[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.74,0:00:05.39,Default,,0000,0000,0000,,>> Let's take a look at how we do\Nadding, subtracting, multiplying,
Dialogue: 0,0:00:05.39,0:00:07.65,Default,,0000,0000,0000,,and dividing complex numbers and see
Dialogue: 0,0:00:07.65,0:00:10.26,Default,,0000,0000,0000,,how different forms of\Nthe complex number lends
Dialogue: 0,0:00:10.26,0:00:15.60,Default,,0000,0000,0000,,themselves to easier algebraic\Nor arithmetic computations.
Dialogue: 0,0:00:15.60,0:00:17.82,Default,,0000,0000,0000,,We have two different\Ncomplex numbers here Z1,
Dialogue: 0,0:00:17.82,0:00:21.27,Default,,0000,0000,0000,,which is equal to four plus J3\Nin rectangular coordinates.
Dialogue: 0,0:00:21.27,0:00:22.78,Default,,0000,0000,0000,,In its polar form,
Dialogue: 0,0:00:22.78,0:00:26.78,Default,,0000,0000,0000,,Z1 is also equal to\Nfive angle 37.87 degrees.
Dialogue: 0,0:00:26.78,0:00:31.58,Default,,0000,0000,0000,,Of course, in it's complex exponential\Nform which is still polar,
Dialogue: 0,0:00:31.58,0:00:34.46,Default,,0000,0000,0000,,it still has the polar values magnitude
Dialogue: 0,0:00:34.46,0:00:40.82,Default,,0000,0000,0000,,five angle e to the j angle 37.87 degrees.
Dialogue: 0,0:00:40.82,0:00:46.35,Default,,0000,0000,0000,,Z2 equal to 2 minus J5,
Dialogue: 0,0:00:46.35,0:00:50.90,Default,,0000,0000,0000,,in polar form it's that and it's\Ncomplex exponential form is that.
Dialogue: 0,0:00:50.90,0:00:53.58,Default,,0000,0000,0000,,Let's just plot these out right here,
Dialogue: 0,0:00:53.58,0:00:56.56,Default,,0000,0000,0000,,tag it in just so that we\Ncan see what we have here.
Dialogue: 0,0:00:56.56,0:01:00.11,Default,,0000,0000,0000,,In the complex plane we're\Ngoing over 4 and up 3.
Dialogue: 0,0:01:00.11,0:01:02.37,Default,,0000,0000,0000,,So it's about like that.
Dialogue: 0,0:01:02.75,0:01:09.28,Default,,0000,0000,0000,,On this one, we're coming\Nover two and going down five.
Dialogue: 0,0:01:09.28,0:01:17.70,Default,,0000,0000,0000,,So maybe it's down here something\Nlike that for Z2 and Z1. All right.
Dialogue: 0,0:01:17.70,0:01:20.67,Default,,0000,0000,0000,,Let's start this off by\Njust adding Z1 plus Z2.
Dialogue: 0,0:01:20.67,0:01:25.55,Default,,0000,0000,0000,,Z1 plus Z2, addition is most easily\Ndone in rectangular coordinates.
Dialogue: 0,0:01:25.55,0:01:29.22,Default,,0000,0000,0000,,Because in that you simply add\Nthe real parts and the imaginary parts.
Dialogue: 0,0:01:29.22,0:01:34.71,Default,,0000,0000,0000,,So Z1 plus Z2 will be 4 plus J3 plus Z2,
Dialogue: 0,0:01:34.71,0:01:37.86,Default,,0000,0000,0000,,which is 2 minus J5.
Dialogue: 0,0:01:37.86,0:01:41.28,Default,,0000,0000,0000,,When you do that, you get, let's see,
Dialogue: 0,0:01:41.28,0:01:49.60,Default,,0000,0000,0000,,4 plus 2 is 6 and J3\Nminus J5 is a minus J2.
Dialogue: 0,0:01:49.60,0:01:52.11,Default,,0000,0000,0000,,Pretty easy. All right.
Dialogue: 0,0:01:52.11,0:01:53.32,Default,,0000,0000,0000,,How about multiplying.
Dialogue: 0,0:01:53.32,0:01:57.16,Default,,0000,0000,0000,,Let's do Z1 times Z2.
