IIT JEE Differentiability and Boundedness

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Kind: captions
Language: en

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Let f be a real-valued function
defined on the interval from 0

00:00:04.835 --> 00:00:05.680
to infinity.

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It does not include 0.

00:00:07.770 --> 00:00:09.490
And f is defined
as the natural log

00:00:09.490 --> 00:00:11.780
of x plus the definite
integral from 0

00:00:11.780 --> 00:00:15.270
to x of the square root of
1 plus sine of t, d of t.

00:00:15.270 --> 00:00:17.650
Then which of the following
statements are true?

00:00:17.650 --> 00:00:19.909
And so more than one
of these might be true.

00:00:19.909 --> 00:00:20.950
So let's figure them out.

00:00:20.950 --> 00:00:23.160
So A says, the
second derivative--

00:00:23.160 --> 00:00:24.840
there's two apostrophes here.

00:00:24.840 --> 00:00:28.860
The second derivative of f
exists for all x between 0,

00:00:28.860 --> 00:00:31.230
or really, all x greater than 0.

00:00:31.230 --> 00:00:33.340
Then they say the
first derivative exists

00:00:33.340 --> 00:00:35.840
for all x greater than
0, and the derivative

00:00:35.840 --> 00:00:39.234
is continuous on that interval,
but not differentiable.

00:00:39.234 --> 00:00:40.900
OK, so everything
they're doing, they're

00:00:40.900 --> 00:00:43.627
relating the function
to the first derivative

00:00:43.627 --> 00:00:44.710
and the second derivative.

00:00:44.710 --> 00:00:46.293
So let's just figure
those things out.

00:00:46.293 --> 00:00:48.320
So let's just write
the function first.

00:00:48.320 --> 00:00:50.320
I'll just rewrite it here.

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The function f is equal
to the natural log

00:00:52.620 --> 00:00:55.140
of x plus the definite
integral from 0

00:00:55.140 --> 00:01:02.410
to x of the square root
of 1 plus sine of t dt.

00:01:02.410 --> 00:01:08.620
The derivative of f-- so
let me write it over here.

00:01:08.620 --> 00:01:12.650
So f prime of x is going to
be equal to the derivative

00:01:12.650 --> 00:01:13.900
of this first term.

00:01:13.900 --> 00:01:17.390
Derivative of natural
log of x is 1/x.

00:01:17.390 --> 00:01:18.860
And then derivative
of this, that's

00:01:18.860 --> 00:01:20.680
just the fundamental
theorem of calculus.

00:01:20.680 --> 00:01:23.890
It's going to be just this
inside expression with respect

00:01:23.890 --> 00:01:28.800
to x, as a function of x,
so plus the square root of 1

00:01:28.800 --> 00:01:32.330
plus sine of x.

00:01:34.910 --> 00:01:36.570
Let's get the second
derivative now.

00:01:36.570 --> 00:01:38.550
The second
derivative-- I'll do it

00:01:38.550 --> 00:01:41.459
over here-- f
prime prime of x is

00:01:41.459 --> 00:01:43.500
going to be equal to the
derivative of this term.

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This is x to the negative 1.

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So it's going to be negative x
to the negative 2 or negative 1

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over x squared.

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That's negative x
to the negative 2.

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You could view it that way.

00:01:54.360 --> 00:01:56.490
And then plus-- so
let's do the chain rule.

00:01:56.490 --> 00:01:57.990
What's the derivative
of the inside?

00:01:57.990 --> 00:02:01.400
Derivative of 1 plus
sine of x is cosine of x.

00:02:01.400 --> 00:02:05.570
So plus cosine of
x times-- now, it's

00:02:05.570 --> 00:02:07.690
1 plus sine of x to the 1/2.

00:02:07.690 --> 00:02:10.090
So it's going to
be times 1/2 times

00:02:10.090 --> 00:02:15.070
1 plus sine of x to
the negative 1/2.

00:02:15.070 --> 00:02:16.610
So I took the derivatives.

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Now let's look at the choices.

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So the first choice says
the second derivative

00:02:21.650 --> 00:02:25.190
of the function exists
for all x greater than 0.

00:02:25.190 --> 00:02:28.080
If you're going between 0 and
infinity and not including 0,

00:02:28.080 --> 00:02:30.460
this is essentially for
all x greater than 0.

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Let's see if that is true.

00:02:32.430 --> 00:02:35.570
So over here, the
one area-- so this

00:02:35.570 --> 00:02:37.540
would be undefined
if x is equal to 0,

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but that's not in the
interval over here.

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So let's just be clear.

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When you take something to
the negative 1/2 power--

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let me rewrite it like this.

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I could rewrite it as--
instead of writing it

00:02:49.850 --> 00:02:51.670
to the negative 1/2
power, I could write it

00:02:51.670 --> 00:02:58.080
in the denominator as the
square root of 1 plus sine of x.

00:02:58.080 --> 00:03:00.330
So if this denominator
becomes undefined,

00:03:00.330 --> 00:03:03.050
the second derivative
won't be undefined.

00:03:03.050 --> 00:03:05.919
So let's think about when that
denominator might be undefined.

00:03:05.919 --> 00:03:07.460
So it'll be undefined
if it equals 0.

00:03:07.460 --> 00:03:11.530
So 1 plus sine of
x is equal to 0.

00:03:11.530 --> 00:03:13.700
Well, it's going to be
equal to 0 if sine of x

00:03:13.700 --> 00:03:14.900
is equal to negative 1.

00:03:18.380 --> 00:03:23.010
And sine of x can equal negative
1 at x is equal to-- let's see.

00:03:23.010 --> 00:03:25.040
If you go around
the unit circle,

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sine is the
y-coordinate, so it has

00:03:27.590 --> 00:03:29.680
to go around-- you
can either go down.

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It would be negative 90 degrees
or positive 270 degrees.

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Or if in radians, if x is equal
to-- because we're assuming,

00:03:37.921 --> 00:03:40.420
or at least I'm assuming that
we're dealing in radians here.

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If x is equal to 3 pi
over 2, assuming we're

00:03:46.290 --> 00:03:50.270
in radians, that will
make this undefined.

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So this f prime prime
not defined at x

00:03:58.360 --> 00:04:03.360
is equal to 3 pi over 2, which
is clearly in this interval.

00:04:03.360 --> 00:04:05.680
So choice A is false.

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Choice A does not work.

00:04:07.417 --> 00:04:08.000
Now let's see.

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Choice B says f
prime of x exists

00:04:10.250 --> 00:04:12.900
for all x's greater than 0.

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Well, that seems to be the case.

00:04:14.900 --> 00:04:18.050
You can't put x equal 0, but
that's not in the interval.

00:04:18.050 --> 00:04:21.050
And f prime is continuous
on 0 to infinity.

00:04:21.050 --> 00:04:24.530
Yeah, it's continuous-- no
obvious discontinuities here--

00:04:24.530 --> 00:04:27.231
but not differentiable
on 0 to infinity.

00:04:27.231 --> 00:04:28.980
Well, it's clearly not
differentiable at 0

00:04:28.980 --> 00:04:29.590
to infinity.

00:04:32.685 --> 00:04:34.560
The first derivative or
the second derivative

00:04:34.560 --> 00:04:37.750
does not exist at x is
equal to 3 pi over 2.

00:04:37.750 --> 00:04:38.460
We just saw that.

00:04:38.460 --> 00:04:42.770
f prime of x is not
differentiable at 3 pi over 2.

00:04:42.770 --> 00:04:45.220
So everything in this is true.

00:04:45.220 --> 00:04:46.372
Everything is true.

00:04:46.372 --> 00:04:48.830
It's not differentiable there
because the second derivative

00:04:48.830 --> 00:04:51.340
is not defined at 3 pi over 2.

00:04:51.340 --> 00:04:54.230
Let's do part C
now, part C. There

00:04:54.230 --> 00:04:56.590
exists an alpha
greater than 1 such

00:04:56.590 --> 00:05:01.940
that the absolute
value of f prime of x

00:05:01.940 --> 00:05:06.700
is less than the absolute value
of f of x for all x essentially

00:05:06.700 --> 00:05:10.280
greater than-- for all x
greater than that alpha.

00:05:10.280 --> 00:05:11.780
So let's think about this.

00:05:11.780 --> 00:05:13.500
Let's think about
this a little bit.

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What we want to show is
that, beyond some threshold,

00:05:18.880 --> 00:05:21.330
the derivative is always
going to be smaller

00:05:21.330 --> 00:05:24.400
than the original f of x.

00:05:24.400 --> 00:05:27.060
The derivative will always
be smaller than f of x.

00:05:27.060 --> 00:05:29.339
So the first thing we might
want to see-- so let's

00:05:29.339 --> 00:05:31.130
just compare the parts
that are comparable.

00:05:31.130 --> 00:05:34.480
So let's compare, first of
all, the natural log of x.

00:05:34.480 --> 00:05:36.450
Let's compare the
natural log of x to 1/x.

00:05:38.997 --> 00:05:40.330
Let's just set a threshold here.

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So is there a number
so at any number

00:05:44.700 --> 00:05:46.300
larger than that,
the natural log

00:05:46.300 --> 00:05:49.577
of x is always going
to be larger than 1/x?

00:05:49.577 --> 00:05:51.160
And the number that
pops into my head,

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just because it's an easy number
to work with with natural logs,

00:05:54.870 --> 00:05:56.630
is e.

00:05:56.630 --> 00:06:02.300
So for x is greater than
e, it's pretty clear

00:06:02.300 --> 00:06:06.650
that the natural log of x is
going to be greater than 1/x.

00:06:06.650 --> 00:06:07.490
Think about it.

00:06:07.490 --> 00:06:10.730
At e, it's going to evaluate
to the natural log of e, which

00:06:10.730 --> 00:06:11.596
is 1.

00:06:11.596 --> 00:06:14.400
And 1 is clearly
greater than 1/e.

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e is 2.71 and so
on and so forth.

00:06:16.420 --> 00:06:18.130
So this is clearly true.

00:06:18.130 --> 00:06:21.680
And then as you get beyond e, as
you get numbers larger than e,

00:06:21.680 --> 00:06:23.970
as this becomes
e plus something,

00:06:23.970 --> 00:06:26.160
this number will become larger.

00:06:26.160 --> 00:06:28.940
And then this number over
here will become smaller

00:06:28.940 --> 00:06:31.500
because we'll be dividing
by a larger number.

00:06:31.500 --> 00:06:33.870
So it's clear that it
exists just for this part,

00:06:33.870 --> 00:06:37.214
for the first part, the
natural log of x beyond e.

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And we just have to
find some threshold.

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That might not be
the first threshold.

00:06:40.620 --> 00:06:45.440
But the natural log of x is
definitely greater than 1/x

00:06:45.440 --> 00:06:47.770
when x is greater than e.

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Now let's see if we can find a
threshold for the second part.

00:06:50.520 --> 00:06:53.250
And if we can, then
we can literally just

00:06:53.250 --> 00:06:55.310
pick the larger of
the two thresholds,

00:06:55.310 --> 00:06:58.200
and then that will be a
threshold above which f of x

00:06:58.200 --> 00:07:01.260
will always be larger
than f prime of x.

00:07:01.260 --> 00:07:02.770
And to think about
it, we just have

00:07:02.770 --> 00:07:07.140
to see that essentially
this part of the function--

00:07:07.140 --> 00:07:09.910
let me write it this way.

00:07:09.910 --> 00:07:17.820
If I define this as h of x,
this right here is h prime of x.

00:07:20.890 --> 00:07:25.300
We've already shown that for
x is greater than e, h of x

00:07:25.300 --> 00:07:28.870
is greater than h prime of x.

00:07:28.870 --> 00:07:30.430
We've seen that already.

00:07:30.430 --> 00:07:33.070
Now let me define
this over here.

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Let me define that as g of x.

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Let me call that g of x.

00:07:38.320 --> 00:07:40.130
And then this is its derivative.

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This is g prime of x.

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That is g prime of x.

00:07:46.010 --> 00:07:51.500
Now, let's just draw g of x.

00:07:51.500 --> 00:07:54.580
Actually, let me draw g
prime of x right over here.

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Let me draw it.

00:07:55.390 --> 00:07:59.300
It's actually a pretty
straightforward graph to draw.

00:07:59.300 --> 00:08:01.480
So let me draw it over here.

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So when x is equal to 0,
sine of 0 is 0 plus 1.

00:08:05.620 --> 00:08:06.750
You take the square root.

00:08:06.750 --> 00:08:10.590
So 0, 1, that's the first point.

00:08:10.590 --> 00:08:14.980
That is f of x equal to
1 when x is equal to 0.

00:08:14.980 --> 00:08:17.300
And then the next interesting
point, maybe we'll

00:08:17.300 --> 00:08:20.080
take x is equal to pi over 2.

00:08:20.080 --> 00:08:25.740
So when x is pi over 2,
sine of pi over 2 is 1.

00:08:25.740 --> 00:08:27.752
1 plus 1 is 2, take
the square root.

00:08:27.752 --> 00:08:29.210
So it's going to
be the square root

00:08:29.210 --> 00:08:33.450
of 2, which is someplace around
here, so the square root of 2.

00:08:33.450 --> 00:08:40.120
And then let's just take when
x is pi, sine of pi is 0.

00:08:40.120 --> 00:08:42.130
This becomes the
square root of 1 again,

00:08:42.130 --> 00:08:44.059
so we're back over here.

00:08:44.059 --> 00:08:47.100
Then let's go to 3 pi over 2.

00:08:47.100 --> 00:08:49.610
Sine of 3 pi over
2 is negative 1.

00:08:49.610 --> 00:08:51.420
Negative 1 plus 1
is going to be 0,

00:08:51.420 --> 00:08:53.450
so we're going to be down here.

00:08:53.450 --> 00:08:57.670
And then we go again to 2 pi.

00:08:57.670 --> 00:09:00.840
Sine of 2 pi is 0.

00:09:00.840 --> 00:09:02.617
0 plus 1 is 1.

00:09:02.617 --> 00:09:04.450
Take the square root,
you're back over here.

00:09:04.450 --> 00:09:09.080
So this graph, g prime
of x we're calling it.

00:09:09.080 --> 00:09:11.700
This graph over here
looks like this.

