Detectable civilizations in our galaxy 3

WEBVTT
Kind: captions
Language: en

00:00:00.504 --> 00:00:01.920
What I want to do
in this video is

00:00:01.920 --> 00:00:04.520
reconcile the more
traditional Drake equation

00:00:04.520 --> 00:00:07.710
with the stuff that we derived,
or that we came up with.

00:00:07.710 --> 00:00:10.740
We kind of thought through it
in the last several videos.

00:00:10.740 --> 00:00:13.630
So the more known
Drake equation is this.

00:00:13.630 --> 00:00:17.230
The number of detectable
civilizations in the galaxy

00:00:17.230 --> 00:00:20.700
is equal to-- and
they'll have this.

00:00:20.700 --> 00:00:23.580
And this is not the number
of stars in the galaxy.

00:00:23.580 --> 00:00:26.700
This is the average
rate of star formation

00:00:26.700 --> 00:00:29.060
per year in the galaxy.

00:00:29.060 --> 00:00:32.140
So star-- so let
me write this down.

00:00:32.140 --> 00:00:38.410
Average rate of star
formation, which

00:00:38.410 --> 00:00:41.530
seems kind of
unintuitive, and frankly,

00:00:41.530 --> 00:00:43.915
it is, but hopefully we'll
reconcile to show you

00:00:43.915 --> 00:00:45.540
that this, and what
we're going to show

00:00:45.540 --> 00:00:46.998
with the traditional
Drake equation

00:00:46.998 --> 00:00:48.386
are actually the same thing.

00:00:48.386 --> 00:00:50.260
So that's the average
rate of star formation.

00:00:50.260 --> 00:00:51.385
So I don't know what it is.

00:00:51.385 --> 00:00:53.940
Maybe it's 10 stars a year,
or something on that order.

00:00:53.940 --> 00:00:56.650
And then the rest of it
looks pretty similar.

00:00:56.650 --> 00:01:00.450
So times the fraction of
stars that have planets.

00:01:00.450 --> 00:01:04.500
So this product would
give you the average stars

00:01:04.500 --> 00:01:06.820
with planet formations per year.

00:01:06.820 --> 00:01:10.180
You multiply that times
the average number

00:01:10.180 --> 00:01:14.970
of planets capable
of sustaining life

00:01:14.970 --> 00:01:18.370
for a star that has planets,
for a solar system that

00:01:18.370 --> 00:01:19.579
has planets.

00:01:19.579 --> 00:01:21.120
So essentially, if
you multiply this,

00:01:21.120 --> 00:01:24.020
this is the average
new planets per year

00:01:24.020 --> 00:01:27.810
in our galaxy capable
of sustaining life.

00:01:27.810 --> 00:01:31.390
You multiply that
times this, which

00:01:31.390 --> 00:01:33.534
is the same exact fraction.

00:01:33.534 --> 00:01:34.950
The fraction of
those planets that

00:01:34.950 --> 00:01:38.620
are capable of sustaining
life, that actually, this

00:01:38.620 --> 00:01:40.630
is capable of sustaining
life, now we're

00:01:40.630 --> 00:01:44.030
saying that the fraction that
actually do develop life.

00:01:44.030 --> 00:01:47.000
And then of the life, we
care about the fraction

00:01:47.000 --> 00:01:49.880
that actually does
become intelligent.

00:01:49.880 --> 00:01:51.450
And of the fraction
that actually

00:01:51.450 --> 00:01:56.470
does become intelligent,
we care about the fraction

00:01:56.470 --> 00:01:58.940
that eventually
becomes detectable.

00:01:58.940 --> 00:02:00.710
That can actually communicate.

00:02:00.710 --> 00:02:02.860
And then in the
traditional Drake equation,

00:02:02.860 --> 00:02:06.120
we multiply that times
this L over here.

00:02:06.120 --> 00:02:09.440
Times the detectable
life of the civilization.

00:02:09.440 --> 00:02:11.880
So how long is that
civilization detectable?

00:02:11.880 --> 00:02:14.420
Are they releasing radio
waves, or something like it

00:02:14.420 --> 00:02:17.181
that a civilization
like ours can detect?

00:02:17.181 --> 00:02:18.930
Maybe there are other
ways to communicate,

00:02:18.930 --> 00:02:20.060
and we're just not
advanced enough.

