[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.19,0:00:06.26,Default,,0000,0000,0000,,>> So complex numbers consist of two\Nparts: A real part and an imaginary part,
Dialogue: 0,0:00:06.26,0:00:12.24,Default,,0000,0000,0000,,and we typically write them then as z a\Ncomplex number is equal to a plus j B.
Dialogue: 0,0:00:12.24,0:00:14.48,Default,,0000,0000,0000,,Because there's two parts it requires
Dialogue: 0,0:00:14.48,0:00:18.10,Default,,0000,0000,0000,,a complete plane to\Nrepresent them graphically.
Dialogue: 0,0:00:18.10,0:00:20.62,Default,,0000,0000,0000,,A real number we can show\Non the real number line,
Dialogue: 0,0:00:20.62,0:00:24.32,Default,,0000,0000,0000,,an imaginary number we can show\Nalong an imaginary number line.
Dialogue: 0,0:00:24.32,0:00:26.55,Default,,0000,0000,0000,,But when you've got\Nboth the real and imaginary part,
Dialogue: 0,0:00:26.55,0:00:29.64,Default,,0000,0000,0000,,we form what is known as\Nthe complex plane where
Dialogue: 0,0:00:29.64,0:00:33.27,Default,,0000,0000,0000,,the horizontal axis of the plane\Nis referred to as the real axis,
Dialogue: 0,0:00:33.27,0:00:37.42,Default,,0000,0000,0000,,and we plot the real value of the\Ncomplex number along the horizontal axis.
Dialogue: 0,0:00:37.42,0:00:40.91,Default,,0000,0000,0000,,The vertical axis becomes\Nthe imaginary axis
Dialogue: 0,0:00:40.91,0:00:44.11,Default,,0000,0000,0000,,where we plot the imaginary part\Nof our complex number.
Dialogue: 0,0:00:44.11,0:00:47.98,Default,,0000,0000,0000,,So, for example here we have\Nz is equal to two plus j.
Dialogue: 0,0:00:47.98,0:00:49.50,Default,,0000,0000,0000,,We come over one,
Dialogue: 0,0:00:49.50,0:00:56.11,Default,,0000,0000,0000,,two units along the real part because\Nin this case a the real part is two,
Dialogue: 0,0:00:56.11,0:01:02.00,Default,,0000,0000,0000,,and we go up one unit along
Dialogue: 0,0:01:02.00,0:01:08.54,Default,,0000,0000,0000,,the imaginary axis because\Nour imaginary part is one times j.
Dialogue: 0,0:01:08.54,0:01:10.84,Default,,0000,0000,0000,,So, here we'd have one j,
Dialogue: 0,0:01:10.84,0:01:12.89,Default,,0000,0000,0000,,two j, three j.
Dialogue: 0,0:01:12.89,0:01:14.63,Default,,0000,0000,0000,,You can think of J almost as
Dialogue: 0,0:01:14.63,0:01:19.78,Default,,0000,0000,0000,,a unit vector or a quantity\Nshowing us which axis we're on.
Dialogue: 0,0:01:19.78,0:01:23.14,Default,,0000,0000,0000,,The distance along this axis is two.
Dialogue: 0,0:01:23.14,0:01:29.89,Default,,0000,0000,0000,,the distance along this axis\Nis one times j. All right.
Dialogue: 0,0:01:29.89,0:01:32.72,Default,,0000,0000,0000,,We here are representing
Dialogue: 0,0:01:32.72,0:01:34.21,Default,,0000,0000,0000,,this real number or
Dialogue: 0,0:01:34.21,0:01:37.72,Default,,0000,0000,0000,,this complex number in what is\Nknown as rectangular coordinates,
Dialogue: 0,0:01:37.72,0:01:41.18,Default,,0000,0000,0000,,a horizontal value and a vertical value.
Dialogue: 0,0:01:41.18,0:01:43.75,Default,,0000,0000,0000,,As you know we can also represent points in
Dialogue: 0,0:01:43.75,0:01:46.92,Default,,0000,0000,0000,,a plane using polar coordinates where
Dialogue: 0,0:01:46.92,0:01:50.65,Default,,0000,0000,0000,,the two coordinate values are
Dialogue: 0,0:01:50.65,0:01:55.70,Default,,0000,0000,0000,,a angle of rotation away from\Nthe horizontal axis which we call theta,
Dialogue: 0,0:01:55.70,0:01:59.02,Default,,0000,0000,0000,,and the distance from the origin\Nout to the point along
Dialogue: 0,0:01:59.02,0:02:04.90,Default,,0000,0000,0000,,that ray is known as the magnitude\Nof the complex number.