Dialogue: 0,0:01:57.16,0:01:59.94,Default,,0000,0000,0000,,Well, before we go on, let me\Njust say that multiplying or
Dialogue: 0,0:01:59.94,0:02:04.54,Default,,0000,0000,0000,,adding in polar form really isn't very,
Dialogue: 0,0:02:04.54,0:02:09.24,Default,,0000,0000,0000,,polar form isn't conducive to adding\Nunless you want to do it graphically.
Dialogue: 0,0:02:09.24,0:02:15.60,Default,,0000,0000,0000,,Of course, adding graphically you take\NZ1 and Z2 and you tip to tail them.
Dialogue: 0,0:02:15.60,0:02:17.52,Default,,0000,0000,0000,,So if that's Z2,
Dialogue: 0,0:02:17.52,0:02:25.16,Default,,0000,0000,0000,,then you'd add in Z1 like that and the\Nresulting complex number would be that.
Dialogue: 0,0:02:25.16,0:02:28.14,Default,,0000,0000,0000,,Generally speaking, we\Ndon't add in polar form
Dialogue: 0,0:02:28.14,0:02:31.58,Default,,0000,0000,0000,,unless you're using your calculator\Nand your calculator can do all things.
Dialogue: 0,0:02:31.58,0:02:36.12,Default,,0000,0000,0000,,So now, let's look at\Nmultiplying Z1 times Z2.
Dialogue: 0,0:02:36.12,0:02:39.45,Default,,0000,0000,0000,,We get then Z1 is
Dialogue: 0,0:02:39.45,0:02:48.88,Default,,0000,0000,0000,,4 plus J3 times Z2 which is 2 minus J5.
Dialogue: 0,0:02:48.88,0:02:52.67,Default,,0000,0000,0000,,We see here that we have\Ntwo binomials multiplied together.
Dialogue: 0,0:02:52.67,0:02:54.34,Default,,0000,0000,0000,,That means we need to foil them.
Dialogue: 0,0:02:54.34,0:02:56.90,Default,,0000,0000,0000,,Distribute both of these terms\Nto both of those terms.
Dialogue: 0,0:02:56.90,0:03:02.28,Default,,0000,0000,0000,,When you do that you get 4 times 2 is 8,
Dialogue: 0,0:03:02.28,0:03:07.84,Default,,0000,0000,0000,,4 times a minus J5 is a minus J20.
Dialogue: 0,0:03:08.27,0:03:13.59,Default,,0000,0000,0000,,J3 times 2 is a plus J6.
Dialogue: 0,0:03:13.59,0:03:18.42,Default,,0000,0000,0000,,J3 times minus J5,
Dialogue: 0,0:03:18.42,0:03:19.56,Default,,0000,0000,0000,,we've go to be careful here.
Dialogue: 0,0:03:19.56,0:03:22.04,Default,,0000,0000,0000,,3 times 5 is 15.
Dialogue: 0,0:03:22.04,0:03:26.84,Default,,0000,0000,0000,,But we've go to be careful with\Nthe sign plus times a minus is minus.
Dialogue: 0,0:03:26.84,0:03:30.72,Default,,0000,0000,0000,,But J times J is a negative one.
Dialogue: 0,0:03:30.72,0:03:34.78,Default,,0000,0000,0000,,So we have a minus times\Nminus becomes plus.
Dialogue: 0,0:03:34.78,0:03:36.92,Default,,0000,0000,0000,,So we have two imaginary terms,
Dialogue: 0,0:03:36.92,0:03:42.09,Default,,0000,0000,0000,,two real terms.8 plus 15 is 23.
Dialogue: 0,0:03:42.56,0:03:50.83,Default,,0000,0000,0000,,The negative J20 plus J6 is a minus J14.
Dialogue: 0,0:03:52.39,0:03:57.56,Default,,0000,0000,0000,,So multiplying two complex numbers\Nin rectangular form requires
Dialogue: 0,0:03:57.56,0:04:02.93,Default,,0000,0000,0000,,us to foil them.
Dialogue: 0,0:04:02.93,0:04:05.42,Default,,0000,0000,0000,,I'll leave it to you and you stop
Dialogue: 0,0:04:05.42,0:04:07.72,Default,,0000,0000,0000,,the video right here and\Ngo ahead and convert
Dialogue: 0,0:04:07.72,0:04:13.12,Default,,0000,0000,0000,,this rectangular complex number\Ninto its polar form.