00:02:20.060 --> 00:02:21.650
Maybe in a few
years, we'll discover

00:02:21.650 --> 00:02:23.524
in a few decades, or a
few hundreds of years,

00:02:23.524 --> 00:02:26.210
we'll discover that all the
other advanced civilizations

00:02:26.210 --> 00:02:27.990
are using a much more
sophisticated way

00:02:27.990 --> 00:02:30.590
of communicating that doesn't
involve electromagnetic waves.

00:02:30.590 --> 00:02:31.560
Who knows?

00:02:31.560 --> 00:02:33.310
But this is what we're
thinking right now.

00:02:33.310 --> 00:02:34.768
But anyway, the
whole point here is

00:02:34.768 --> 00:02:37.650
to reconcile this thing which
is less intuitive-- for me

00:02:37.650 --> 00:02:39.140
at least-- than with this thing.

00:02:39.140 --> 00:02:41.098
Because I started up here
with the total number

00:02:41.098 --> 00:02:42.339
of stars in the galaxy.

00:02:42.339 --> 00:02:43.880
The traditional
Drake equation starts

00:02:43.880 --> 00:02:47.036
with the average rate
of star formation.

00:02:47.036 --> 00:02:49.660
So it's like, well, how does the
average rate of star formation

00:02:49.660 --> 00:02:52.350
gel with the total
number of stars,

00:02:52.350 --> 00:02:54.650
or civilizations that
are now detectable?

00:02:54.650 --> 00:02:57.270
What I want to do is diagram
that out a little bit,

00:02:57.270 --> 00:02:59.140
and I'm going to make
a few assumptions.

00:02:59.140 --> 00:03:01.850
I'm going to assume that this
is kind of constant, that we're

00:03:01.850 --> 00:03:03.350
in a steady state.

00:03:03.350 --> 00:03:06.329
So this is constant, and
we are in a steady state.

00:03:06.329 --> 00:03:07.870
The reality is that
what would matter

00:03:07.870 --> 00:03:11.910
is the rate of star
formation maybe 4, 5,

00:03:11.910 --> 00:03:13.370
6 billion years
ago, I don't know

00:03:13.370 --> 00:03:16.270
how long it has to be
ago, so that now it

00:03:16.270 --> 00:03:18.850
starts to become realistic
for real intelligence

00:03:18.850 --> 00:03:21.010
and real detectable
intelligence to exist.

00:03:21.010 --> 00:03:22.830
But let's just assume
that this number

00:03:22.830 --> 00:03:25.052
is constant for most of
the life of the galaxy.

00:03:25.052 --> 00:03:27.510
Obviously we're making all
sorts of crazy assumptions here,

00:03:27.510 --> 00:03:29.530
so why not make another one?

00:03:29.530 --> 00:03:31.850
But what I want you
to show is that this

00:03:31.850 --> 00:03:36.800
is equivalent to the number
of stars in the galaxy divided

00:03:36.800 --> 00:03:43.260
by the average life of a
star, or the average life

00:03:43.260 --> 00:03:44.200
of a solar system.

00:03:44.200 --> 00:03:47.160
And if n divided
by this t sub s,

00:03:47.160 --> 00:03:49.250
if that's the same
thing as our star, then

00:03:49.250 --> 00:03:51.640
essentially we have
the same formulas.

00:03:51.640 --> 00:03:55.160
And to see that they're
the same, imagine this.

00:03:55.160 --> 00:03:57.310
Imagine this.

00:03:57.310 --> 00:04:02.120
That this year, so this is
this year, so this is-- well,

00:04:02.120 --> 00:04:04.000
let me say this year.

00:04:04.000 --> 00:04:05.540
This year.

00:04:05.540 --> 00:04:07.670
Let's say that we have our star.

00:04:07.670 --> 00:04:09.700
Let's say that
this number is 10.

00:04:09.700 --> 00:04:12.880
We have 10 new
stars in the galaxy.

00:04:12.880 --> 00:04:16.149
So this is-- I'll
just say it's 10.

00:04:16.149 --> 00:04:19.390
So our star is equal to 10.

00:04:19.390 --> 00:04:20.959
So this height over here is 10.

00:04:20.959 --> 00:04:22.220
That's what I'm depicting.

00:04:22.220 --> 00:04:23.595
So if I were to
slide it, I could

00:04:23.595 --> 00:04:25.720
show that this is 10
units high, or whatever.

00:04:25.720 --> 00:04:29.660
And then last year, there was
also 10, so on and so forth.