00:09:11.700 --> 00:09:13.150
It looks something like.

00:09:13.150 --> 00:09:15.670
This is my best
attempt at drawing it.

00:09:15.670 --> 00:09:16.910
It just keeps oscillating.

00:09:16.910 --> 00:09:18.618
If its maximum point
is square root of 2,

00:09:18.618 --> 00:09:21.150
its minimum point is 0,
and it keeps oscillating.

00:09:21.150 --> 00:09:23.272
So this is g prime of x.

00:09:23.272 --> 00:09:26.140
G prime of x is equal
to the square root

00:09:26.140 --> 00:09:28.410
of 1 plus sine of x.

00:09:28.410 --> 00:09:29.600
Now, what is g of x?

00:09:29.600 --> 00:09:31.290
What is this thing over here?

00:09:31.290 --> 00:09:33.260
Well, this is the
definite integral from 0

00:09:33.260 --> 00:09:34.630
to x of this thing.

00:09:34.630 --> 00:09:39.900
So you take any x-value,
this thing over here, g of x,

00:09:39.900 --> 00:09:44.350
is the area over here.

00:09:44.350 --> 00:09:45.490
So what's happening here?

00:09:45.490 --> 00:09:47.410
So let's think about this.

00:09:47.410 --> 00:09:49.880
As we get to larger
and larger x-values,

00:09:49.880 --> 00:09:51.990
the area gets bigger
because this line never

00:09:51.990 --> 00:09:53.990
drops below the x-axis.

00:09:53.990 --> 00:09:56.640
So as we get larger
and larger x-values,

00:09:56.640 --> 00:10:00.860
this area just keeps getting
bigger and bigger and bigger.

00:10:00.860 --> 00:10:12.130
So we can write g of x
unbounded as x goes to infinity.

00:10:12.130 --> 00:10:13.840
G of x approaches infinity.

00:10:13.840 --> 00:10:15.626
What about g prime of x?

00:10:15.626 --> 00:10:17.960
g prime of x, it maxes out.

00:10:17.960 --> 00:10:21.030
It just keeps oscillating
between 0 and square root of 2.

00:10:21.030 --> 00:10:25.160
So g prime of x max-- or maybe
I should write it this way.

00:10:25.160 --> 00:10:30.140
The max of g prime of x is
equal to the square root of 2.

00:10:30.140 --> 00:10:32.010
So it's pretty clear
that at some point

00:10:32.010 --> 00:10:34.800
this area is going to be either
some x-value or this area

00:10:34.800 --> 00:10:36.860
is going to be more than
the square root of 2.

00:10:36.860 --> 00:10:39.000
Let's call that point alpha.

00:10:39.000 --> 00:10:42.070
And after that alpha,
where maybe it's over here,

00:10:42.070 --> 00:10:46.460
I mean, clearly we could
say, at x equals 10,

00:10:46.460 --> 00:10:49.750
we definitely have a lot
of area under this curve.

00:10:49.750 --> 00:10:52.200
The area is definitely bigger
than the square root of 2.

00:10:52.200 --> 00:10:54.075
So let me just write it
there because we just

00:10:54.075 --> 00:10:56.610
need to find some boundary.

00:10:56.610 --> 00:11:01.354
Let's just say at x
greater than-- I'm just

00:11:01.354 --> 00:11:02.520
picking an arbitrary number.

00:11:02.520 --> 00:11:07.170
At x is greater
than 10, it's clear

00:11:07.170 --> 00:11:13.020
that the integral from 0 to
x of the square root of 1

00:11:13.020 --> 00:11:19.760
plus sine of x, or I should
say 1 plus sine of t dt,

00:11:19.760 --> 00:11:22.570
it's clear that that
is going to be greater

00:11:22.570 --> 00:11:26.169
than the square root
of 1 plus sine of x,

00:11:26.169 --> 00:11:27.210
because this is the area.

00:11:30.000 --> 00:11:32.300
At any value above
10, this thing at most

00:11:32.300 --> 00:11:34.534
can ever get to
square root of 2.

00:11:34.534 --> 00:11:36.700
This thing's just going to
keep growing and growing.

00:11:36.700 --> 00:11:38.500
At 10, it's already
going to be larger

00:11:38.500 --> 00:11:39.740
than the square root of 2.

00:11:39.740 --> 00:11:41.570
So we know that
there's some boundary.

00:11:41.570 --> 00:11:45.610
So for x is greater than 10--
so let's write it this place.

00:11:45.610 --> 00:11:50.560
So for x greater
than 10, well, 10

00:11:50.560 --> 00:11:51.770
is definitely greater than e.

00:11:51.770 --> 00:11:56.130
We know that h of x is
greater than h prime of x.

00:11:56.130 --> 00:11:57.340
We know that's true.

00:11:57.340 --> 00:11:59.920
10 is clearly greater than
e, so we can state that.

00:11:59.920 --> 00:12:08.450
And we also know that g of x
is greater than g prime of x.

00:12:08.450 --> 00:12:15.350
So we know that f of x, which
is equal to h of x plus g of x,

00:12:15.350 --> 00:12:19.994
is going to be greater
than f prime of x, which

00:12:19.994 --> 00:12:21.910
was the derivative, the
sum of the derivative,

00:12:21.910 --> 00:12:26.040
which is equal to h prime
of x plus g prime of x.

00:12:26.040 --> 00:12:29.350
So we know that this is true.

00:12:29.350 --> 00:12:32.850
We can make alpha equal 10 and
this thing is going to be true.

00:12:32.850 --> 00:12:35.560
So this is true.

00:12:35.560 --> 00:12:39.780
Now let's do part D. There
exists a beta greater than 0

00:12:39.780 --> 00:12:42.440
such that the
absolute value of f

00:12:42.440 --> 00:12:47.040
of x plus the absolute
value of the derivative of f

00:12:47.040 --> 00:12:50.160
is going to be less than
or equal to beta for all x.

00:12:50.160 --> 00:12:51.940
Well, we already
know that f of x

00:12:51.940 --> 00:12:57.870
is unbounded, because
this integral part of it

00:12:57.870 --> 00:13:00.090
right here just
keeps increasing as x

00:13:00.090 --> 00:13:01.527
gets larger and
larger and larger.

00:13:01.527 --> 00:13:02.360
It never dips below.

00:13:02.360 --> 00:13:04.640
We never have negative area.

00:13:04.640 --> 00:13:06.860
So this area just keeps growing.

00:13:06.860 --> 00:13:08.590
So this is unbounded.

00:13:08.590 --> 00:13:10.980
And actually, so is
the natural log of x.

00:13:10.980 --> 00:13:15.360
The natural log of x is
also going to be unbounded.

00:13:15.360 --> 00:13:18.210
So you cannot say that it's
going to be less than some

00:13:18.210 --> 00:13:21.030
value because it's going
to approach infinity.

00:13:21.030 --> 00:13:26.180
So it's not going to
be less than some value

00:13:26.180 --> 00:13:29.380
for all x from 0 to infinity.

00:13:29.380 --> 00:13:30.370
So you can't bound it.

00:13:30.370 --> 00:13:31.940
This is an unbounded function.

00:13:31.940 --> 00:13:37.030
Just this part alone is
an unbounded function.

00:13:37.030 --> 00:13:41.690
f prime of x is-- well, even
there you have to be careful.

00:13:41.690 --> 00:13:44.650
But f of x is clearly unbounded,
so this can't be true.

00:13:44.650 --> 00:13:51.770
There's not any kind of roof
you can put on the-- well,

00:13:51.770 --> 00:13:53.160
I think you get
the general idea.

00:13:53.160 --> 00:13:55.340
So this is definitely not true.

00:13:55.340 --> 00:13:58.923
So the answer here is B and C.


IIT JEE complex numbers (part 3)

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:01.920
So we're in the home stretch.

00:00:01.920 --> 00:00:05.850
Let's see if part C is true, and
I'm running out of real estate.

00:00:05.850 --> 00:00:09.610
So let me copy and
paste it actually.

00:00:09.610 --> 00:00:12.290
Actually, maybe I could do
it over here to the left.

00:00:12.290 --> 00:00:13.390
I had some doodles here.

00:00:13.390 --> 00:00:15.060
Let me clear those out.

00:00:15.060 --> 00:00:18.620
So let's see if we can do
it over here to the left.

00:00:18.620 --> 00:00:21.750
Part C. So we have
this determinant,

00:00:21.750 --> 00:00:23.420
and it's claiming
that it equals 0,

00:00:23.420 --> 00:00:24.930
so we have to see
if that's true.

00:00:24.930 --> 00:00:27.263
And so we can actually try
to multiply these out and see

00:00:27.263 --> 00:00:30.690
if anything interesting happens,
but we could even better

00:00:30.690 --> 00:00:32.759
leverage some of the
algebra that we've already

00:00:32.759 --> 00:00:43.040
done for part A. In part A, we
figured out that z minus z1 is

00:00:43.040 --> 00:00:46.220
equal to t times z2 minus z1.

00:00:51.580 --> 00:00:53.750
So this thing right
over here, this

00:00:53.750 --> 00:00:56.779
is the same thing as
t times z2 minus z1,

00:00:56.779 --> 00:00:58.320
and the reason why
that's interesting

00:00:58.320 --> 00:01:00.520
is I have a z1 minus
z1 here, so I'm

00:01:00.520 --> 00:01:02.400
starting to have similar things.

00:01:02.400 --> 00:01:04.560
And then this thing
over here-- so

00:01:04.560 --> 00:01:10.140
if I take the conjugate of each
of these, the conjugate of z

00:01:10.140 --> 00:01:12.730
minus the conjugate
of z1, this is

00:01:12.730 --> 00:01:15.920
going to be equal to t
times the conjugate of z2

00:01:15.920 --> 00:01:18.350
minus the conjugate of z1.

00:01:18.350 --> 00:01:21.430
And I know that
because when I do this,

00:01:21.430 --> 00:01:24.830
I'm swapping all of the
signs on the imaginary parts

00:01:24.830 --> 00:01:26.182
of the complex numbers.

00:01:26.182 --> 00:01:28.390
So if you were to go through
this entire process just

00:01:28.390 --> 00:01:31.300
swapping the imaginary
parts of the complex number,

00:01:31.300 --> 00:01:35.230
the end result would have
it's imaginary parts swapped,

00:01:35.230 --> 00:01:37.150
and that's exactly
what we have over here.

00:01:37.150 --> 00:01:39.880
We have the imaginary
parts swapped over here.

00:01:39.880 --> 00:01:43.310
So if you want to work it out,
set this equal to A plus BI

00:01:43.310 --> 00:01:46.390
or set this equal to A1 plus
B1, you can work it out,

00:01:46.390 --> 00:01:49.070
but I think it's a pretty
intuitive idea that we're just

00:01:49.070 --> 00:01:52.170
making all of the-- we're
just swapping the signs on all

00:01:52.170 --> 00:01:55.300
of the imaginary parts of
each of these complex numbers.

00:01:55.300 --> 00:01:56.920
Now, with this said,
this determinant

00:01:56.920 --> 00:01:58.840
becomes pretty simple.

00:01:58.840 --> 00:02:04.980
This becomes t
times z2 minus z1.

00:02:04.980 --> 00:02:13.370
This becomes t times the
conjugate of z2 minus z1.

00:02:13.370 --> 00:02:17.300
This down here is z2 minus z1.

00:02:17.300 --> 00:02:21.340
And then this over here is
the conjugate of z2 minus z1.

00:02:21.340 --> 00:02:22.760
And so what is this determinant?

00:02:22.760 --> 00:02:24.520
It's going to be
this times this.

00:02:24.520 --> 00:02:30.440
So it's t times z2
minus z1 times z2

00:02:30.440 --> 00:02:32.730
minus z1 times the
conjugate of those.

00:02:32.730 --> 00:02:37.840
So it's that times that
minus this times this, so

00:02:37.840 --> 00:02:45.220
minus t z2 z1, z2 minus
z1 times the conjugate

00:02:45.220 --> 00:02:48.880
of z2 minus the conjugate of z1.

00:02:48.880 --> 00:02:51.410
Now, this is exactly
equal to this.

00:02:51.410 --> 00:02:53.750
These things are obviously
going to cancel out,

00:02:53.750 --> 00:02:56.930
and we're clearly
going to get 0.

00:02:56.930 --> 00:02:59.870
So C is also true.

00:02:59.870 --> 00:03:04.540
So the correct answers
to this original problem

00:03:04.540 --> 00:03:08.138
were A, C, and D.


IIT JEE complex numbers (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.670
In the last video, we saw
that choice A is true.

00:00:03.670 --> 00:00:07.250
Now let's see if B, C,
and D are also true.

00:00:07.250 --> 00:00:09.820
So choice B, they're telling
us that the argument,

00:00:09.820 --> 00:00:11.260
let me rewrite it
over here-- I'll

00:00:11.260 --> 00:00:14.170
do it in another color,
their choice B is telling us

00:00:14.170 --> 00:00:19.750
that the argument
of z minus z1 is

00:00:19.750 --> 00:00:25.914
equal to the argument
of z minus z2.

00:00:25.914 --> 00:00:27.580
Now let's think about
this a little bit.

00:00:27.580 --> 00:00:28.621
We already did some work.

00:00:28.621 --> 00:00:31.870
We simplified what z
minus z1 is equal to.

00:00:31.870 --> 00:00:35.770
It's equal to t
times z2 minus z1.

00:00:35.770 --> 00:00:42.370
So this is equivalent to saying
that the argument of t times

00:00:42.370 --> 00:00:44.020
z2 minus z1.

00:00:44.020 --> 00:00:45.690
I'm just leveraging
some of the algebra

00:00:45.690 --> 00:00:48.040
that we already did,
because this thing is

00:00:48.040 --> 00:00:50.930
the same thing as what's
inside of the absolute value

00:00:50.930 --> 00:00:54.580
sign, which is the same thing
as this thing over here.

00:00:54.580 --> 00:00:58.310
So the argument, t
times z2 minus z1,

00:00:58.310 --> 00:01:00.440
is equal to-- so I'm
just rewriting statement

00:01:00.440 --> 00:01:02.390
B. Statement B is just
saying that this should

00:01:02.390 --> 00:01:07.025
be equal to the argument
of-- now what's z minus z2?