Dialogue: 0,0:02:04.90,0:02:08.20,Default,,0000,0000,0000,,So, we can write z in polar form,
Dialogue: 0,0:02:08.20,0:02:12.50,Default,,0000,0000,0000,,in terms of this magnitude and an angle.
Dialogue: 0,0:02:12.50,0:02:17.08,Default,,0000,0000,0000,,Now, how did the polar and rectangular\Ncoordinates relate to each other?
Dialogue: 0,0:02:17.08,0:02:21.94,Default,,0000,0000,0000,,We can go from rectangular to\Npolar using Pythagorean's theorem.
Dialogue: 0,0:02:21.94,0:02:30.41,Default,,0000,0000,0000,,So, from rectangular to polar we\Ncan calculate the magnitude of z.
Dialogue: 0,0:02:30.41,0:02:32.94,Default,,0000,0000,0000,,That's simply equal to,
Dialogue: 0,0:02:32.94,0:02:37.16,Default,,0000,0000,0000,,using Pythagorean's theorem, the\Nsquare root of the two sides.
Dialogue: 0,0:02:37.16,0:02:44.13,Default,,0000,0000,0000,,Well this is A, so it would be\Nthe square root of A squared,
Dialogue: 0,0:02:44.13,0:02:48.44,Default,,0000,0000,0000,,and the distance along\Nthe imaginary axis is b.
Dialogue: 0,0:02:48.44,0:02:53.27,Default,,0000,0000,0000,,So, A squared plus B squared and
Dialogue: 0,0:02:53.27,0:02:58.55,Default,,0000,0000,0000,,theta the angle is equal\Nto the inverse tangent
Dialogue: 0,0:02:58.55,0:03:00.23,Default,,0000,0000,0000,,of the opposite over the adjacent or
Dialogue: 0,0:03:00.23,0:03:02.76,Default,,0000,0000,0000,,the inverse of the imaginary over the real
Dialogue: 0,0:03:02.76,0:03:10.46,Default,,0000,0000,0000,,or the inverse tangent of b over a.
Dialogue: 0,0:03:10.46,0:03:15.36,Default,,0000,0000,0000,,Now, to go from polar coordinates\Nback to rectangular coordinates,
Dialogue: 0,0:03:15.36,0:03:18.26,Default,,0000,0000,0000,,we again use the right\Ntriangle and recognize
Dialogue: 0,0:03:18.26,0:03:22.04,Default,,0000,0000,0000,,that the distance along\Nthe horizontal axis is A,
Dialogue: 0,0:03:22.04,0:03:24.86,Default,,0000,0000,0000,,that's simply equal to the projection\Nof that point down onto
Dialogue: 0,0:03:24.86,0:03:26.45,Default,,0000,0000,0000,,the real axis which of course is
Dialogue: 0,0:03:26.45,0:03:29.02,Default,,0000,0000,0000,,the magnitude of z times\Nthe cosine of the angle.
Dialogue: 0,0:03:29.02,0:03:32.38,Default,,0000,0000,0000,,So, A then, is equal to
Dialogue: 0,0:03:32.38,0:03:37.57,Default,,0000,0000,0000,,the magnitude of z times\Nthe cosine of the angle theta,
Dialogue: 0,0:03:37.57,0:03:41.93,Default,,0000,0000,0000,,and b, the imaginary component is equal
Dialogue: 0,0:03:41.93,0:03:46.74,Default,,0000,0000,0000,,to the magnitude of z\Ntimes the sine of theta.
Dialogue: 0,0:03:46.74,0:03:49.44,Default,,0000,0000,0000,,Again, is just for\Nthe projection of this point,
Dialogue: 0,0:03:49.44,0:03:51.55,Default,,0000,0000,0000,,onto the imaginary axis.
Dialogue: 0,0:03:51.55,0:03:54.82,Default,,0000,0000,0000,,So, that takes us from polar coordinates,
Dialogue: 0,0:03:54.82,0:03:56.90,Default,,0000,0000,0000,,back to rectangular coordinates.
Dialogue: 0,0:03:56.90,0:03:59.62,Default,,0000,0000,0000,,Now let's just do an example real\Nquick to show how this works.
Dialogue: 0,0:03:59.62,0:04:03.94,Default,,0000,0000,0000,,In fact let's do it for\Nthis quantity right there.