00:04:29.660 --> 00:04:33.740
Now, let's go to
whatever-- let's

00:04:33.740 --> 00:04:35.860
say that this number, this
number right over here

00:04:35.860 --> 00:04:36.830
is 10 billion years.

00:04:36.830 --> 00:04:39.220
The average star life
is 10 billion years.

00:04:39.220 --> 00:04:44.430
So let's go back 10 billion
years into the past.

00:04:44.430 --> 00:04:50.720
So the average life of a star
is equal to 10 billion years.

00:04:50.720 --> 00:04:53.210
And we're assuming
that this is constant.

00:04:53.210 --> 00:04:56.360
So 10 billion years
ago this year,

00:04:56.360 --> 00:05:01.990
there were also 10
new stars came about.

00:05:01.990 --> 00:05:07.310
And every year in between
you had 10 stars come about.

00:05:07.310 --> 00:05:12.550
Now, how many total stars
would there be in our galaxy?

00:05:12.550 --> 00:05:15.560
Well, any star that came about--
so we could go beyond that.

00:05:15.560 --> 00:05:19.190
We could go to stars that were
born more than 10 billion years

00:05:19.190 --> 00:05:21.740
ago, more than this
t sub s years ago.

00:05:21.740 --> 00:05:25.480
So you could have a
star that was born 10

00:05:25.480 --> 00:05:27.590
billion and 1 years
ago, on average.

00:05:27.590 --> 00:05:29.220
We're talking about
on averages here.

00:05:29.220 --> 00:05:31.770
On average, that star
will not exist anymore,

00:05:31.770 --> 00:05:34.580
so that star is
not in existence.

00:05:34.580 --> 00:05:37.520
The stars that are in existence,
once again, on average,

00:05:37.520 --> 00:05:40.990
are the ones that were born 10
billion years ago, all the way

00:05:40.990 --> 00:05:43.090
to the ones that
were born this year.

00:05:43.090 --> 00:05:46.200
So you have 10 billion
years of star birth,

00:05:46.200 --> 00:05:47.760
the ones that are still around.

00:05:47.760 --> 00:05:50.040
Each year there's
10 of those years,

00:05:50.040 --> 00:05:54.430
so the total number
of stars should

00:05:54.430 --> 00:05:57.240
be equal to the
number of stars that

00:05:57.240 --> 00:06:00.300
are born each year-- assuming
that that is constant--

00:06:00.300 --> 00:06:05.530
times the average
lifespan of the stars.

00:06:05.530 --> 00:06:10.160
Times the average
lifespan of the stars.

00:06:10.160 --> 00:06:15.910
And once again, this works
because the stars that

00:06:15.910 --> 00:06:19.570
were born before this
lifespan don't exist anymore.

00:06:19.570 --> 00:06:21.020
They've died out, on average.

00:06:21.020 --> 00:06:23.590
We care about kind of
this area right over here.

00:06:23.590 --> 00:06:26.237
10 stars per year
times 10 billion years.

00:06:26.237 --> 00:06:28.070
And now if you manipulate
this a little bit,

00:06:28.070 --> 00:06:29.945
you'll see that we'll
get the result we want.

00:06:29.945 --> 00:06:30.880
Let's solve for r.

00:06:30.880 --> 00:06:33.980
So we could just divide
both sides by this t.

00:06:33.980 --> 00:06:38.670
So you get n star, so the number
of stars in our galaxy now,

00:06:38.670 --> 00:06:40.320
making a bunch of
assumptions, divided

00:06:40.320 --> 00:06:43.360
by the average life
of the stars is

00:06:43.360 --> 00:06:49.930
equal to the average number
of new stars per year.

00:06:49.930 --> 00:06:53.240
Is equal to the average
number of new stars per year,

00:06:53.240 --> 00:06:54.250
and we get our result.

00:06:54.250 --> 00:07:00.590
If you replace this with the
total number of stars divided

00:07:00.590 --> 00:07:05.900
by t, you get the exact same
result that we had before.

00:07:05.900 --> 00:07:07.260
You just change the order a bit.

00:07:07.260 --> 00:07:10.900
We can take this divided
by t, put it under this n,

00:07:10.900 --> 00:07:13.880
take it out up here, and then
you get the exact same thing.

00:07:13.880 --> 00:07:17.580
So hopefully that reconciles
it a little bit for you.