00:01:07.025 --> 00:01:12.430
Z minus z2, we figured out
was this thing over here.

00:01:12.430 --> 00:01:15.130
Actually, let's say it's
this thing right over here.

00:01:15.130 --> 00:01:18.560
It's equal to this
thing right over here.

00:01:18.560 --> 00:01:22.400
And we could rewrite
this thing as being

00:01:22.400 --> 00:01:25.990
equal to-- I'm running
out of real estate here--

00:01:25.990 --> 00:01:28.520
but we can rewrite
this thing right here

00:01:28.520 --> 00:01:34.160
as B equal to 1 minus t,
I'm just factoring out

00:01:34.160 --> 00:01:40.660
the 1 minus t, times
z1 one minus z2.

00:01:45.080 --> 00:01:49.210
Or so we can get in
the same form here,

00:01:49.210 --> 00:01:52.230
let's multiply this times
negative 1 and this times

00:01:52.230 --> 00:01:52.736
negative 1.

00:01:52.736 --> 00:01:54.360
So I'm multiplying
by negative 1 twice.

00:01:54.360 --> 00:01:56.190
So I'm not changing the number.

00:01:56.190 --> 00:01:58.630
So this is going to be
equal to, or the statement

00:01:58.630 --> 00:02:05.520
is claiming this is equal to,
the argument of t minus 1--

00:02:05.520 --> 00:02:07.732
let me write it a little
bit neater than that.

00:02:07.732 --> 00:02:09.190
So the claim is
that this should be

00:02:09.190 --> 00:02:19.877
equal to the argument of t
minus 1 times z2 minus z1.

00:02:19.877 --> 00:02:20.960
So let's think about this.

00:02:20.960 --> 00:02:22.335
And just remember,
all I did here

00:02:22.335 --> 00:02:25.370
is I multiplied this by negative
1, and this by negative 1.

00:02:25.370 --> 00:02:27.337
If I multiply two
things times negative 1,

00:02:27.337 --> 00:02:29.170
it's equivalent to
multiplying by negative 1

00:02:29.170 --> 00:02:32.350
twice, which is just
multiplying it by one.

00:02:32.350 --> 00:02:34.640
So I was able to
swap both of these.

00:02:34.640 --> 00:02:36.190
So is this true?

00:02:36.190 --> 00:02:40.680
Is it true that the argument
of t times z2 minus z1--

00:02:40.680 --> 00:02:43.340
is it true that that's the
same thing as the argument of t

00:02:43.340 --> 00:02:46.980
minus 1 times z2 minus z1?

00:02:46.980 --> 00:02:48.980
So let's just think
about it a little bit.

00:02:48.980 --> 00:02:55.080
Let's draw an
Argand diagram here.

00:02:55.080 --> 00:02:57.950
That's the imaginary axis.

00:02:57.950 --> 00:03:00.540
This is the real axis.

00:03:00.540 --> 00:03:03.870
And let's draw the
vector z2 minus z1.

00:03:03.870 --> 00:03:06.620
So let's say that this
right here is the vector.

00:03:06.620 --> 00:03:08.180
Let me draw it like this.

00:03:08.180 --> 00:03:12.730
This right here is the
vector, z2 minus z1.

00:03:12.730 --> 00:03:16.140
Now what would t times
z2 minus z1 look like?

00:03:16.140 --> 00:03:19.750
So t, we learned,
is between 0 and 1.

00:03:19.750 --> 00:03:24.300
So it's going to be a scaled
down version of z2 minus z1.

00:03:24.300 --> 00:03:26.960
So this right here-- so
who knows what it is.

00:03:26.960 --> 00:03:32.840
This right here would
be t times z2 minus z1.

00:03:32.840 --> 00:03:37.230
Now what would t minus 1
times z2 minus z1 look like?

00:03:37.230 --> 00:03:41.170
Well t minus 1, remember
t is between 0 and 1,

00:03:41.170 --> 00:03:43.320
so t is less than 1.

00:03:43.320 --> 00:03:46.220
So t minus 1--
this right here is

00:03:46.220 --> 00:03:48.470
going to be a negative number.

00:03:48.470 --> 00:03:51.520
It's going to be
a negative number.

00:03:51.520 --> 00:03:54.100
So we're going to be scaling
it by a negative number.

00:03:54.100 --> 00:03:57.140
So we're going to be scaling z2
minus z1 by a negative number.

00:03:57.140 --> 00:04:04.250
So this thing right over here,
t minus 1 times z2 minus z1

00:04:04.250 --> 00:04:07.350
is going to look like this.

00:04:07.350 --> 00:04:14.510
This is t minus 1
times z2 minus z1.

00:04:14.510 --> 00:04:17.709
Now the arguments
are just the angles

00:04:17.709 --> 00:04:22.390
between each of these
numbers and the real axis.

00:04:22.390 --> 00:04:24.650
So the argument for this
thing right over here

00:04:24.650 --> 00:04:25.745
is going to be this angle.

00:04:25.745 --> 00:04:29.130
It's going to be
phi right over here.

00:04:29.130 --> 00:04:31.660
So I could call that phi.

00:04:31.660 --> 00:04:33.650
But what's the argument
for this thing?

00:04:33.650 --> 00:04:38.580
It's going to be that plus
pi, or plus 180 degrees.

00:04:38.580 --> 00:04:42.560
We have to go all
the way around.

00:04:42.560 --> 00:04:45.050
So they are not the same angle.

00:04:45.050 --> 00:04:47.320
Whatever number this
is right over here,

00:04:47.320 --> 00:04:51.600
this number is going to be
that, plus or even minus pi.

00:04:51.600 --> 00:04:53.130
You could even go that way.

00:04:53.130 --> 00:04:56.480
But they're definitely not
going to be the same angle.

00:04:56.480 --> 00:05:03.980
So we can cross out B
choice B as an option.

00:05:03.980 --> 00:05:05.940
But now that we're
in this mode, let's

00:05:05.940 --> 00:05:08.502
see if we can tackle choice D,
because it looks very similar.

00:05:08.502 --> 00:05:10.710
And then I'll probably do
choice C in the next video,

00:05:10.710 --> 00:05:14.380
because I don't want to spend
too much time in each video.

00:05:14.380 --> 00:05:16.300
So what answer
choice D telling us?

00:05:16.300 --> 00:05:18.460
So let me write it down here.

00:05:18.460 --> 00:05:20.864
Choice D is telling
us-- so we don't

00:05:20.864 --> 00:05:22.280
know, we have to
see if it's true,

00:05:22.280 --> 00:05:26.240
that the argument
of z minus z1 is

00:05:26.240 --> 00:05:34.320
equal to the argument
of z2 minus z1.

00:05:34.320 --> 00:05:38.700
So let's think a little
bit about this right now.

00:05:38.700 --> 00:05:40.870
So once again, they
tell us, we already

00:05:40.870 --> 00:05:43.740
figured out what z
minus z1 is equal to.

00:05:43.740 --> 00:05:45.420
It's equal to this
business over here.

00:05:45.420 --> 00:05:47.710
So this statement is
equivalent to the statement

00:05:47.710 --> 00:05:54.110
that the argument of z minus z1,
which is t times z2 minus z1,

00:05:54.110 --> 00:05:55.920
leveraging our algebra
in the last video,

00:05:55.920 --> 00:06:00.810
t times z2 minus z1.

00:06:00.810 --> 00:06:06.750
Is this equal to the
argument of z2 minus z1?

00:06:06.750 --> 00:06:11.640
Well, once again, let's draw
our Argand diagram here.

00:06:11.640 --> 00:06:14.460
Actually, we could leverage
the same Argand diagram.

00:06:14.460 --> 00:06:19.860
In green right here,
we have z2 minus z1.

00:06:19.860 --> 00:06:22.797
It's argument would be
this angle right over here.

00:06:22.797 --> 00:06:26.480
It would be this angle, phi.

00:06:26.480 --> 00:06:28.350
This is z2 minus z1.

00:06:28.350 --> 00:06:29.840
That's that angle.

00:06:29.840 --> 00:06:32.240
T times z2 minus
z1 is just going

00:06:32.240 --> 00:06:35.400
to be the scaled down version
of it, what we have an orange.

00:06:35.400 --> 00:06:38.170
Now clearly, this is
just-- t is positive,

00:06:38.170 --> 00:06:39.920
so it's not pointing
in another direction.

00:06:39.920 --> 00:06:43.255
This is just a scaled
down version of this.

00:06:43.255 --> 00:06:47.184
This vector and this vector,
or this complex number

00:06:47.184 --> 00:06:48.600
and this complex
number, are going

00:06:48.600 --> 00:06:52.730
to point in the same direction,
so their angles are the same.

00:06:52.730 --> 00:06:57.680
T-- let me do it-- this orange
vector is this right over here,

00:06:57.680 --> 00:07:00.510
or that orange complex number
is this right over here.

00:07:00.510 --> 00:07:03.350
This is t times z2 minus z1.

00:07:03.350 --> 00:07:07.507
And then the green one, just to
be clear, z2 minus z1, is that.

00:07:07.507 --> 00:07:09.090
They're clearly in
the same direction.

00:07:09.090 --> 00:07:11.180
They clearly have
the same argument.

00:07:11.180 --> 00:07:19.735
So choice D another
correct choice.

00:07:19.735 --> 00:07:21.110
Now I'm going to
leave you there.

00:07:21.110 --> 00:07:24.927
In the next video, we're going
to see if choice C works.


IIT JEE complex numbers (part 1)

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:04.330
Let z1 and z2 be two
distinct complex numbers.

00:00:04.330 --> 00:00:06.210
And let z equal,
and they say it's

00:00:06.210 --> 00:00:11.080
"1 minus t times z1 plus t times
z2, for some real number with t

00:00:11.080 --> 00:00:12.700
being between 0 and 1."

00:00:12.700 --> 00:00:15.610
And they say, "If the argument
w denotes the principal argument

00:00:15.610 --> 00:00:18.190
of a nonzero complex
number w, then?"

00:00:18.190 --> 00:00:21.730
And we know the argument is the
angle of kind of the position

00:00:21.730 --> 00:00:26.290
vector specifying the complex
number of the complex plane,

00:00:26.290 --> 00:00:28.184
and the real axis.

00:00:28.184 --> 00:00:29.100
Let me just draw that.

00:00:29.100 --> 00:00:33.080
So if we do an Argand diagram,
this is the imaginary axis.

00:00:33.080 --> 00:00:35.310
And this is the real axis.

00:00:35.310 --> 00:00:38.680
If this is our complex
number, if that is z,

00:00:38.680 --> 00:00:41.000
it's argument--
the argument of z--

00:00:41.000 --> 00:00:43.280
is going to be this
angle right over here.

00:00:43.280 --> 00:00:48.430
So phi is equal to
the argument of z.

00:00:48.430 --> 00:00:51.200
That's all they're telling
us in this second part.

00:00:51.200 --> 00:00:53.660
Now, let's go through each of
these and see are they true.

00:00:53.660 --> 00:00:54.710
And just to be
clear, this is one

00:00:54.710 --> 00:00:56.960
of those problems where more
than one of these choices

00:00:56.960 --> 00:00:57.690
might be correct.

00:00:57.690 --> 00:00:59.510
So let's see if they're true.

00:00:59.510 --> 00:01:04.959
So let's first figure out what
the magnitude of z minus z1 is.

00:01:04.959 --> 00:01:08.870
So the magnitude
of z minus z1 is

00:01:08.870 --> 00:01:11.940
going to be the magnitude of--
so let me write-- this is going

00:01:11.940 --> 00:01:14.900
to be the magnitude of--
let's do the z part first.

00:01:14.900 --> 00:01:17.030
Let me write in the same colors.

00:01:17.030 --> 00:01:19.830
So if I would just
focus on the z first,

00:01:19.830 --> 00:01:22.010
z is this thing up here.

00:01:22.010 --> 00:01:23.510
And let me distribute the z1.

00:01:23.510 --> 00:01:31.630
So it's z1 minus tz1 minus
tz1, and then plus tz2.

00:01:35.120 --> 00:01:38.160
And from that, we
want to subtract z1.

00:01:38.160 --> 00:01:42.780
So from that, we want to
subtract z1, so minus Z1.

00:01:42.780 --> 00:01:44.980
And that and that cancel out.

00:01:47.777 --> 00:01:48.360
And let's see.

00:01:48.360 --> 00:01:52.490
We can factor out a t over here.

00:01:52.490 --> 00:01:57.450
So this is going to be equal to
the magnitude of-- let's factor

00:01:57.450 --> 00:02:01.110
out a t times z2 minus z1.

00:02:06.640 --> 00:02:10.490
So that is the
magnitude of z minus z1,

00:02:10.490 --> 00:02:12.270
this first term over here.

00:02:12.270 --> 00:02:15.430
Let's figure out the
magnitude of z minus z2.

00:02:19.060 --> 00:02:22.610
I'm going to color
code it. z minus z2

00:02:22.610 --> 00:02:24.610
is equal to the
magnitude-- well,

00:02:24.610 --> 00:02:27.040
z is just this thing up here.

00:02:27.040 --> 00:02:28.140
Let me just write it out.

00:02:28.140 --> 00:02:36.370
So it is 1 minus t times z1
plus t times z2, that's z.

00:02:36.370 --> 00:02:41.964
And from that, we want to
subtract z2, so minus z2.

00:02:41.964 --> 00:02:43.130
So what can we do over here?

00:02:43.130 --> 00:02:48.740
We can take a t minus
1 out from here.

00:02:48.740 --> 00:02:51.970
We can factor out the z2
out of these two terms.

00:02:51.970 --> 00:02:56.470
So this is going to be
equal to the magnitude of 1

00:02:56.470 --> 00:03:03.360
minus t times z1,
1 minus t times z1.

00:03:03.360 --> 00:03:05.470
That's this term
right over here.

00:03:05.470 --> 00:03:16.320
Plus-- let me do it in another
color-- plus t minus 1,

00:03:16.320 --> 00:03:20.080
times z2.

00:03:20.080 --> 00:03:25.122
The magnitude of this part over
here is this thing over there.