Dialogue: 0,0:04:13.12,0:04:23.37,Default,,0000,0000,0000,,Let me just give the answer. The answer\Nis 26.93 angle negative 31.33.
Dialogue: 0,0:04:24.22,0:04:30.68,Default,,0000,0000,0000,,All right. Now, let's multiply Z1 times Z2.
Dialogue: 0,0:04:30.68,0:04:35.83,Default,,0000,0000,0000,,Only doing it in polar coordinates\Nusing our polar form of these.
Dialogue: 0,0:04:35.83,0:04:40.92,Default,,0000,0000,0000,,In polar we have
Dialogue: 0,0:04:40.92,0:04:47.68,Default,,0000,0000,0000,,Z1 times Z2 that's equal to 5.
Dialogue: 0,0:04:47.68,0:04:51.13,Default,,0000,0000,0000,,I'm going to write it using\Nits complex exponential form.
Dialogue: 0,0:04:51.13,0:04:57.84,Default,,0000,0000,0000,,5e to the J37.87 times
Dialogue: 0,0:04:57.84,0:05:05.47,Default,,0000,0000,0000,,Z2 which is 5.39 angle negative,
Dialogue: 0,0:05:05.47,0:05:08.13,Default,,0000,0000,0000,,I'm doing an exponential form,
Dialogue: 0,0:05:08.13,0:05:12.78,Default,,0000,0000,0000,,e to the negative J68.2.
Dialogue: 0,0:05:14.90,0:05:19.31,Default,,0000,0000,0000,,Of course, these two forms\Nare equivalent forms.
Dialogue: 0,0:05:19.31,0:05:20.75,Default,,0000,0000,0000,,It's just that it's more obvious what we're
Dialogue: 0,0:05:20.75,0:05:23.09,Default,,0000,0000,0000,,doing when we write them as exponentials.
Dialogue: 0,0:05:23.09,0:05:28.61,Default,,0000,0000,0000,,So when you do that 5 times
Dialogue: 0,0:05:28.61,0:05:35.68,Default,,0000,0000,0000,,5.39 is equal to 26.93.
Dialogue: 0,0:05:35.68,0:05:44.02,Default,,0000,0000,0000,,Now we have e to the J positive 37.887\Nand we have e to the minus J68.2.
Dialogue: 0,0:05:44.02,0:05:46.17,Default,,0000,0000,0000,,So when we add the exponents,
Dialogue: 0,0:05:46.17,0:05:55.26,Default,,0000,0000,0000,,the negative 68.2 plus the\NJ37.87 gives us an angle of,
Dialogue: 0,0:05:55.26,0:05:57.52,Default,,0000,0000,0000,,so the minus J angle.
Dialogue: 0,0:05:57.52,0:06:00.94,Default,,0000,0000,0000,,31.33.
Dialogue: 0,0:06:01.34,0:06:05.09,Default,,0000,0000,0000,,What we see is we get the same answer
Dialogue: 0,0:06:05.09,0:06:08.90,Default,,0000,0000,0000,,foiling and then converting to polar as we
Dialogue: 0,0:06:08.90,0:06:12.17,Default,,0000,0000,0000,,did by just starting out in\Npolar form and multiplying
Dialogue: 0,0:06:12.17,0:06:15.71,Default,,0000,0000,0000,,the coefficients and adding the exponents.
Dialogue: 0,0:06:15.71,0:06:18.80,Default,,0000,0000,0000,,I'm going to leave it to\Nyou to take and convert
Dialogue: 0,0:06:18.80,0:06:21.08,Default,,0000,0000,0000,,this expression here in
Dialogue: 0,0:06:21.08,0:06:22.79,Default,,0000,0000,0000,,polar back to rectangular
Dialogue: 0,0:06:22.79,0:06:24.52,Default,,0000,0000,0000,,and convince yourself that\Nyou get the same thing there.
Dialogue: 0,0:06:24.52,0:06:28.40,Default,,0000,0000,0000,,Go and stop the video and\Ndo that now. All right.
