is 5.39 angle negative, 0:05:05.470,0:05:08.130 I'm doing an exponential form, 0:05:08.130,0:05:12.780 e to the negative J68.2. 0:05:14.900,0:05:19.310 Of course, these two forms[br]are equivalent forms. 0:05:19.310,0:05:20.750 It's just that it's more obvious what we're 0:05:20.750,0:05:23.090 doing when we write them as exponentials. 0:05:23.090,0:05:28.610 So when you do that 5 times 0:05:28.610,0:05:35.675 5.39 is equal to 26.93. 0:05:35.675,0:05:44.025 Now we have e to the J positive 37.887[br]and we have e to the minus J68.2. 0:05:44.025,0:05:46.170 So when we add the exponents, 0:05:46.170,0:05:55.260 the negative 68.2 plus the[br]J37.87 gives us an angle of, 0:05:55.260,0:05:57.525 so the minus J angle. 0:05:57.525,0:06:00.940 31.33. 0:06:01.340,0:06:05.090 What we see is we get the same answer 0:06:05.090,0:06:08.900 foiling and then converting to polar as we 0:06:08.900,0:06:12.170 did by just starting out in[br]polar form and multiplying 0:06:12.170,0:06:15.710 the coefficients and adding the exponents. 0:06:15.710,0:06:18.800 I'm going to leave it to[br]you to take and convert 0:06:18.800,0:06:21.080 this expression here in 0:06:21.080,0:06:22.790 polar back to rectangular 0:06:22.790,0:06:24.515 and convince yourself that[br]you get the same thing there. 0:06:24.515,0:06:28.405 Go and stop the video and[br]do that now. All right. 0:06:28.405,0:06:30.645 Now, we've got addition, 0:06:30.645,0:06:32.005 by the way, I should have mentioned it, 0:06:32.005,0:06:34.010 subtraction is just the same[br]as addition or when you're 0:06:34.010,0:06:37.535 subtracting the real parts and then[br]the subtracting the imaginary parts. 0:06:37.535,0:06:42.105 We've now done multiplication in[br]both rectangular form and in polar form. 0:06:42.105,0:06:47.975 Now, let's look at at[br]dividing two complex numbers. 0:06:47.975,0:06:54.695 Let's do Z1 divided by Z2. 0:06:54.695,0:06:57.590 You'll recall from your[br]college algebra that 0:06:57.590,0:07:00.665 this gets to be a little bit ugly[br]and then at least a little involved. 0:07:00.665,0:07:04.250 We've got one in doing[br]this in rectangular form. 0:07:04.250,0:07:14.610 We have 4 plus J3 divided by 2 minus J5. 0:07:14.610,0:07:17.870 You'll recall that in college algebra, 0:07:17.870,0:07:21.980 they taught us to do this division[br]by multiplying numerator and 0:07:21.980,0:07:26.755 denominator by the complex conjugate[br]of the denominator. 0:07:26.755,0:07:29.840 Effectively rationalizing[br]the denominator so 0:07:29.840,0:07:32.780 that we get a pure real number[br]in the denominator, 0:07:32.780,0:07:34.580 and then the numerator[br]falls wherever it may. 0:07:34.580,0:07:40.175 So we're going to multiply[br]numerator and denominator 0:07:40.175,0:07:47.280 by 2 plus J5 over 2 plus J5. 0:07:47.280,0:07:52.695 2 plus J5 is the conjugate of 2 minus J5. 0:07:52.695,0:07:55.970 Then you can go through[br]and do the foiling because 0:07:55.970,0:07:58.730 we have this complex number times[br]this complex number, 0:07:58.730,0:08:00.140 it's got to be foiled. 0:08:00.140,0:08:05.300 I'm going to leave that to you[br]to show that it turns out to 0:08:05.300,0:08:14.295 be negative 7 plus J26 divided by, 0:08:14.295,0:08:18.245 now let me show the details down here[br]just to remind you what happens here. 0:08:18.245,0:08:21.575 We've got 2 times 2 is 4. 0:08:21.575,0:08:25.185 We have 2 times a positive J5, 0:08:25.185,0:08:28.840 that's a positive J10. 0:08:29.450,0:08:32.610 Now we have a negative J5 times 2, 0:08:32.610,0:08:35.385 that gives me a minus J10. 0:08:35.385,0:08:44.085 Then we have a negative[br]J5 times a positive J5. 0:08:44.085,0:08:46.590 Again, we've got to be[br]careful with the signs here. 0:08:46.590,0:08:49.225 5 times 5 is 25. 0:08:49.225,0:08:54.020 J times J is a negative 1[br]minus times a plus is a minus. 