00:03:25.122 --> 00:03:26.580
Now, let's see what
happens when we

00:03:26.580 --> 00:03:27.890
add this thing to this thing.

00:03:27.890 --> 00:03:29.390
That's what choice A is doing.

00:03:29.390 --> 00:03:30.980
We're adding this
thing to this thing.

00:03:30.980 --> 00:03:33.010
Let's see if we can simplify it.

00:03:33.010 --> 00:03:38.630
So this becomes the absolute
value, or the magnitude,

00:03:38.630 --> 00:03:46.090
of t times z2 minus z1 plus--
I'll do the whole thing here

00:03:46.090 --> 00:03:49.600
in magenta-- plus
this thing over here.

00:03:49.600 --> 00:03:52.050
And this thing over here--
actually, before I even

00:03:52.050 --> 00:03:54.460
write it out-- how
can we simplify that?

00:03:54.460 --> 00:03:56.720
We'll have a 1 minus
t and a t minus 1.

00:03:56.720 --> 00:03:58.320
Well, I'll just
write it out here.

00:03:58.320 --> 00:04:00.111
But I'm going to change
it up a little bit.

00:04:00.111 --> 00:04:02.000
So this is equal
to the magnitude

00:04:02.000 --> 00:04:08.490
of 1 minus t times z1.

00:04:08.490 --> 00:04:10.000
And so that I have
a 1 minus t here,

00:04:10.000 --> 00:04:11.791
I'm just going to put
a negative out front.

00:04:11.791 --> 00:04:15.220
So minus 1, minus t,
I just swapped them.

00:04:15.220 --> 00:04:17.920
Negative 1 minus t is the
same thing as positive t

00:04:17.920 --> 00:04:20.589
minus 1, z2.

00:04:20.589 --> 00:04:23.670
And then this thing over here,
since I have the 1 minus t

00:04:23.670 --> 00:04:25.440
out there, this
becomes the magnitude

00:04:25.440 --> 00:04:28.241
of, I'm just factoring
out the 1 minus t,

00:04:28.241 --> 00:04:32.670
1 minus t times z1 minus z2.

00:04:36.380 --> 00:04:38.340
So we have this thing.

00:04:38.340 --> 00:04:40.210
Let me copy and paste it.

00:04:40.210 --> 00:04:42.810
So, copy and paste.

00:04:42.810 --> 00:04:44.810
We have this thing
plus this thing.

00:04:44.810 --> 00:04:48.800
This is what this expression
A has simplified to.

00:04:48.800 --> 00:04:51.530
Now let's see if we can
simplify that even more.

00:04:51.530 --> 00:04:53.440
Let's see if we can
simplify it even more.

00:04:53.440 --> 00:04:57.180
So t is just a scalar.

00:04:57.180 --> 00:04:59.270
And t is between 0 and 1.

00:04:59.270 --> 00:05:02.440
They have told us that over
here, t is between 0 and 1.

00:05:02.440 --> 00:05:03.720
So this is positive.

00:05:03.720 --> 00:05:05.880
This right here is
a positive value.

00:05:05.880 --> 00:05:11.370
And then this right here is also
going to be a positive value.

00:05:11.370 --> 00:05:15.605
t is greater than 0,
and it is less than 1.

00:05:15.605 --> 00:05:17.480
So this is also going
to be a positive value.

00:05:17.480 --> 00:05:18.860
And these are just scalars.

00:05:18.860 --> 00:05:21.050
So this is going to
be the same thing.

00:05:21.050 --> 00:05:23.087
These are just
scaling the magnitude.

00:05:23.087 --> 00:05:24.670
So this is going to
be the same thing.

00:05:24.670 --> 00:05:26.380
These are positive values.

00:05:26.380 --> 00:05:30.602
So this is going to be t times
the absolute value-- actually

00:05:30.602 --> 00:05:31.560
let me not skip a step.

00:05:31.560 --> 00:05:33.893
This is going to be the same
thing as the absolute value

00:05:33.893 --> 00:05:38.840
of t times the absolute
value of z2 minus z1--

00:05:38.840 --> 00:05:40.460
because this is
just scaling it--

00:05:40.460 --> 00:05:48.770
plus the absolute
value of 1 minus t,

00:05:48.770 --> 00:05:51.980
times the absolute
value of z1 minus z2.

00:05:51.980 --> 00:05:55.600
Now let me be clear, the
absolute value of z1 minus z2

00:05:55.600 --> 00:06:00.240
is going to be equal to the
absolute value of z2 minus z1.

00:06:00.240 --> 00:06:02.739
These vectors are just pointing
in different directions,

00:06:02.739 --> 00:06:03.780
or these complex numbers.

00:06:03.780 --> 00:06:05.702
One is just the
negative of the other.

00:06:05.702 --> 00:06:07.660
But their absolute values,
or their magnitudes,

00:06:07.660 --> 00:06:09.100
are going to be the same.

00:06:09.100 --> 00:06:13.050
So let me write z2 minus z1, the
absolute value of z2 minus z1,

00:06:13.050 --> 00:06:13.550
there.

00:06:13.550 --> 00:06:14.660
And the reason why
I'm doing it is

00:06:14.660 --> 00:06:15.900
so I get the same thing here.

00:06:15.900 --> 00:06:17.320
And I can factor it out.

00:06:17.320 --> 00:06:19.940
So the absolute
value of z1 minus z1

00:06:19.940 --> 00:06:21.660
is the same thing as
the absolute value

00:06:21.660 --> 00:06:24.760
of z1 minus z2,
because it's just

00:06:24.760 --> 00:06:26.760
going in the other direction.

00:06:26.760 --> 00:06:28.380
Now, what can we do here?

00:06:28.380 --> 00:06:30.460
Well, the absolute
value of t, remember

00:06:30.460 --> 00:06:34.875
t is positive, so this is
just going to be equal to t.

00:06:34.875 --> 00:06:37.000
The absolute value of 1
minus t, once again, that's

00:06:37.000 --> 00:06:39.650
positive, so that's just
going to be 1 minus t.

00:06:39.650 --> 00:06:43.800
So we can factor
this business out.

00:06:43.800 --> 00:06:50.360
So we're going to
get t plus 1 minus t,

00:06:50.360 --> 00:06:54.970
times the absolute
value of z2 minus z1.

00:06:58.360 --> 00:06:59.720
Now, the t's cancel out here.

00:06:59.720 --> 00:07:00.980
We just got a one out front.

00:07:00.980 --> 00:07:06.210
So this is just equal to the
absolute value of z2 minus z1.

00:07:06.210 --> 00:07:08.890
So we see that
choice A does work.

00:07:08.890 --> 00:07:12.620
This plus this does
indeed equal that.

00:07:12.620 --> 00:07:13.910
I'm going to leave you there.

00:07:13.910 --> 00:07:15.326
And in the next
video, we're going

00:07:15.326 --> 00:07:17.410
to try out some of these
other possibilities

00:07:17.410 --> 00:07:20.740
to see if they
might also be true.


Complex conjugates

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.040
I want to make a
quick clarification

00:00:02.040 --> 00:00:05.210
and then add more tools in
our complex number toolkit.

00:00:05.210 --> 00:00:09.040
In the first video, I said that
if I had a complex number z,

00:00:09.040 --> 00:00:14.222
and it's equal to a
plus bi, I used a word.

00:00:14.222 --> 00:00:15.930
And I have to be
careful about that word,

00:00:15.930 --> 00:00:18.870
because I used in
the everyday sense.

00:00:18.870 --> 00:00:21.460
But it also has a
formal reality to it.

00:00:21.460 --> 00:00:26.100
So clearly, the real part
of this complex number is a.

00:00:26.100 --> 00:00:29.200
Clearly, that is the real part.

00:00:29.200 --> 00:00:32.250
And clearly, this
complex number is made up

00:00:32.250 --> 00:00:35.510
of a real number plus
an imaginary number.

00:00:35.510 --> 00:00:37.970
So just kind of talking
in everyday terms,

00:00:37.970 --> 00:00:41.460
I called this the
imaginary part.

00:00:41.460 --> 00:00:47.780
I called this imaginary
number the imaginary part.

00:00:47.780 --> 00:00:49.880
But I want to just
be careful there.

00:00:49.880 --> 00:00:52.012
I did make it clear
that if you were

00:00:52.012 --> 00:00:54.770
to see the function
the real part of z,

00:00:54.770 --> 00:00:56.442
this would spit out the a.

00:00:56.442 --> 00:01:00.030
And the function the
imaginary part of z,

00:01:00.030 --> 00:01:01.520
this would spit
out-- and we talked

00:01:01.520 --> 00:01:03.561
about this in the first
video-- it would spit out

00:01:03.561 --> 00:01:05.450
the number that's scaling the i.

00:01:05.450 --> 00:01:07.540
So it would spit out the b.

00:01:07.540 --> 00:01:10.130
So if someone is talking
in the formal sense

00:01:10.130 --> 00:01:11.900
about the imaginary
part, they're

00:01:11.900 --> 00:01:15.670
really talking about the
number that is scaling the i.

00:01:15.670 --> 00:01:18.620
But in my brain, when I
think of a complex number,

00:01:18.620 --> 00:01:21.296
I think of it having a real
number and an imaginary number.

00:01:21.296 --> 00:01:23.420
And if someone were to say,
well, what part of that

00:01:23.420 --> 00:01:24.530
is the imaginary number?

00:01:24.530 --> 00:01:25.750
I would have given
this whole thing.

00:01:25.750 --> 00:01:27.920
But if someone says just
what's the imaginary part,

00:01:27.920 --> 00:01:31.680
where they give you this
function, just give them the b.

00:01:31.680 --> 00:01:33.330
Hopefully, that
clarifies things.

00:01:33.330 --> 00:01:36.040
Frankly, I think the word
"imaginary part" is badly

00:01:36.040 --> 00:01:38.380
named because clearly,
this whole thing is

00:01:38.380 --> 00:01:39.660
an imaginary number.

00:01:39.660 --> 00:01:41.810
This right here is not
an imaginary number.

00:01:41.810 --> 00:01:43.300
It's just a real number.

00:01:43.300 --> 00:01:46.610
It's the real number
scaling the i.

00:01:46.610 --> 00:01:48.970
So they should call
this the number scaling

00:01:48.970 --> 00:01:50.990
the imaginary part of z.

00:01:50.990 --> 00:01:52.770
Anyway, with that
said, what I want

00:01:52.770 --> 00:01:55.910
to introduce you to is the
idea of a complex number's

00:01:55.910 --> 00:01:57.070
conjugate.

00:01:57.070 --> 00:02:01.950
So if this is z, the
conjugate of z-- it'd

00:02:01.950 --> 00:02:03.550
be denoted with z
with a bar over it.

00:02:03.550 --> 00:02:07.000
Sometimes it's z with a little
asterisk right over there.

00:02:07.000 --> 00:02:11.270
That would just be
equal to a minus bi.

00:02:11.270 --> 00:02:15.580
So let's see how they
look on an Argand diagram.

00:02:15.580 --> 00:02:17.375
So that's my real axis.

00:02:19.910 --> 00:02:23.220
And then that is
my imaginary axis.

00:02:23.220 --> 00:02:26.620
And then if I have
z-- this is z over

00:02:26.620 --> 00:02:29.200
here-- this height
over here is b.

00:02:29.200 --> 00:02:32.010
This base, or this
length, right here is a.

00:02:32.010 --> 00:02:33.840
That's z.

00:02:33.840 --> 00:02:36.860
The conjugate of
z is a minus bi.

00:02:36.860 --> 00:02:38.590
So it comes out a
on the real axis,

00:02:38.590 --> 00:02:42.350
but it has minus b as
its imaginary part,

00:02:42.350 --> 00:02:46.580
so just like this.

00:02:46.580 --> 00:02:48.320
So this is the conjugate of z.

00:02:48.320 --> 00:02:52.270
So just to visualize it, a
conjugate of a complex number

00:02:52.270 --> 00:02:55.920
is really the mirror image of
that complex number reflected

00:02:55.920 --> 00:02:57.807
over the x-axis.

00:02:57.807 --> 00:02:59.640
You can imagine if this
was a pool of water,

00:02:59.640 --> 00:03:01.612
we're seeing its
reflection over here.

00:03:01.612 --> 00:03:03.070
And so we can
actually look at this

00:03:03.070 --> 00:03:09.220
to visually add the complex
number and its conjugate.

00:03:09.220 --> 00:03:11.220
So we said these are just
like position vectors.

00:03:11.220 --> 00:03:13.870
So if we were to add
z and its conjugate,

00:03:13.870 --> 00:03:16.010
we could essentially
just take this vector,

00:03:16.010 --> 00:03:17.860
shift it up here,
do heads to tails.

00:03:20.980 --> 00:03:24.530
So this right here, we are
adding z to its conjugate.

00:03:24.530 --> 00:03:29.220
And so this point right
here, or the vector

00:03:29.220 --> 00:03:34.270
that specifies that point,
is z plus z's conjugate.

00:03:34.270 --> 00:03:36.400
And you can see right
here, just visually, this

00:03:36.400 --> 00:03:37.380
is going to be 2a.

00:03:40.660 --> 00:03:42.130
And to do that algebraically.

00:03:42.130 --> 00:03:47.370
If we were to add z--
that's a plus bi--

00:03:47.370 --> 00:03:51.889
and add that to its
conjugate, so plus a minus bi,

00:03:51.889 --> 00:03:52.930
what are we going to get?

00:03:52.930 --> 00:03:54.670
These two guys cancel out.

00:03:54.670 --> 00:03:56.214
We're just going to have 2a.

00:03:56.214 --> 00:03:57.630
Or another way to
think about it--

00:03:57.630 --> 00:03:59.680
and really, we're just
playing around with math--

00:03:59.680 --> 00:04:05.970
if I take any complex number,
and to it I add its conjugate,

00:04:05.970 --> 00:04:10.760
I'm going to get 2
times the real part

00:04:10.760 --> 00:04:12.410
of the complex number.

00:04:12.410 --> 00:04:14.660
Oh, and this is also going
to be 2 times the real part

00:04:14.660 --> 00:04:19.070
of the conjugate because they
have the exact same real part.