Detectable civilizations in our galaxy 2

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:02.680
Some of you might
be wondering, why

00:00:02.680 --> 00:00:06.210
are we even worried about
this Drake equation,

00:00:06.210 --> 00:00:09.190
or why are we even tempted
to go through this thought

00:00:09.190 --> 00:00:11.610
experiment of the number
of detectable civilizations

00:00:11.610 --> 00:00:13.996
in the galaxy when we
don't have a clue of some

00:00:13.996 --> 00:00:14.870
of these assumptions?

00:00:14.870 --> 00:00:18.080
We don't know what fraction of
planets capable of sustaining

00:00:18.080 --> 00:00:20.050
life actually do generate life.

00:00:20.050 --> 00:00:23.142
We don't know of all of
the planets that have life,

00:00:23.142 --> 00:00:25.725
what fraction of those planets
go on to have intelligent life,

00:00:25.725 --> 00:00:27.870
and what fractions of
those civilizations

00:00:27.870 --> 00:00:31.460
go on to using
electromagnetic radiation

00:00:31.460 --> 00:00:32.924
as a form of communication.

00:00:32.924 --> 00:00:34.090
We don't know these answers.

00:00:34.090 --> 00:00:35.590
In fact, we probably
won't know some

00:00:35.590 --> 00:00:37.620
of these answers for some time.

00:00:37.620 --> 00:00:40.430
So what's the point of
going through this exercise?

00:00:40.430 --> 00:00:42.870
And that is a valid
point of view.

00:00:42.870 --> 00:00:44.870
The Drake equation, or
even this little equation

00:00:44.870 --> 00:00:47.230
that we've set up here,
it's not an equation

00:00:47.230 --> 00:00:49.770
in the traditional sense, where
we can immediately apply it

00:00:49.770 --> 00:00:52.590
to some engineering problem,
or some physical problem,

00:00:52.590 --> 00:00:54.010
or anything like that.

00:00:54.010 --> 00:00:56.800
I view it more as a bit
of a thought experiment.

00:00:56.800 --> 00:00:58.270
And what's interesting
about it is

00:00:58.270 --> 00:01:01.332
it kind of can structure our
thought around the problem,

00:01:01.332 --> 00:01:03.290
and I think that's where
it has the most value.

00:01:03.290 --> 00:01:08.220
We'll probably not get a solid
number on this any time soon,

00:01:08.220 --> 00:01:11.240
but it does lead us to thinking
about these interesting

00:01:11.240 --> 00:01:14.050
problems of what
does it mean, or what

00:01:14.050 --> 00:01:18.492
do we think has to happen for
a planet to start getting life,

00:01:18.492 --> 00:01:20.200
even if it has all
the right ingredients?

00:01:20.200 --> 00:01:22.990
And then what does it mean
for things to eventually get

00:01:22.990 --> 00:01:25.360
to the point that you
have intelligent life?

00:01:25.360 --> 00:01:29.641
And in all fairness to this is
that probably 200 years ago,

00:01:29.641 --> 00:01:31.140
there would have
been no way to even

00:01:31.140 --> 00:01:33.850
have a decent estimate of the
number of stars in the galaxy.

00:01:33.850 --> 00:01:36.940
Now we're starting to
do an OK job on that.

00:01:36.940 --> 00:01:38.620
20 or 30 years
ago, it would have

00:01:38.620 --> 00:01:41.510
been viewed impossible to say
the fraction of stars that

00:01:41.510 --> 00:01:44.620
have planets, but now
we're finding exoplanets.

00:01:44.620 --> 00:01:46.409
We're seeing stars wobble.

00:01:46.409 --> 00:01:48.450
We're getting more and
more accurate instruments.

00:01:48.450 --> 00:01:51.310
So we can start to think
about planets that are closer

00:01:51.310 --> 00:01:54.310
to the size of Earth's.

00:01:54.310 --> 00:01:55.790
We're making headway there.

00:01:55.790 --> 00:01:57.800
There's other indirect
methods to think about,

00:01:57.800 --> 00:01:59.370
well, some of these
exoplanets look

00:01:59.370 --> 00:02:00.680
like they're in the
right zone, and they

00:02:00.680 --> 00:02:02.930
look like they have the right
chemical signature based

00:02:02.930 --> 00:02:04.870
on other information
that we're getting,

00:02:04.870 --> 00:02:07.560
and that maybe they are
capable of sustaining life.

00:02:07.560 --> 00:02:10.350
So as time goes on, and
as technology improves,

00:02:10.350 --> 00:02:13.022
we might be able to get better
and better and better at this.