0:08:54.020,0:08:56.240 So we've got a minus from[br]the signs a minus from 0:08:56.240,0:08:59.645 the J squared that gives us a positive. 0:08:59.645,0:09:10.300 So this then turns out to be[br]negative 7 plus J26 divided by 29, 0:09:10.300,0:09:20.880 which works out to be equal[br]to negative 0.24 plus J0.9. 0:09:20.880,0:09:23.870 So the division of Z1 divided by Z2 in 0:09:23.870,0:09:29.430 rectangular coordinates gives[br]us this rectangular coordinate. 0:09:30.980,0:09:39.905 Now, hub goodness, let's go ahead and[br]convert this to polar coordinates. 0:09:39.905,0:09:45.485 In polar coordinates, we're going to[br]have the magnitude of that as equal to 0:09:45.485,0:09:52.115 the square root of negative[br]0.24 squared plus 0.9 squared. 0:09:52.115,0:09:56.490 That turns out to be 0.928. 0:09:59.110,0:10:01.580 That's the magnitude. 0:10:01.580,0:10:05.360 Now, the angle here gets[br]to be a little bit tricky. 0:10:05.360,0:10:11.105 The angle is going to be the arc tangent 0:10:11.105,0:10:19.070 of 0.9 divided by negative 0.24. 0:10:22.430,0:10:27.920 Now, the arc tangent button on[br]your calculator returns a value between 0:10:27.920,0:10:33.630 plus or minus Pi halves or[br]plus or minus 90 degrees. 0:10:33.690,0:10:36.880 This becomes ambiguous. 0:10:36.880,0:10:40.075 To find out what the actual angle is, 0:10:40.075,0:10:45.785 you take the arc tangent of[br]0.9 divided by negative 0.24 0:10:45.785,0:10:52.460 and you come up with a negative 75 degrees. 0:10:52.460,0:10:54.940 So this turns out to be 0:10:54.940,0:11:03.390 0.928 angle negative 75 degrees. 0:11:03.390,0:11:05.885 But that's not exactly right. 0:11:05.885,0:11:09.835 Coming back here and looking[br]at our complex number, 0:11:09.835,0:11:14.675 we see that we are in[br]not the fourth quadrant, 0:11:14.675,0:11:17.330 negative 75 it run out of room, 0:11:17.330,0:11:19.620 let's just draw it down here. 0:11:21.740,0:11:26.105 It has this down here[br]at negative 75 degrees. 0:11:26.105,0:11:28.415 But when we look at this number here, 0:11:28.415,0:11:30.200 we're at negative 0.24. 0:11:30.200,0:11:31.550 The real part is negative, 0:11:31.550,0:11:35.270 so we're over here and up 0.9. 0:11:35.270,0:11:40.685 In reality where we're at[br]with this complex number 0:11:40.685,0:11:51.135 is 180 degrees off from[br]this negative 75 degrees. 0:11:51.135,0:11:56.840 So the actual angle is[br]negative 75 plus 180. 0:11:56.840,0:12:04.010 This angle right here is[br]actually 105 degrees. 0:12:04.010,0:12:06.320 So when you're doing it in 0:12:06.320,0:12:08.300 rectangular coordinates using[br]your arc tangent button 0:12:08.300,0:12:09.380 on your calculator you got be 0:12:09.380,0:12:14.260 careful and look at the actual[br]coordinates that you're working with, 0:12:14.260,0:12:17.960 because your arc tangent button[br]can cause you some grief. 0:12:17.960,0:12:24.095 Let's do this calculation[br]directly from polar form. 0:12:24.095,0:12:29.130 We have as Z1 over Z2 is equal to, 0:12:29.130,0:12:38.670 in polar form Z1 is 5e to[br]the J37.87 divided by Z2, 0:12:38.670,0:12:49.035 which is 5.39e to the minus J68.2. 0:12:49.035,0:12:52.590 Now we can do that directly[br]and 5 divided by 5.39, 0:12:52.590,0:13:00.585 that is infact equal to 0.928e to the, 0:13:00.585,0:13:05.010 now the exponent is going to be J, 0:13:05.010,0:13:11.535 and we've got 37.87 minus 68.82, 0:13:11.535,0:13:15.005 that turns out to be the 105[br]and some round off error there. 0:13:15.005,0:13:20.375 But it's positive J105 degrees. 0:13:20.375,0:13:23.275 When you do it this way, 0:13:23.275,0:13:26.305 already in the polar form, 0:13:26.305,0:13:28.940 the ambiguity that the arc tangent button 0:13:28.940,0:13:33.060 introduces is we avoid that ambiguity.