Dialogue: 0,0:06:28.40,0:06:30.64,Default,,0000,0000,0000,,Now, we've got addition,
Dialogue: 0,0:06:30.64,0:06:32.00,Default,,0000,0000,0000,,by the way, I should have mentioned it,
Dialogue: 0,0:06:32.00,0:06:34.01,Default,,0000,0000,0000,,subtraction is just the same\Nas addition or when you're
Dialogue: 0,0:06:34.01,0:06:37.54,Default,,0000,0000,0000,,subtracting the real parts and then\Nthe subtracting the imaginary parts.
Dialogue: 0,0:06:37.54,0:06:42.10,Default,,0000,0000,0000,,We've now done multiplication in\Nboth rectangular form and in polar form.
Dialogue: 0,0:06:42.10,0:06:47.98,Default,,0000,0000,0000,,Now, let's look at at\Ndividing two complex numbers.
Dialogue: 0,0:06:47.98,0:06:54.70,Default,,0000,0000,0000,,Let's do Z1 divided by Z2.
Dialogue: 0,0:06:54.70,0:06:57.59,Default,,0000,0000,0000,,You'll recall from your\Ncollege algebra that
Dialogue: 0,0:06:57.59,0:07:00.66,Default,,0000,0000,0000,,this gets to be a little bit ugly\Nand then at least a little involved.
Dialogue: 0,0:07:00.66,0:07:04.25,Default,,0000,0000,0000,,We've got one in doing\Nthis in rectangular form.
Dialogue: 0,0:07:04.25,0:07:14.61,Default,,0000,0000,0000,,We have 4 plus J3 divided by 2 minus J5.
Dialogue: 0,0:07:14.61,0:07:17.87,Default,,0000,0000,0000,,You'll recall that in college algebra,
Dialogue: 0,0:07:17.87,0:07:21.98,Default,,0000,0000,0000,,they taught us to do this division\Nby multiplying numerator and
Dialogue: 0,0:07:21.98,0:07:26.76,Default,,0000,0000,0000,,denominator by the complex conjugate\Nof the denominator.
Dialogue: 0,0:07:26.76,0:07:29.84,Default,,0000,0000,0000,,Effectively rationalizing\Nthe denominator so
Dialogue: 0,0:07:29.84,0:07:32.78,Default,,0000,0000,0000,,that we get a pure real number\Nin the denominator,
Dialogue: 0,0:07:32.78,0:07:34.58,Default,,0000,0000,0000,,and then the numerator\Nfalls wherever it may.
Dialogue: 0,0:07:34.58,0:07:40.18,Default,,0000,0000,0000,,So we're going to multiply\Nnumerator and denominator
Dialogue: 0,0:07:40.18,0:07:47.28,Default,,0000,0000,0000,,by 2 plus J5 over 2 plus J5.
Dialogue: 0,0:07:47.28,0:07:52.70,Default,,0000,0000,0000,,2 plus J5 is the conjugate of 2 minus J5.
Dialogue: 0,0:07:52.70,0:07:55.97,Default,,0000,0000,0000,,Then you can go through\Nand do the foiling because
Dialogue: 0,0:07:55.97,0:07:58.73,Default,,0000,0000,0000,,we have this complex number times\Nthis complex number,
Dialogue: 0,0:07:58.73,0:08:00.14,Default,,0000,0000,0000,,it's got to be foiled.
Dialogue: 0,0:08:00.14,0:08:05.30,Default,,0000,0000,0000,,I'm going to leave that to you\Nto show that it turns out to
Dialogue: 0,0:08:05.30,0:08:14.30,Default,,0000,0000,0000,,be negative 7 plus J26 divided by,
Dialogue: 0,0:08:14.30,0:08:18.24,Default,,0000,0000,0000,,now let me show the details down here\Njust to remind you what happens here.
Dialogue: 0,0:08:18.24,0:08:21.58,Default,,0000,0000,0000,,We've got 2 times 2 is 4.
Dialogue: 0,0:08:21.58,0:08:25.18,Default,,0000,0000,0000,,We have 2 times a positive J5,
Dialogue: 0,0:08:25.18,0:08:28.84,Default,,0000,0000,0000,,that's a positive J10.
Dialogue: 0,0:08:29.45,0:08:32.61,Default,,0000,0000,0000,,Now we have a negative J5 times 2,
Dialogue: 0,0:08:32.61,0:08:35.38,Default,,0000,0000,0000,,that gives me a minus J10.