00:04:19.070 --> 00:04:21.190
Now with that said,
let's think about where

00:04:21.190 --> 00:04:25.280
the conjugate could be useful.

00:04:25.280 --> 00:04:27.490
So let's say I had
something like 1

00:04:27.490 --> 00:04:37.220
plus 2i divided by 4 minus 5i.

00:04:37.220 --> 00:04:40.154
So it's no real obvious way
to simplify this expression.

00:04:40.154 --> 00:04:42.320
Maybe I don't like having
this i in the denominator.

00:04:42.320 --> 00:04:45.030
Maybe I just want to write
this as one complex number.

00:04:45.030 --> 00:04:46.780
If I divide one complex
number by another,

00:04:46.780 --> 00:04:48.280
I should get another
complex number.

00:04:48.280 --> 00:04:49.680
But how do I do that?

00:04:49.680 --> 00:04:52.005
Well, one thing to
do is to multiply

00:04:52.005 --> 00:04:53.380
the numerator and
the denominator

00:04:53.380 --> 00:04:55.710
by the conjugate
of the denominator,

00:04:55.710 --> 00:05:01.620
so 4 plus 5i over 4 plus 5i.

00:05:01.620 --> 00:05:03.390
And clearly, I'm just
multiplying by 1,

00:05:03.390 --> 00:05:05.598
because this is the same
number over the same number.

00:05:05.598 --> 00:05:07.230
But the reason why
this is valuable

00:05:07.230 --> 00:05:10.700
is if I multiply a number
times its conjugate,

00:05:10.700 --> 00:05:13.120
I'm going to get a real number.

00:05:13.120 --> 00:05:15.020
So let me just
show you that here.

00:05:15.020 --> 00:05:18.420
So let's just multiply this out.

00:05:18.420 --> 00:05:23.850
So we're going to get 1
times 4 plus 5i is 4 plus 5i.

00:05:23.850 --> 00:05:29.360
And then 2i times 4 is plus 8i.

00:05:29.360 --> 00:05:32.250
And then 2i times
5i-- that would

00:05:32.250 --> 00:05:35.890
be 10i squared, or negative 10.

00:05:35.890 --> 00:05:38.130
And then that will
be over-- now,

00:05:38.130 --> 00:05:41.190
this has the form a
minus b times a plus b.

00:05:44.120 --> 00:05:47.710
Well, a plus b times a minus b
is a squared minus b squared.

00:05:47.710 --> 00:05:50.890
So it's going to be equal
to 4 squared, which is 16,

00:05:50.890 --> 00:05:54.450
minus 4 squared-- oh, why
did I have 4 plus 4i here?

00:05:54.450 --> 00:05:57.460
This should be 4 plus 5i.

00:05:57.460 --> 00:05:58.170
What am I doing?

00:05:58.170 --> 00:06:02.450
4 plus 5i, the same number
over the same number.

00:06:02.450 --> 00:06:05.834
This was a 10 right over there.

00:06:05.834 --> 00:06:06.750
This is the conjugate.

00:06:06.750 --> 00:06:07.291
I don't know.

00:06:07.291 --> 00:06:09.030
My brain must have
been thinking in 4's.

00:06:09.030 --> 00:06:11.071
So obviously, I don't want
to change the number--

00:06:11.071 --> 00:06:14.075
4 plus 5i over 4 plus 5i.

00:06:14.075 --> 00:06:14.950
So let's multiply it.

00:06:14.950 --> 00:06:18.690
This is a minus b times
a plus b, so 4 times 4.

00:06:18.690 --> 00:06:21.710
So this is going to be 4
squared minus 5i squared.

00:06:26.700 --> 00:06:29.210
And so this is going to
be equal to 4 minus 10.

00:06:29.210 --> 00:06:30.720
Let's add the real parts.

00:06:30.720 --> 00:06:33.490
4 minus 10 is negative 6.

00:06:33.490 --> 00:06:37.050
5i plus 8i is 13i.

00:06:37.050 --> 00:06:38.560
Add the imaginary parts.

00:06:38.560 --> 00:06:41.980
And then you have
16 minus 5i squared.

00:06:41.980 --> 00:06:45.610
Well, 5i squared-- i
squared is negative 1.

00:06:45.610 --> 00:06:47.490
5 squared's just going
to be negative 25.

00:06:47.490 --> 00:06:49.240
The negative and the
negative cancel out,

00:06:49.240 --> 00:06:52.360
so you have 16 plus 25.

00:06:52.360 --> 00:06:54.852
So that is 41.

00:06:54.852 --> 00:06:56.560
So we can write this
as a complex number.

00:06:56.560 --> 00:07:03.270
This is negative
6/41 plus 13/41 i.

00:07:03.270 --> 00:07:06.220
We were able to divide
these two complex numbers.

00:07:06.220 --> 00:07:08.660
So the useful thing
here is the property

00:07:08.660 --> 00:07:12.190
that if I take any
complex number,

00:07:12.190 --> 00:07:17.465
and I multiply it
by its conjugate--

00:07:17.465 --> 00:07:19.340
and obviously, the
conjugate of the conjugate

00:07:19.340 --> 00:07:20.882
is the original number.

00:07:20.882 --> 00:07:22.340
But I would take
any complex number

00:07:22.340 --> 00:07:24.090
and I multiply it
by its conjugate,

00:07:24.090 --> 00:07:28.789
so this would be a plus
bi times a minus bi.

00:07:28.789 --> 00:07:30.080
I'm going to get a real number.

00:07:30.080 --> 00:07:37.120
It's going to be a squared
minus bi squared, difference

00:07:37.120 --> 00:07:40.180
of squares, which is
equal to a squared-- now,

00:07:40.180 --> 00:07:41.810
this is going to be
negative b squared.

00:07:41.810 --> 00:07:43.351
But we have a negative
sign out here,

00:07:43.351 --> 00:07:46.319
so they cancel out-- a
squared plus b squared.

00:07:46.319 --> 00:07:48.360
And just out of curiosity,
this is the same thing

00:07:48.360 --> 00:07:52.680
as the magnitude of our
complex number squared.

00:07:52.680 --> 00:07:53.930
So this is a neat property.

00:07:53.930 --> 00:07:55.970
This is what makes
conjugates really useful,

00:07:55.970 --> 00:07:58.370
especially when you
want to simplify

00:07:58.370 --> 00:08:00.760
division of complex numbers.

00:08:00.760 --> 00:08:03.230
Anyway, hopefully,
you found that useful.


Exponential form to find complex roots

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:06.110
In this video, we're going
to hopefully understand

00:00:06.110 --> 00:00:08.980
why the exponential
form of a complex number

00:00:08.980 --> 00:00:10.635
is actually useful.

00:00:13.320 --> 00:00:15.610
So let's say we want
to solve the equation x

00:00:15.610 --> 00:00:18.210
to the third power
is equal to 1.

00:00:18.210 --> 00:00:22.900
So we want to find all of
the real and/or complex roots

00:00:22.900 --> 00:00:24.540
of this equation
right over here.

00:00:24.540 --> 00:00:27.070
This is the same thing
as x to the third minus 1

00:00:27.070 --> 00:00:27.950
is equal to 0.

00:00:27.950 --> 00:00:31.470
So we're looking for all the
real and complex roots of this.

00:00:31.470 --> 00:00:34.170
And there are ways to do
this without exponential form

00:00:34.170 --> 00:00:35.530
of a complex number.

00:00:35.530 --> 00:00:38.370
But the technique we're
going to see in this video

00:00:38.370 --> 00:00:40.750
could be applied if this
was x to the fifth minus 1,

00:00:40.750 --> 00:00:42.840
or x to the 13th minus 1.

00:00:42.840 --> 00:00:46.020
And it's also going to
show us the patterns that

00:00:46.020 --> 00:00:49.530
emerge when you start looking
at things on an Argand diagram.

00:00:49.530 --> 00:00:54.220
So to do this, let's think about
the exponential representation

00:00:54.220 --> 00:00:55.330
of 1.

00:00:55.330 --> 00:00:57.800
So let's just say
z is equal to 1.

00:00:57.800 --> 00:00:59.140
1 is a complex number.

00:00:59.140 --> 00:01:00.220
It's a real number.

00:01:00.220 --> 00:01:02.660
All real numbers are
also complex numbers.

00:01:02.660 --> 00:01:04.150
They're a subset.

00:01:04.150 --> 00:01:07.860
They're in the complex plane.

00:01:07.860 --> 00:01:10.110
They just don't have
an imaginary part.

00:01:10.110 --> 00:01:13.200
So let's draw this
on an Argand diagram.

00:01:13.200 --> 00:01:15.850
So that's my real axis.

00:01:15.850 --> 00:01:17.955
This is my imaginary axis.

00:01:22.830 --> 00:01:24.870
And so this is the real.

00:01:24.870 --> 00:01:26.660
This is the imaginary.

00:01:26.660 --> 00:01:29.720
And if I wanted to
represent z equals 1,

00:01:29.720 --> 00:01:31.570
it only has a real part.

00:01:31.570 --> 00:01:33.860
So let me just
draw 1 all around.

00:01:33.860 --> 00:01:35.260
Negative 1.

00:01:35.260 --> 00:01:36.994
Negative 1.

00:01:36.994 --> 00:01:38.720
z looks like this.

00:01:42.080 --> 00:01:44.560
z would look like
that, a position

00:01:44.560 --> 00:01:47.150
vector that just goes to 1, 0.

00:01:47.150 --> 00:01:51.500
One way to view it-- this is
the same thing as equal to 1

00:01:51.500 --> 00:01:54.270
plus 0i.

00:01:54.270 --> 00:01:57.610
Now, let's put this
in exponential form.

00:01:57.610 --> 00:02:00.000
Well, its magnitude is
pretty straightforward.

00:02:00.000 --> 00:02:03.780
The magnitude of z is just
the length of this vector,

00:02:03.780 --> 00:02:06.690
or it's the absolute value of 1.

00:02:06.690 --> 00:02:09.060
That's just going to be 1.

00:02:09.060 --> 00:02:14.030
Now, what's the argument of z?

00:02:14.030 --> 00:02:15.670
What's the angle
that this vector

00:02:15.670 --> 00:02:19.050
makes with the
positive real axis?

00:02:19.050 --> 00:02:20.790
Well, it's on the
positive real axis.

00:02:20.790 --> 00:02:22.140
It's a real number.

00:02:22.140 --> 00:02:23.340
So it has no angle.

00:02:23.340 --> 00:02:26.960
So the arg of z is 0.

00:02:26.960 --> 00:02:30.420
So that might not be
too interesting so far.

00:02:30.420 --> 00:02:40.550
We just figured out that 1 is
equal to 1 times e to the 0i.

00:02:40.550 --> 00:02:41.770
And this is kind of obvious.

00:02:41.770 --> 00:02:44.060
e to the 0-- this is
just going to be 0.

00:02:44.060 --> 00:02:45.270
0 times i is 0.

00:02:45.270 --> 00:02:47.770
e to the 0 is going to be
1, times 1 is equal to 1.

00:02:47.770 --> 00:02:49.570
Not a big deal there.

00:02:49.570 --> 00:02:52.690
But what is neat is that this
argument-- you could view it

00:02:52.690 --> 00:02:55.800
as 0 radians, or you could
go all the way around

00:02:55.800 --> 00:02:58.100
and add 2 pi to it and
get to the same point.

00:03:02.500 --> 00:03:07.540
So the argument of our complex
number-- or of the number 1,

00:03:07.540 --> 00:03:12.910
really-- could also be an angle
of 2 pi, or an angle of 4 pi,

00:03:12.910 --> 00:03:15.700
or an angle of 6 pi,
or an angle of 8 pi.

00:03:15.700 --> 00:03:20.860
So we can write 1
as 1 times e-- I

00:03:20.860 --> 00:03:22.520
won't write the 1
anymore-- 1 times

00:03:22.520 --> 00:03:31.120
e to the 2 pi i, or 1
times e to the 4 pi i.

00:03:31.120 --> 00:03:32.740
And the reason why
this is interesting

00:03:32.740 --> 00:03:35.544
is then this equation
right here can

00:03:35.544 --> 00:03:36.710
be written in multiple ways.

00:03:36.710 --> 00:03:42.200
It can be written as x to
the third is equal to 1.

00:03:42.200 --> 00:03:44.970
It could be written
as x to the third

00:03:44.970 --> 00:03:49.550
is equal to e to the 2 pi i.

00:03:49.550 --> 00:03:52.370
Or it could be written
as x to the third

00:03:52.370 --> 00:03:54.737
is equal to e to the 4 pi i.

00:03:57.844 --> 00:03:59.260
And this is
interesting, and we're

00:03:59.260 --> 00:04:00.030
going to see this in a second.

00:04:00.030 --> 00:04:02.140
Let's take both sides
of all these equations

00:04:02.140 --> 00:04:04.900
to the one-third
power to solve for x.

00:04:04.900 --> 00:04:06.810
So to the one-third.

00:04:06.810 --> 00:04:08.560
We're going to take
that to the one-third.

00:04:08.560 --> 00:04:10.690
We're going to do that
same thing over here.

00:04:10.690 --> 00:04:12.900
We're just taking everything
to the one-third power

00:04:12.900 --> 00:04:16.839
to solve for the x's in
each of these equations.

00:04:16.839 --> 00:04:18.160
To the one-third power.

00:04:18.160 --> 00:04:21.649
So this first equation over
here becomes x is equal to 1

00:04:21.649 --> 00:04:25.510
to the one-third power,
which is just equal to 1.

00:04:25.510 --> 00:04:27.740
Now, what's the second
equation become?

00:04:27.740 --> 00:04:31.680
This second equation-- x is
equal to e to the-- well,

00:04:31.680 --> 00:04:34.325
this is going to be the
2 pi over 3, i power.

00:04:41.480 --> 00:04:43.160
And then this
equation over here is

00:04:43.160 --> 00:04:46.430
going to be-- so x is going
to be equal to-- obviously,

00:04:46.430 --> 00:04:50.220
the 3 to the one-third, that
just becomes x to the 1.

00:04:50.220 --> 00:04:52.620
Let me do that same blue.

00:04:52.620 --> 00:05:03.590
x over here is going to be equal
to e to the 4 pi over 3, i.