Dialogue: 0,0:04:03.94,0:04:08.08,Default,,0000,0000,0000,,The complex number z equals two plus j\Ngiven to us in rectangular coordinates,
Dialogue: 0,0:04:08.08,0:04:10.22,Default,,0000,0000,0000,,how do we get the polar\Nrepresentation of it?
Dialogue: 0,0:04:10.22,0:04:11.51,Default,,0000,0000,0000,,While coming down here,
Dialogue: 0,0:04:11.51,0:04:14.81,Default,,0000,0000,0000,,the magnitude of z is going to equal
Dialogue: 0,0:04:14.81,0:04:18.78,Default,,0000,0000,0000,,the square root of the real part\Nsquared which is two squared,
Dialogue: 0,0:04:18.78,0:04:20.74,Default,,0000,0000,0000,,plus the imaginary part.
Dialogue: 0,0:04:20.74,0:04:23.28,Default,,0000,0000,0000,,The imaginary part is\Ncoming up here one unit,
Dialogue: 0,0:04:23.28,0:04:25.07,Default,,0000,0000,0000,,again we don't square the j.
Dialogue: 0,0:04:25.07,0:04:29.44,Default,,0000,0000,0000,,The distance there, for\Nthe geometry in this circuit,
Dialogue: 0,0:04:29.44,0:04:33.76,Default,,0000,0000,0000,,in this problem, the distance\Nthere is one unit,
Dialogue: 0,0:04:33.76,0:04:39.10,Default,,0000,0000,0000,,so it'll be one squared and that\Nequals the square root of five.
Dialogue: 0,0:04:39.10,0:04:45.92,Default,,0000,0000,0000,,Theta is equal to the arc-tangent\Nof the imaginary part which is one,
Dialogue: 0,0:04:45.92,0:04:47.69,Default,,0000,0000,0000,,it's not J, it's one.
Dialogue: 0,0:04:47.69,0:04:50.36,Default,,0000,0000,0000,,Again, this distance is one unit.
Dialogue: 0,0:04:50.36,0:04:55.64,Default,,0000,0000,0000,,So, the one imaginary part\Nover the real part two,
Dialogue: 0,0:04:55.64,0:05:03.06,Default,,0000,0000,0000,,and that turns out to be 26.6 degrees.
Dialogue: 0,0:05:03.06,0:05:03.72,Default,,0000,0000,0000,,Now to show how we take it and go back.
Dialogue: 0,0:05:03.72,0:05:07.94,Default,,0000,0000,0000,,First of all, this number\Nthen in polar coordinates,
Dialogue: 0,0:05:07.94,0:05:16.76,Default,,0000,0000,0000,,we would represent as square root\Nof five angle 26.6 degrees.
Dialogue: 0,0:05:16.76,0:05:19.79,Default,,0000,0000,0000,,Okay? Now, how do we take
Dialogue: 0,0:05:19.79,0:05:24.16,Default,,0000,0000,0000,,this number in polar form and go\Nback into rectangular coordinates?
Dialogue: 0,0:05:24.16,0:05:26.77,Default,,0000,0000,0000,,Well, it tells us right here that A,
Dialogue: 0,0:05:26.77,0:05:30.02,Default,,0000,0000,0000,,the real part of our number\Nis equal to A is
Dialogue: 0,0:05:30.02,0:05:34.24,Default,,0000,0000,0000,,equal to the magnitude of Z which we\Nfound to be the square root of five,
Dialogue: 0,0:05:34.24,0:05:41.02,Default,,0000,0000,0000,,times the cosine of the angle which\Nis in this case is 26.6 degrees.
Dialogue: 0,0:05:41.02,0:05:43.22,Default,,0000,0000,0000,,When you're doing this make sure\Nyou got your calculator is set in
Dialogue: 0,0:05:43.22,0:05:45.65,Default,,0000,0000,0000,,degrees and not radians if\Nyou're working in degrees.
Dialogue: 0,0:05:45.65,0:05:49.08,Default,,0000,0000,0000,,That then of course is going to equal A,
Dialogue: 0,0:05:49.08,0:05:51.98,Default,,0000,0000,0000,,the distance along\Nthe real axis which we know
Dialogue: 0,0:05:51.98,0:05:55.43,Default,,0000,0000,0000,,and when you can't do that calculation\Nif you find that is equal to two,
Dialogue: 0,0:05:55.43,0:05:59.33,Default,,0000,0000,0000,,and b the imaginary part then\Nis equal to the square root of
Dialogue: 0,0:05:59.33,0:06:07.45,Default,,0000,0000,0000,,five times the sine of 26.6\Ndegrees which equals one.