Dialogue: 0,0:08:35.38,0:08:44.08,Default,,0000,0000,0000,,Then we have a negative\NJ5 times a positive J5.
Dialogue: 0,0:08:44.08,0:08:46.59,Default,,0000,0000,0000,,Again, we've got to be\Ncareful with the signs here.
Dialogue: 0,0:08:46.59,0:08:49.22,Default,,0000,0000,0000,,5 times 5 is 25.
Dialogue: 0,0:08:49.22,0:08:54.02,Default,,0000,0000,0000,,J times J is a negative 1\Nminus times a plus is a minus.
Dialogue: 0,0:08:54.02,0:08:56.24,Default,,0000,0000,0000,,So we've got a minus from\Nthe signs a minus from
Dialogue: 0,0:08:56.24,0:08:59.64,Default,,0000,0000,0000,,the J squared that gives us a positive.
Dialogue: 0,0:08:59.64,0:09:10.30,Default,,0000,0000,0000,,So this then turns out to be\Nnegative 7 plus J26 divided by 29,
Dialogue: 0,0:09:10.30,0:09:20.88,Default,,0000,0000,0000,,which works out to be equal\Nto negative 0.24 plus J0.9.
Dialogue: 0,0:09:20.88,0:09:23.87,Default,,0000,0000,0000,,So the division of Z1 divided by Z2 in
Dialogue: 0,0:09:23.87,0:09:29.43,Default,,0000,0000,0000,,rectangular coordinates gives\Nus this rectangular coordinate.
Dialogue: 0,0:09:30.98,0:09:39.90,Default,,0000,0000,0000,,Now, hub goodness, let's go ahead and\Nconvert this to polar coordinates.
Dialogue: 0,0:09:39.90,0:09:45.48,Default,,0000,0000,0000,,In polar coordinates, we're going to\Nhave the magnitude of that as equal to
Dialogue: 0,0:09:45.48,0:09:52.12,Default,,0000,0000,0000,,the square root of negative\N0.24 squared plus 0.9 squared.
Dialogue: 0,0:09:52.12,0:09:56.49,Default,,0000,0000,0000,,That turns out to be 0.928.
Dialogue: 0,0:09:59.11,0:10:01.58,Default,,0000,0000,0000,,That's the magnitude.
Dialogue: 0,0:10:01.58,0:10:05.36,Default,,0000,0000,0000,,Now, the angle here gets\Nto be a little bit tricky.
Dialogue: 0,0:10:05.36,0:10:11.10,Default,,0000,0000,0000,,The angle is going to be the arc tangent
Dialogue: 0,0:10:11.10,0:10:19.07,Default,,0000,0000,0000,,of 0.9 divided by negative 0.24.
Dialogue: 0,0:10:22.43,0:10:27.92,Default,,0000,0000,0000,,Now, the arc tangent button on\Nyour calculator returns a value between
Dialogue: 0,0:10:27.92,0:10:33.63,Default,,0000,0000,0000,,plus or minus Pi halves or\Nplus or minus 90 degrees.
Dialogue: 0,0:10:33.69,0:10:36.88,Default,,0000,0000,0000,,This becomes ambiguous.
Dialogue: 0,0:10:36.88,0:10:40.08,Default,,0000,0000,0000,,To find out what the actual angle is,
Dialogue: 0,0:10:40.08,0:10:45.78,Default,,0000,0000,0000,,you take the arc tangent of\N0.9 divided by negative 0.24
Dialogue: 0,0:10:45.78,0:10:52.46,Default,,0000,0000,0000,,and you come up with a negative 75 degrees.
Dialogue: 0,0:10:52.46,0:10:54.94,Default,,0000,0000,0000,,So this turns out to be
Dialogue: 0,0:10:54.94,0:11:03.39,Default,,0000,0000,0000,,0.928 angle negative 75 degrees.
Dialogue: 0,0:11:03.39,0:11:05.88,Default,,0000,0000,0000,,But that's not exactly right.