00:05:03.590 --> 00:05:06.580
So let's think about
this for a little bit.

00:05:06.580 --> 00:05:08.530
What is this?

00:05:08.530 --> 00:05:11.070
So immediately, what's
the argument here?

00:05:11.070 --> 00:05:14.670
So these are three
different roots.

00:05:14.670 --> 00:05:18.649
Let me call this x1, x2, and x3.

00:05:18.649 --> 00:05:20.190
So these are three
different numbers.

00:05:20.190 --> 00:05:22.130
One of the roots is 1.

00:05:22.130 --> 00:05:23.510
That's pretty clear over here.

00:05:23.510 --> 00:05:26.760
1 is one of the cube
roots of itself.

00:05:26.760 --> 00:05:28.270
But these are other numbers.

00:05:28.270 --> 00:05:30.370
And these are going
to be complex numbers.

00:05:30.370 --> 00:05:33.545
So let's visualize these
numbers a little bit.

00:05:33.545 --> 00:05:34.545
So what is the argument?

00:05:39.280 --> 00:05:41.910
The magnitude of x2
is still clearly 1.

00:05:41.910 --> 00:05:44.015
It's the coefficient
out in front of the e.

00:05:44.015 --> 00:05:46.465
It's clearly 1.

00:05:46.465 --> 00:05:48.410
Let me do that same color.

00:05:48.410 --> 00:05:52.880
The magnitude of x3 is
also clearly going to be 1.

00:05:52.880 --> 00:05:55.780
But what is the argument of x2?

00:05:55.780 --> 00:05:57.090
What is phi?

00:05:57.090 --> 00:05:59.000
What is the argument?

00:05:59.000 --> 00:06:02.190
Well, it's 2 pi over 3.

00:06:02.190 --> 00:06:06.620
So how would we draw x2?

00:06:06.620 --> 00:06:09.680
So the angle is 2 pi over 3.

00:06:09.680 --> 00:06:11.830
It's easier for me to
visualize in degrees.

00:06:11.830 --> 00:06:14.050
So 2 pi is 360 degrees.

00:06:14.050 --> 00:06:18.530
360 degrees divided
by 3 is 120 degrees.

00:06:18.530 --> 00:06:20.300
So this is going
to be-- 120 degrees

00:06:20.300 --> 00:06:23.345
is 60 short of-- so it's
going to look like this.

00:06:29.240 --> 00:06:31.210
So this angle right
here, its argument

00:06:31.210 --> 00:06:34.760
is going to be
120 degrees, which

00:06:34.760 --> 00:06:38.686
is the same thing
as 2 pi over 3.

00:06:38.686 --> 00:06:40.560
And it's going to have
the exact same length.

00:06:40.560 --> 00:06:41.976
So let me do it
in the same color.

00:06:41.976 --> 00:06:43.030
So this is x1.

00:06:43.030 --> 00:06:46.980
So that is this green
color right over here.

00:06:46.980 --> 00:06:50.349
x2 is this magenta
one right over here.

00:06:50.349 --> 00:06:51.890
And they all have
the same magnitude.

00:06:51.890 --> 00:06:53.056
So we really just rotate it.

00:06:53.056 --> 00:06:54.690
We rotate it 120 degrees.

00:06:54.690 --> 00:06:55.750
And what about x3?

00:06:55.750 --> 00:06:57.770
What's its argument?

00:06:57.770 --> 00:07:00.100
What's x3's argument?

00:07:00.100 --> 00:07:04.480
Its argument is 4 pi over 3.

00:07:04.480 --> 00:07:09.400
That's the same thing
as 720 degrees over 3,

00:07:09.400 --> 00:07:11.130
if we were to put
it into degrees.

00:07:11.130 --> 00:07:16.890
And so 3 goes into
720-- what is it?

00:07:16.890 --> 00:07:18.710
240?

00:07:18.710 --> 00:07:19.780
720.

00:07:19.780 --> 00:07:22.350
240.

00:07:22.350 --> 00:07:22.890
Right.

00:07:22.890 --> 00:07:24.230
I should have known that.

00:07:24.230 --> 00:07:27.710
So 240 degrees-- we're
going to go 180 degrees,

00:07:27.710 --> 00:07:29.950
and then go another 60 degrees.

00:07:29.950 --> 00:07:31.680
So it's going to
be right over here.

00:07:35.152 --> 00:07:36.740
So let me draw it like this.

00:07:36.740 --> 00:07:39.310
It's going to be
right over here.

00:07:41.860 --> 00:07:44.000
That angle right
over there is 4 pi

00:07:44.000 --> 00:07:52.400
over 3 radians, which
is equal to 240 degrees.

00:07:52.400 --> 00:07:55.290
And once again, it has
the same magnitude.

00:07:55.290 --> 00:07:57.310
So what we just saw is
when I take the cube

00:07:57.310 --> 00:08:00.210
roots of this real
number, I'm essentially

00:08:00.210 --> 00:08:02.120
taking the entire--
I guess we could

00:08:02.120 --> 00:08:05.350
call it the entire
circle or the entire 360

00:08:05.350 --> 00:08:07.480
degrees or the
entire 2 pi radians--

00:08:07.480 --> 00:08:12.180
and I'm dividing it
into three, essentially.

00:08:12.180 --> 00:08:13.250
This is one third.

00:08:13.250 --> 00:08:15.120
Then we have
another 120 degrees.

00:08:15.120 --> 00:08:17.310
And then we have
another 120 degrees.

00:08:17.310 --> 00:08:20.216
And so you see the pattern of
where all of the roots are.

00:08:20.216 --> 00:08:21.840
And in case you're
still not satisfied,

00:08:21.840 --> 00:08:24.680
you're just like, well, you said
you would find complex roots.

00:08:24.680 --> 00:08:27.870
Yeah, I'm not used
to this or this

00:08:27.870 --> 00:08:30.350
as actually being
complex numbers.

00:08:30.350 --> 00:08:34.020
I actually want it to be in the
form a plus bi-- we can easily

00:08:34.020 --> 00:08:36.820
figure it out from
this right over here.

00:08:36.820 --> 00:08:47.510
So x2-- it's going to be equal
to the cosine of 2 pi over 3

00:08:47.510 --> 00:08:52.090
plus i times the
sine of 2 pi over 3.

00:08:55.624 --> 00:08:57.040
And if you look
at this over here,

00:08:57.040 --> 00:08:59.206
we can figure out what those
things are going to be.

00:08:59.206 --> 00:09:01.060
This is the angle
right over here.

00:09:01.060 --> 00:09:04.120
If this angle right over
here is 60 degrees-- which

00:09:04.120 --> 00:09:05.550
it is, because
this up here is 30

00:09:05.550 --> 00:09:09.860
degrees-- the hypotenuse,
or the length, is 1, then

00:09:09.860 --> 00:09:12.690
this over here is
square root of 3 over 2.

00:09:12.690 --> 00:09:16.870
And then this distance right
over here is negative 1/2.

00:09:16.870 --> 00:09:23.340
So x2 is going to be equal
to-- cosine of 2 pi over 3

00:09:23.340 --> 00:09:26.237
is-- negative 1/2.

00:09:26.237 --> 00:09:27.070
Did I do that right?

00:09:27.070 --> 00:09:32.102
Yep, negative 1/2, plus i
times sine of 2 pi over 3.

00:09:32.102 --> 00:09:33.560
That's this height
over here, which

00:09:33.560 --> 00:09:37.330
is square root of 3 over 2, i.

00:09:37.330 --> 00:09:38.770
So that's x2.

00:09:38.770 --> 00:09:42.110
And we could do the
exact same thing with x3.

00:09:42.110 --> 00:09:45.965
x3 is going to be
equal to its x value.

00:09:45.965 --> 00:09:47.340
Or I should say
its real value is

00:09:47.340 --> 00:09:48.715
going to be the
exact same thing.

00:09:48.715 --> 00:09:51.550
It's going to be negative 1/2.

00:09:51.550 --> 00:09:56.690
And then, its imaginary
value, so this angle right

00:09:56.690 --> 00:10:00.750
over here-- this just from
the negative real axis down

00:10:00.750 --> 00:10:03.736
to the vector-- is going
to be negative 60 degrees.

00:10:03.736 --> 00:10:05.110
So this height
right over here is

00:10:05.110 --> 00:10:08.130
going to be negative
square root of 3 over 2.

00:10:08.130 --> 00:10:13.050
So it's negative 1/2 minus the
square root of 3 over 2, i.

00:10:13.050 --> 00:10:17.580
So using this technique,
we were able to find

00:10:17.580 --> 00:10:20.300
the three complex roots of 1.

00:10:20.300 --> 00:10:22.250
This is one of them.

00:10:22.250 --> 00:10:24.100
This is another one.

00:10:24.100 --> 00:10:27.015
And of course, 1 is
one of them as well.

00:10:27.015 --> 00:10:27.890
Where did we do that?

00:10:27.890 --> 00:10:29.379
1 is one of them as well.

00:10:29.379 --> 00:10:31.170
And you could use this
exact same technique

00:10:31.170 --> 00:10:32.700
if we were finding
the fourth roots.

00:10:32.700 --> 00:10:36.170
We would take the 2 pi
radians, or the 360 degrees,

00:10:36.170 --> 00:10:37.356
and divide it into 4.

00:10:37.356 --> 00:10:38.730
And so it would
actually be this.

00:10:38.730 --> 00:10:40.220
It would be i.

00:10:40.220 --> 00:10:41.225
It would be negative 1.

00:10:41.225 --> 00:10:42.350
And it would be negative i.

00:10:42.350 --> 00:10:44.890
And we know if you take i
to the fourth, you get 1.

00:10:44.890 --> 00:10:48.270
If you take negative i
to the fourth, you get 1.

00:10:48.270 --> 00:10:50.600
And if you take negative 1
to the fourth, you get 1.

00:10:50.600 --> 00:10:52.391
And if you take 1 to
the fourth, you get 1.

00:10:52.391 --> 00:10:54.610
And so you can find
the eighth roots of 1

00:10:54.610 --> 00:10:56.120
using this technique.

00:10:56.120 --> 00:10:59.290
Now, the other question that
might be popping in your brain

00:10:59.290 --> 00:11:03.686
is, why did I stop
at e to the 4 pi i?

00:11:03.686 --> 00:11:05.060
Why didn't I go
on and say, well,

00:11:05.060 --> 00:11:08.350
this is equal to e to the 6 pi
i and look for another root?

00:11:08.350 --> 00:11:11.910
And so if I did that, if I
said x to the third-- let's say

00:11:11.910 --> 00:11:13.900
I wanted to find a
fourth root here, maybe.

00:11:13.900 --> 00:11:18.430
x to the third is equal
to e to the 6 pi i.

00:11:18.430 --> 00:11:20.140
And I take both sides
of this equation

00:11:20.140 --> 00:11:22.570
to the one-third power.

00:11:22.570 --> 00:11:28.200
Taking this to the one third,
I would get e to the 2 pi i.

00:11:28.200 --> 00:11:30.470
Well, what's e to the
2 pi i? e to the 2 pi

00:11:30.470 --> 00:11:32.930
i would just get us back to 1.

00:11:32.930 --> 00:11:35.580
So when I added 2 pi again, it
just gets us back to this root

00:11:35.580 --> 00:11:36.810
again.

00:11:36.810 --> 00:11:39.400
If I took e to the 6 pi,
if I took e to the 8 pi,

00:11:39.400 --> 00:11:40.860
I would get this root again.

00:11:40.860 --> 00:11:46.800
So you're going to get
only three roots if you're

00:11:46.800 --> 00:11:49.050
finding the third
roots of something.

00:11:49.050 --> 00:11:51.678
Anything beyond that, it
just becomes redundant.


Basic complex analysis

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:02.300
What I want to do
in this video is

00:00:02.300 --> 00:00:07.350
make sure we're comfortable with
ways to represent and visualize

00:00:07.350 --> 00:00:11.650
complex numbers.

00:00:11.650 --> 00:00:13.860
So you're probably
familiar with the idea.

00:00:13.860 --> 00:00:17.240
A complex number,
let's call it z-- and z

00:00:17.240 --> 00:00:21.830
is the variable we do tend to
use for complex numbers-- let's

00:00:21.830 --> 00:00:25.870
say that z is
equal to a plus bi.

00:00:25.870 --> 00:00:31.220
We call it complex
because it has a real part

00:00:31.220 --> 00:00:32.820
and it has an imaginary part.

00:00:36.640 --> 00:00:38.580
And just so you're
used to the notation,

00:00:38.580 --> 00:00:41.420
sometimes you'll see
someone write the real part,

00:00:41.420 --> 00:00:43.230
give me the real part of z.

00:00:43.230 --> 00:00:45.680
This is a function, that
you input a complex number,

00:00:45.680 --> 00:00:48.360
and it will output the real
part, and in this case,

00:00:48.360 --> 00:00:50.920
the real part is equal to a.

00:00:50.920 --> 00:00:53.260
And you could have
another function

00:00:53.260 --> 00:00:56.870
called the imaginary part of z.

00:00:56.870 --> 00:00:58.810
You input some
complex number it'll

00:00:58.810 --> 00:01:03.510
output the imaginary
part, or it'll

00:01:03.510 --> 00:01:06.730
say how much are scaling
up i, and in this case,

00:01:06.730 --> 00:01:08.794
it would be b.

00:01:08.794 --> 00:01:10.460
This is a real number,
but this tells us

00:01:10.460 --> 00:01:14.970
how much the i is scaled
up in the complex number z

00:01:14.970 --> 00:01:16.270
right over there.

00:01:16.270 --> 00:01:18.795
Now, one way to visualize
complex numbers,

00:01:18.795 --> 00:01:21.170
and this is actually a very
helpful way of visualizing it

00:01:21.170 --> 00:01:23.750
when we start thinking about
the roots of numbers, especially

00:01:23.750 --> 00:01:27.500
the complex roots, is using
something called an Argand

00:01:27.500 --> 00:01:28.000
diagram.

00:01:33.570 --> 00:01:34.270
So this is this.

00:01:34.270 --> 00:01:36.186
And so it looks a lot
like the coordinate axes

00:01:36.186 --> 00:01:37.940
and it is a coordinate axes.