Dialogue: 0,0:11:05.88,0:11:09.84,Default,,0000,0000,0000,,Coming back here and looking\Nat our complex number,
Dialogue: 0,0:11:09.84,0:11:14.68,Default,,0000,0000,0000,,we see that we are in\Nnot the fourth quadrant,
Dialogue: 0,0:11:14.68,0:11:17.33,Default,,0000,0000,0000,,negative 75 it run out of room,
Dialogue: 0,0:11:17.33,0:11:19.62,Default,,0000,0000,0000,,let's just draw it down here.
Dialogue: 0,0:11:21.74,0:11:26.10,Default,,0000,0000,0000,,It has this down here\Nat negative 75 degrees.
Dialogue: 0,0:11:26.10,0:11:28.42,Default,,0000,0000,0000,,But when we look at this number here,
Dialogue: 0,0:11:28.42,0:11:30.20,Default,,0000,0000,0000,,we're at negative 0.24.
Dialogue: 0,0:11:30.20,0:11:31.55,Default,,0000,0000,0000,,The real part is negative,
Dialogue: 0,0:11:31.55,0:11:35.27,Default,,0000,0000,0000,,so we're over here and up 0.9.
Dialogue: 0,0:11:35.27,0:11:40.68,Default,,0000,0000,0000,,In reality where we're at\Nwith this complex number
Dialogue: 0,0:11:40.68,0:11:51.14,Default,,0000,0000,0000,,is 180 degrees off from\Nthis negative 75 degrees.
Dialogue: 0,0:11:51.14,0:11:56.84,Default,,0000,0000,0000,,So the actual angle is\Nnegative 75 plus 180.
Dialogue: 0,0:11:56.84,0:12:04.01,Default,,0000,0000,0000,,This angle right here is\Nactually 105 degrees.
Dialogue: 0,0:12:04.01,0:12:06.32,Default,,0000,0000,0000,,So when you're doing it in
Dialogue: 0,0:12:06.32,0:12:08.30,Default,,0000,0000,0000,,rectangular coordinates using\Nyour arc tangent button
Dialogue: 0,0:12:08.30,0:12:09.38,Default,,0000,0000,0000,,on your calculator you got be
Dialogue: 0,0:12:09.38,0:12:14.26,Default,,0000,0000,0000,,careful and look at the actual\Ncoordinates that you're working with,
Dialogue: 0,0:12:14.26,0:12:17.96,Default,,0000,0000,0000,,because your arc tangent button\Ncan cause you some grief.
Dialogue: 0,0:12:17.96,0:12:24.10,Default,,0000,0000,0000,,Let's do this calculation\Ndirectly from polar form.
Dialogue: 0,0:12:24.10,0:12:29.13,Default,,0000,0000,0000,,We have as Z1 over Z2 is equal to,
Dialogue: 0,0:12:29.13,0:12:38.67,Default,,0000,0000,0000,,in polar form Z1 is 5e to\Nthe J37.87 divided by Z2,
Dialogue: 0,0:12:38.67,0:12:49.04,Default,,0000,0000,0000,,which is 5.39e to the minus J68.2.
Dialogue: 0,0:12:49.04,0:12:52.59,Default,,0000,0000,0000,,Now we can do that directly\Nand 5 divided by 5.39,
Dialogue: 0,0:12:52.59,0:13:00.58,Default,,0000,0000,0000,,that is infact equal to 0.928e to the,
Dialogue: 0,0:13:00.58,0:13:05.01,Default,,0000,0000,0000,,now the exponent is going to be J,
Dialogue: 0,0:13:05.01,0:13:11.54,Default,,0000,0000,0000,,and we've got 37.87 minus 68.82,
Dialogue: 0,0:13:11.54,0:13:15.00,Default,,0000,0000,0000,,that turns out to be the 105\Nand some round off error there.
Dialogue: 0,0:13:15.00,0:13:20.38,Default,,0000,0000,0000,,But it's positive J105 degrees.
Dialogue: 0,0:13:20.38,0:13:23.28,Default,,0000,0000,0000,,When you do it this way,
Dialogue: 0,0:13:23.28,0:13:26.30,Default,,0000,0000,0000,,already in the polar form,
Dialogue: 0,0:13:26.30,0:13:28.94,Default,,0000,0000,0000,,the ambiguity that the arc tangent button
Dialogue: 0,0:13:28.94,0:13:33.06,Default,,0000,0000,0000,,introduces is we avoid that ambiguity.