00:01:37.940 --> 00:01:40.670
But instead of having
an x and y-axis

00:01:40.670 --> 00:01:46.360
it has a real and
an imaginary axis.

00:01:46.360 --> 00:01:49.990
So in the example of
z being a plus bi,

00:01:49.990 --> 00:01:53.100
we would plot it really as
a position vector, where

00:01:53.100 --> 00:01:55.480
you have the real part
on the horizontal axis.

00:01:55.480 --> 00:01:58.830
So let's say this is a and
then the imaginary part

00:01:58.830 --> 00:02:02.320
along the vertical axis,
or the imaginary axis.

00:02:02.320 --> 00:02:04.400
So let's say that this is b.

00:02:04.400 --> 00:02:06.900
And so we would represent,
in an Argand diagram,

00:02:06.900 --> 00:02:12.100
the vector z as a position
vector that starts at 0

00:02:12.100 --> 00:02:15.300
and that has a tip at
the coordinate a comma b.

00:02:15.300 --> 00:02:22.090
So this right here is
our complex number.

00:02:22.090 --> 00:02:24.070
This right here is
a representation

00:02:24.070 --> 00:02:27.030
in our Argand diagram
of the complex number

00:02:27.030 --> 00:02:29.549
a plus bi, or of z.

00:02:29.549 --> 00:02:32.090
Now when you draw it this way,
when you draw it as a position

00:02:32.090 --> 00:02:35.640
vector, and if you're familiar
with polar coordinates,

00:02:35.640 --> 00:02:38.170
you're probably
thinking, hey, I don't

00:02:38.170 --> 00:02:40.160
have to represent this
complex number just

00:02:40.160 --> 00:02:43.570
as coordinates, just
as an a plus bi.

00:02:43.570 --> 00:02:49.910
Maybe I could represent
this as some angle here,

00:02:49.910 --> 00:02:53.380
let's call that angle phi,
and some the distance here,

00:02:53.380 --> 00:02:56.325
let's call that r, which
is kind of the magnitude

00:02:56.325 --> 00:02:56.950
of this vector.

00:02:56.950 --> 00:02:57.660
And you could.

00:02:57.660 --> 00:02:59.990
If you gave some angle
and some distance,

00:02:59.990 --> 00:03:04.380
that would also specify this
point in the complex plane.

00:03:04.380 --> 00:03:08.030
And this is actually called the
argument of the complex number

00:03:08.030 --> 00:03:11.960
and this right here is called
the magnitude, or sometimes

00:03:11.960 --> 00:03:14.819
the modulus, or the absolute
value of the complex number.

00:03:14.819 --> 00:03:16.360
So let's think about
it a little bit.

00:03:16.360 --> 00:03:17.985
Let's think about
how we would actually

00:03:17.985 --> 00:03:19.670
calculate these values.

00:03:19.670 --> 00:03:22.750
So r, which is the
modulus, or the magnitude.

00:03:22.750 --> 00:03:27.170
It's denoted by the magnitude
or the absolute value of z1.

00:03:27.170 --> 00:03:29.210
What's this going to be.

00:03:29.210 --> 00:03:30.730
Well, we have a
right triangle here.

00:03:33.360 --> 00:03:36.410
This side is b, length b.

00:03:36.410 --> 00:03:39.060
The base right
here has length a.

00:03:39.060 --> 00:03:42.680
So to calculate r, we can just
use the Pythagorean Theorem. r

00:03:42.680 --> 00:03:45.320
squared is going to be equal
to a squared plus b squared.

00:03:45.320 --> 00:03:52.340
Or r is going to be equal to the
square root of a squared plus b

00:03:52.340 --> 00:03:53.180
squared.

00:03:53.180 --> 00:03:59.000
If we want to figure
out the argument,

00:03:59.000 --> 00:04:01.445
this is going to
be equal to what?

00:04:01.445 --> 00:04:03.070
So let's think about
this a little bit.

00:04:03.070 --> 00:04:05.320
We have b and a.

00:04:05.320 --> 00:04:08.730
So what trig function deals
with the opposite side

00:04:08.730 --> 00:04:10.470
of an angle and
the adjacent side?

00:04:10.470 --> 00:04:13.710
So let me write all of, let me
write the famous sohcahtoa up

00:04:13.710 --> 00:04:16.529
here. "Soh-cah-toa."

00:04:16.529 --> 00:04:19.810
Tangent deals with
opposite over adjacent.

00:04:19.810 --> 00:04:23.310
So the tangent of
this angle, which

00:04:23.310 --> 00:04:26.070
we called the argument of the
complex number, the tangent

00:04:26.070 --> 00:04:29.240
of the argument is going to
be equal to the opposite side

00:04:29.240 --> 00:04:30.460
over the adjacent side.

00:04:30.460 --> 00:04:34.440
It is equal to b/a.

00:04:34.440 --> 00:04:36.700
And so if we wanted to
solve for this argument,

00:04:36.700 --> 00:04:38.300
we would say that
the argument is

00:04:38.300 --> 00:04:45.420
equal to the arctan, or the
inverse tangent, of b/a.

00:04:45.420 --> 00:04:47.290
Now, if we wanted
to represent, let's

00:04:47.290 --> 00:05:00.230
say that we were given the
modulus and the argument.

00:05:00.230 --> 00:05:01.570
Let's say we were given that.

00:05:01.570 --> 00:05:02.695
How do we go the other way?

00:05:02.695 --> 00:05:05.470
Right now if we have the a's and
the b's, the real complex part,

00:05:05.470 --> 00:05:07.880
I just showed you how
to get the magnitude

00:05:07.880 --> 00:05:09.659
and how to get the
angle, or the argument.

00:05:09.659 --> 00:05:10.700
But if you're given this.

00:05:10.700 --> 00:05:12.320
How do you go the other way?

00:05:12.320 --> 00:05:14.470
Well here, if you're
trying to figure out

00:05:14.470 --> 00:05:17.860
a, given r and theta--
so you're trying

00:05:17.860 --> 00:05:22.040
to figure out an adjacent side
given angle and the hypotenuse.

00:05:22.040 --> 00:05:25.680
So adjacent over hypotenuse
is equal to cosine.

00:05:25.680 --> 00:05:29.360
So you would have
cosine of the argument

00:05:29.360 --> 00:05:34.920
is equal to the adjacent
over the hypotenuse.

00:05:34.920 --> 00:05:37.420
It is equal to a/r.

00:05:37.420 --> 00:05:43.280
Multiply both sides by r,
you get r cosine of phi

00:05:43.280 --> 00:05:44.980
is equal to a.

00:05:44.980 --> 00:05:47.740
Do something very similar for b.

00:05:47.740 --> 00:05:50.390
If we use sine, opposite
over hypotenuse.

00:05:50.390 --> 00:05:56.160
Sine of the argument
is equal to b/r.

00:05:56.160 --> 00:05:58.330
It is equal to b
over the magnitude.

00:05:58.330 --> 00:06:03.920
Multiply both sides by
r, you get r sine of phi

00:06:03.920 --> 00:06:05.400
is equal to b.

00:06:05.400 --> 00:06:08.220
So how would we write
this complex number.

00:06:08.220 --> 00:06:13.130
So this complex
number z is going

00:06:13.130 --> 00:06:15.710
to be equal to it's
real part, which

00:06:15.710 --> 00:06:23.620
is r cosine of phi plus
the imaginary part times i.

00:06:23.620 --> 00:06:27.820
Plus r-- let me do that
same green-- plus r

00:06:27.820 --> 00:06:33.560
sine of phi times i.

00:06:33.560 --> 00:06:35.900
Now this might pop out
at you as something

00:06:35.900 --> 00:06:38.525
that's pretty interesting, if
you ever seen Euler's formula.

00:06:38.525 --> 00:06:40.510
Let's factor out
this r over here.

00:06:40.510 --> 00:06:44.790
So this is going to be equal
to-- factor out an r-- r times

00:06:44.790 --> 00:06:50.610
cosine of phi plus-- I'll
put the i out front-- i

00:06:50.610 --> 00:06:51.630
sine of phi.

00:06:56.030 --> 00:06:57.410
Now what is this?

00:06:57.410 --> 00:06:59.710
And if you've seen
the video, I do it

00:06:59.710 --> 00:07:02.340
in the Taylor series,
a series of videos

00:07:02.340 --> 00:07:03.590
in the calculus playlist.

00:07:03.590 --> 00:07:06.030
And it's really one of the
most profound results and all

00:07:06.030 --> 00:07:08.310
of mathematics, it
still gives me chills.

00:07:08.310 --> 00:07:10.200
This is Euler's formula.

00:07:10.200 --> 00:07:15.424
Or this, by Euler's
formula, is the same thing.

00:07:15.424 --> 00:07:17.340
And we show it by looking
at the Taylor series

00:07:17.340 --> 00:07:20.730
representations of e to the x.

00:07:20.730 --> 00:07:22.900
And the Taylor series
representations

00:07:22.900 --> 00:07:24.650
of cosine of x and sine of x.

00:07:24.650 --> 00:07:31.300
But this is, if we're dealing
with radians, e to the i phi.

00:07:31.300 --> 00:07:38.814
So z is going to be equal
to r times e to the i phi.

00:07:43.060 --> 00:07:46.620
So there's two ways to
write a complex number.

00:07:46.620 --> 00:07:48.240
You could write it
like this, where

00:07:48.240 --> 00:07:50.190
you have the real and
imaginary part, that's

00:07:50.190 --> 00:07:51.620
maybe what we're used to.

00:07:51.620 --> 00:07:54.580
Or we can write it in
exponential form, where

00:07:54.580 --> 00:07:57.420
you have the modulus,
or the magnitude,

00:07:57.420 --> 00:07:59.714
being multiplied by a
complex exponential.

00:07:59.714 --> 00:08:02.130
And we're going to see that
this going to be super useful,

00:08:02.130 --> 00:08:04.794
especially when we're
trying to find roots.

00:08:04.794 --> 00:08:06.710
Now just to make this
tangible, let's actually

00:08:06.710 --> 00:08:10.110
do this with an actual example.

00:08:10.110 --> 00:08:12.570
So let's say that I
had, I don't know,

00:08:12.570 --> 00:08:21.070
let's say that I had to z1
is equal to square root of,

00:08:21.070 --> 00:08:30.140
let's say it's square
root of 3/2 plus i.

00:08:30.140 --> 00:08:36.500
And so we want to figure
out its magnitude,

00:08:36.500 --> 00:08:39.809
and we want to figure
out its argument.

00:08:39.809 --> 00:08:40.650
So let's do that.

00:08:40.650 --> 00:08:43.940
So the magnitude of
z1 is going to be

00:08:43.940 --> 00:08:46.700
equal to the square
root of this squared.

00:08:46.700 --> 00:08:56.700
So this is going to be equal
to 3/4 plus 1 squared--

00:08:56.700 --> 00:09:00.130
or I should say plus 4/4.

00:09:00.130 --> 00:09:04.760
So this is going to be equal
to square root of 7/4, which

00:09:04.760 --> 00:09:06.530
is equal to the
square root of 7/2.

00:09:09.440 --> 00:09:11.540
And now let's figure
out its argument.

00:09:11.540 --> 00:09:19.490
So if I were to draw this
on an Argand diagram,

00:09:19.490 --> 00:09:21.045
it would look like this.

00:09:21.045 --> 00:09:22.670
It's going to be in
the first quadrant,

00:09:22.670 --> 00:09:25.150
so that's all I
have to worry about.

00:09:25.150 --> 00:09:27.600
So let me draw it.

00:09:27.600 --> 00:09:30.740
Let me draw it like this.

00:09:30.740 --> 00:09:33.689
And so we have a situation.

00:09:33.689 --> 00:09:35.230
So it's going to be
square root of 3,

00:09:35.230 --> 00:09:36.900
actually, let me change
this up a little bit,

00:09:36.900 --> 00:09:38.800
just so the numbers get
a little bit cleaner.

00:09:38.800 --> 00:09:39.670
Sorry about this.

00:09:39.670 --> 00:09:42.240
Let me make it a little
bit, slightly cleaner.

00:09:42.240 --> 00:09:45.720
So just so that we have a
slightly cleaner result,

00:09:45.720 --> 00:09:49.690
because we want to make our
first example a simple one.

00:09:49.690 --> 00:09:52.940
So let's make this square
root of 3/2 plus 1/2i..

00:09:55.880 --> 00:09:58.070
So let's figure
out the magnitude,

00:09:58.070 --> 00:10:02.170
the magnitude here is z1
is equal to the square root

00:10:02.170 --> 00:10:05.270
of, square root
of 3/2 squared, is

00:10:05.270 --> 00:10:09.940
equal to 3/4 plus 1/2
squared is equal to 1/4,.

00:10:09.940 --> 00:10:11.650
This makes things a lot nicer.

00:10:11.650 --> 00:10:14.959
This is equal to the square
root of 1, which is 1.

00:10:14.959 --> 00:10:16.750
And now let's think
about it, let's draw it

00:10:16.750 --> 00:10:21.780
on an Argand diagram to
visualize the argument.

00:10:21.780 --> 00:10:24.115
So this is my imaginary axis.

00:10:29.210 --> 00:10:31.205
This is my real axis.

00:10:34.080 --> 00:10:38.000
And so this complex number
is square root of 3/2.

00:10:38.000 --> 00:10:39.730
The square root
of 3 is like 1.7.

00:10:39.730 --> 00:10:45.250
So if we have like 1, it'll
be like right over here,

00:10:45.250 --> 00:10:46.900
someplace right over here.

00:10:46.900 --> 00:10:50.310
This is square root
of 3/2, the real part.

00:10:50.310 --> 00:10:51.750
The imaginary part is 1/2.

00:10:51.750 --> 00:10:54.760
So if this is 1, this is
1/2, the imaginary part

00:10:54.760 --> 00:10:56.960
is right over here, 1/2.

00:10:56.960 --> 00:10:59.900
And we actually also know
its length, its length,

00:10:59.900 --> 00:11:01.880
or its magnitude is 1.

00:11:01.880 --> 00:11:05.850
So how do we figure
out phi over here?

00:11:05.850 --> 00:11:08.814
We know that this side over
here is square root of 3

00:11:08.814 --> 00:11:11.230
over-- oh let me be careful--
we know that side over there

00:11:11.230 --> 00:11:12.390
is 1/2.

00:11:12.390 --> 00:11:13.510
That's the imaginary part.

00:11:13.510 --> 00:11:16.720
And we know the base is
the square root of 3/2.

00:11:16.720 --> 00:11:19.880
So a bunch of ways
we can do this.

00:11:19.880 --> 00:11:23.352
One, you could just
do the tangent,

00:11:23.352 --> 00:11:25.560
because that involves the
opposite over the adjacent.

00:11:25.560 --> 00:11:28.270
You could say that
the tangent of phi

00:11:28.270 --> 00:11:31.620
is equal to the
opposite, is equal to 1/2

00:11:31.620 --> 00:11:35.430
over the square root of 3/2.

00:11:35.430 --> 00:11:37.620
And then you can take the
inverse tan of both sides.

00:11:37.620 --> 00:11:40.540
So this would be the
same thing as phi

00:11:40.540 --> 00:11:43.040
being equal to the
inverse tangent,

00:11:43.040 --> 00:11:45.136
or the arctangent
of-- if you multiply

00:11:45.136 --> 00:11:46.760
the numerator and
the denominator by 2,

00:11:46.760 --> 00:11:49.790
this is 1 over the
square root of 3.

00:11:49.790 --> 00:11:51.050
You could do it like that.

00:11:51.050 --> 00:11:56.190
You can also say that phi is
equal to the inverse sine of,

00:11:56.190 --> 00:11:58.080
so the sine of
phi is going to be

00:11:58.080 --> 00:12:00.640
equal to the opposite
over the hypotenuse.

00:12:00.640 --> 00:12:03.210
So sine of phi is
equal to 1/2 over 1,

00:12:03.210 --> 00:12:06.610
or phi is equal to
the arcsine of 1/2.

00:12:06.610 --> 00:12:08.590
And you could put that
into your calculator.

00:12:08.590 --> 00:12:11.750
Or you could recognize this
is a 30-60-90 triangle.

00:12:11.750 --> 00:12:14.690
This base right here, square
root of 3/2, this is 1/2,

00:12:14.690 --> 00:12:15.830
this is 1.

00:12:15.830 --> 00:12:18.099
So this angle right here
is going to be 30 degrees.

00:12:18.099 --> 00:12:19.640
And that's just from
pattern matching

00:12:19.640 --> 00:12:21.720
from a 30-60-90 triangle.

00:12:21.720 --> 00:12:24.290
You could look at these and
also get something similar.

00:12:24.290 --> 00:12:25.990
Now I want to put
this in radian form,

00:12:25.990 --> 00:12:27.781
because whenever I use
the exponential form

00:12:27.781 --> 00:12:30.000
you want it to be in radians.

00:12:30.000 --> 00:12:34.730
So phi is equal to
30 degrees, which

00:12:34.730 --> 00:12:39.620
is the same thing as pi over 6.

00:12:39.620 --> 00:12:42.640
So if I wanted to represent
z1 in exponential form,

00:12:42.640 --> 00:12:45.351
it would be the exact same
thing as r, or its magnitude,

00:12:45.351 --> 00:12:47.850
which is 1-- I'll put the 1 out
there even though you really

00:12:47.850 --> 00:12:55.580
don't have to-- 1 times
e to the pi over 6i.

00:13:01.550 --> 00:13:03.400
And we're done.


IIT JEE Hairy Trig and Algebra (Part 3)

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:03.160
So I'm looking forward to the
end of this problem as much

00:00:03.160 --> 00:00:04.390
as I'm guessing you are.

00:00:04.390 --> 00:00:06.000
And I think we're
in the home stretch.

00:00:06.000 --> 00:00:10.970
We just have to simplify this to
get the possible values for x.

00:00:10.970 --> 00:00:13.470
So let's simplify what we
have here in the radical.

00:00:13.470 --> 00:00:14.470
So let's see.

00:00:14.470 --> 00:00:18.809
Square root of 3 minus 2.

00:00:18.809 --> 00:00:20.600
And we're going to
square this whole thing.

00:00:20.600 --> 00:00:22.683
That's going to be square
root of 3 squared, which

00:00:22.683 --> 00:00:25.490
is 3, minus 2 times
their product,

00:00:25.490 --> 00:00:28.870
so that's minus 4
square roots of 3.

00:00:28.870 --> 00:00:33.390
and then plus 2
squared, so plus 4.

00:00:33.390 --> 00:00:36.940
And then we have minus
4 times-- square root

00:00:36.940 --> 00:00:39.410
of 3 times square
root of 3 is 3.

00:00:39.410 --> 00:00:43.050
Square root of 3 times 1 is
plus the square root of 3.

00:00:43.050 --> 00:00:44.800
And then negative 2
times square root of 3

00:00:44.800 --> 00:00:47.280
is negative 2 square roots of 3.

00:00:47.280 --> 00:00:50.870
And then negative 2
times 1 is negative 2.

00:00:50.870 --> 00:00:52.900
So this is going to
be equal 3 plus 4 is

00:00:52.900 --> 00:00:57.410
7 minus 4 square
roots of 3 minus--

00:00:57.410 --> 00:01:00.920
so let's see, this 3 and
this negative 2 over here,

00:01:00.920 --> 00:01:04.120
that becomes a 1.

00:01:04.120 --> 00:01:06.530
And then the square root of
3 minus 2 square roots of 3,

00:01:06.530 --> 00:01:08.950
that just becomes a
minus square root of 3.

00:01:08.950 --> 00:01:13.872
So that just becomes
negative square root of 3.

00:01:13.872 --> 00:01:15.080
My brain is starting to hurt.

00:01:15.080 --> 00:01:16.620
So you have a
negative 4 times 1.

00:01:16.620 --> 00:01:18.239
So it's minus 4.

00:01:18.239 --> 00:01:20.530
And then a negative 4 times
negative square roots of 3.

00:01:20.530 --> 00:01:23.120
So plus 4 square roots of 3.

00:01:23.120 --> 00:01:27.140
And thank God something has
finally simplified here.

00:01:27.140 --> 00:01:30.980
And so that cancels
out with that.

00:01:30.980 --> 00:01:34.302
So you have 7 minus
4 is equal to 3.

00:01:34.302 --> 00:01:36.010
So the whole thing
under the radical sign

00:01:36.010 --> 00:01:38.320
is the square root of 3.

00:01:38.320 --> 00:01:42.270
So then this becomes
our potential values

00:01:42.270 --> 00:01:47.190
for x. x is equal to 2
minus the square root of 3

00:01:47.190 --> 00:01:52.100
plus or minus the square root
of this thing, which simplified

00:01:52.100 --> 00:01:56.010
to the square root
of 3, all of that

00:01:56.010 --> 00:02:00.354
over 2 square
roots of 3 minus 4.

00:02:00.354 --> 00:02:02.270
So let's try to see what
our answers could be.

00:02:02.270 --> 00:02:06.180
If we add a square root
of 3-- so one answer is

00:02:06.180 --> 00:02:10.020
if we add the square root, if
we go with the plus option,

00:02:10.020 --> 00:02:12.190
this would be 2 minus
square roots of 3

00:02:12.190 --> 00:02:13.800
plus square root of 3.

00:02:13.800 --> 00:02:14.990
So it would be 2.

00:02:14.990 --> 00:02:16.800
The square roots of 3
will just cancel out.

00:02:16.800 --> 00:02:20.830
2 over 2 root 3 minus 4.

00:02:20.830 --> 00:02:23.180
We can divide the numerator
and the denominator by 2.

00:02:23.180 --> 00:02:28.220
And we get 1 over the
square root of 3 minus 2.

00:02:28.220 --> 00:02:31.350
And then to rationalize
this, we want

00:02:31.350 --> 00:02:35.290
to multiply it times the
square root of 3 plus 2

00:02:35.290 --> 00:02:37.570
over the square
root of 3 plus 2.

00:02:37.570 --> 00:02:39.850
We do that so that when
we take the product here,

00:02:39.850 --> 00:02:43.070
it takes the form b
squared minus a squared.

00:02:43.070 --> 00:02:47.720
So this is equal to the square
root of 3 plus 2, all of that

00:02:47.720 --> 00:02:59.690
over-- square root of 3
squared is now 3-- 3 minus 4.

00:02:59.690 --> 00:03:02.000
So this is going to be
equal to-- this part here

00:03:02.000 --> 00:03:03.340
is a negative 1.

00:03:03.340 --> 00:03:05.210
And so we could
view it as negative,

00:03:05.210 --> 00:03:07.500
and I'll just write it as 2
plus the square root of 3.

00:03:10.330 --> 00:03:12.830
So that's one
possible value of x.

00:03:12.830 --> 00:03:15.500
And the other
possible value of x,

00:03:15.500 --> 00:03:17.479
if we subtract the
square root of 3.

00:03:17.479 --> 00:03:19.020
So then on the
numerator, we're going

00:03:19.020 --> 00:03:21.054
to have 2 minus--
square root of 3

00:03:21.054 --> 00:03:22.470
minus the square
root of 3, that's

00:03:22.470 --> 00:03:26.260
2 minus 2 square roots of 3
over-- we could write this

00:03:26.260 --> 00:03:30.982
as negative 4 plus
2 square roots of 3.

00:03:30.982 --> 00:03:32.440
The first thing we
might want to do

00:03:32.440 --> 00:03:34.689
is just divide the numerator
and the denominator by 3.

00:03:34.689 --> 00:03:38.660
I swapped these
down here to divide

00:03:38.660 --> 00:03:40.490
the numerator and
the denominator by 2.

00:03:40.490 --> 00:03:44.050
So this is equal to 1
minus the square root of 3

00:03:44.050 --> 00:03:48.750
over negative 2 plus
the square root of 3.

00:03:48.750 --> 00:03:52.710
And now we can rationalize this
by multiplying it by negative 2

00:03:52.710 --> 00:03:55.880
minus the square root
of 3 over negative 2

00:03:55.880 --> 00:03:58.260
minus the square root of 3.

00:03:58.260 --> 00:04:00.175
And what do we get here?

00:04:05.942 --> 00:04:08.150
So this is going to be equal
to-- in the denominator,

00:04:08.150 --> 00:04:10.108
we're going to have
negative 2 times negative 2

00:04:10.108 --> 00:04:13.220
is 4, 4 minus 3.

00:04:13.220 --> 00:04:14.220
So we just get a 1.

00:04:14.220 --> 00:04:16.230
So this is equal to
1 in the denominator.

00:04:16.230 --> 00:04:21.001
In the numerator, so let me
just multiply 1 times negative 2

00:04:21.001 --> 00:04:21.959
minus square root of 3.

00:04:21.959 --> 00:04:25.170
So it's negative 2 minus
the square root of 3.

00:04:25.170 --> 00:04:28.430
And then the negative square
root of 3 times negative 2

00:04:28.430 --> 00:04:31.510
is plus 2 square roots of 3.

00:04:31.510 --> 00:04:33.220
And then negative
square roots of 3 times

00:04:33.220 --> 00:04:36.900
negative square
roots of 3 is plus 3.

00:04:36.900 --> 00:04:39.920
So you have 3 minus 2 is 1.

00:04:39.920 --> 00:04:43.030
And then 2 roots 3 minus
the square root of 3

00:04:43.030 --> 00:04:46.050
is plus the square root of 3.

00:04:46.050 --> 00:04:48.970
So this is kind of exciting.

00:04:48.970 --> 00:04:53.390
This was an unbelievably
hairy problem.

00:04:53.390 --> 00:04:56.500
And if any of y'all know a
faster way to do this problem,

00:04:56.500 --> 00:04:57.370
let me know.

00:04:57.370 --> 00:04:58.650
I've thought about it a lot.

00:04:58.650 --> 00:05:02.060
It doesn't seem like there is a
faster way to do this problem.

00:05:02.060 --> 00:05:05.950
But my possible
x-values are negative 2

00:05:05.950 --> 00:05:10.760
plus the square root of 3 or
1 plus the square root of 3.

00:05:10.760 --> 00:05:13.070
Now, we have to be careful.

00:05:13.070 --> 00:05:16.250
If it wasn't enough that we got
this far, we did so much work,

00:05:16.250 --> 00:05:19.890
we finally got these answers,
we have to be very careful,

00:05:19.890 --> 00:05:22.350
because both of
those are options.

00:05:22.350 --> 00:05:26.027
This is one of the options, and
then this is the other option.

00:05:26.027 --> 00:05:28.610
And once again, these are one
of those multiple correct choice

00:05:28.610 --> 00:05:31.470
answers, so you can actually
select more than one answer

00:05:31.470 --> 00:05:31.970
here.

00:05:31.970 --> 00:05:34.790
Or you maybe need to select
more than one answer.

00:05:34.790 --> 00:05:37.480
So you might, after doing all
of that work, say oh, finally.

00:05:37.480 --> 00:05:39.640
OK, bam and bam.

00:05:39.640 --> 00:05:41.530
But be very careful.

00:05:41.530 --> 00:05:46.610
Remember the context
of the problem.

00:05:46.610 --> 00:05:49.940
These sides had to be positive.

00:05:49.940 --> 00:05:52.140
And if you put this
number in here,

00:05:52.140 --> 00:05:54.390
all the sides will
not be positive.

00:05:54.390 --> 00:05:58.580
In particular, if x was
2 plus the square root

00:05:58.580 --> 00:06:01.390
negative of this value right
here, this value over here

00:06:01.390 --> 00:06:03.839
is going to be, I don't know,
3 point something something.

00:06:03.839 --> 00:06:05.380
And if you took the
negative of that,

00:06:05.380 --> 00:06:07.430
and you put it right
here for C, you're

00:06:07.430 --> 00:06:09.900
going to get a negative
distance for C.

00:06:09.900 --> 00:06:11.110
And C cannot be negative.

00:06:11.110 --> 00:06:13.550
It's the length of a
side of a triangle.

00:06:13.550 --> 00:06:15.640
So A is not a valid choice.

00:06:15.640 --> 00:06:19.110
So these guys are really
trying to trip people up.

00:06:19.110 --> 00:06:26.924
The correct answer is
only B. That was tiring.