00:02:13.022 --> 00:02:15.480
But with that said, it's not
going to happen any time soon,

00:02:15.480 --> 00:02:17.370
and the real value
of all of this

00:02:17.370 --> 00:02:19.380
is really to
structure our thoughts

00:02:19.380 --> 00:02:22.880
about really, a super,
super interesting topic.

00:02:22.880 --> 00:02:24.740
Now, the other thing
I want to talk about

00:02:24.740 --> 00:02:26.650
is a slight
clarification of what

00:02:26.650 --> 00:02:28.310
I talked about in
the last video.

00:02:28.310 --> 00:02:30.000
In the last video
for this l, I said

00:02:30.000 --> 00:02:32.080
it's the civilization's
lifespan,

00:02:32.080 --> 00:02:36.120
but what's actually
relevant is the lifespan

00:02:36.120 --> 00:02:38.990
of the civilization
while it is detectable.

00:02:38.990 --> 00:02:41.160
So, detectable.

00:02:41.160 --> 00:02:44.590
So it doesn't matter if the
civilization is around 100,000

00:02:44.590 --> 00:02:47.310
years, but it's not
releasing any type of thing

00:02:47.310 --> 00:02:48.190
that we can detect.

00:02:48.190 --> 00:02:49.480
That's not what we care about.

00:02:49.480 --> 00:02:52.630
We care about that the 5,000
years, or the 10,000 years,

00:02:52.630 --> 00:02:55.010
or the 100,000 years
when they are actually

00:02:55.010 --> 00:02:57.850
using some type
of communications,

00:02:57.850 --> 00:03:00.800
or some type of
electromagnetic radiation

00:03:00.800 --> 00:03:05.120
that we can eventually detect
once those things reach us.

00:03:05.120 --> 00:03:06.830
Now, the other thing
I want to make clear

00:03:06.830 --> 00:03:08.350
is we're talking
about the number

00:03:08.350 --> 00:03:12.080
of detectable civilizations
in the galaxy right now.

00:03:12.080 --> 00:03:15.620
And I'll write now
in quotation marks,

00:03:15.620 --> 00:03:19.380
because we're not talking about
a civilization that is maybe

00:03:19.380 --> 00:03:23.580
even a peer civilization
with us that developed radio

00:03:23.580 --> 00:03:25.890
communication on the
order of 100 years ago,

00:03:25.890 --> 00:03:28.080
because frankly, they
would have to be no more

00:03:28.080 --> 00:03:29.840
than 100 light years
away for us to be

00:03:29.840 --> 00:03:31.670
able to detect
those signals now.

00:03:31.670 --> 00:03:33.750
If they were on the
other side of the galaxy,

00:03:33.750 --> 00:03:37.650
we wouldn't be able to detect
their for tens of thousands

00:03:37.650 --> 00:03:38.730
of years.

00:03:38.730 --> 00:03:40.230
So when I talk about
now, I'm saying

00:03:40.230 --> 00:03:42.880
that the signals
are getting to us.

00:03:42.880 --> 00:03:47.530
Signals getting,
signals received.

00:03:47.530 --> 00:03:52.370
The signals are being
received right now.

00:03:52.370 --> 00:03:56.390
So you could have a civilization
that developed radio

00:03:56.390 --> 00:04:00.460
70,000 years ago, but they're
70,000 light years away,

00:04:00.460 --> 00:04:03.170
and maybe they collapsed
10,000 years later,

00:04:03.170 --> 00:04:06.220
but we're just receiving
their first radio signal.

00:04:06.220 --> 00:04:08.290
So that would be
as a civilization

00:04:08.290 --> 00:04:13.130
that I would count in this
equation we're setting up.

00:04:13.130 --> 00:04:15.075
And so just to make
sure we understand it,

00:04:15.075 --> 00:04:16.700
and then we can play
with some numbers,

00:04:16.700 --> 00:04:17.720
let's remind ourselves.

00:04:17.720 --> 00:04:19.668
This is the number of
stars, our estimate

00:04:19.668 --> 00:04:21.209
of the number of
stars of the galaxy.

00:04:21.209 --> 00:04:24.480
Multiplied by this,
you now know the number

00:04:24.480 --> 00:04:26.810
of stars in the galaxy
that have planets.

00:04:26.810 --> 00:04:31.040
You multiply by this n sub p,
the average number of planets

00:04:31.040 --> 00:04:33.690
capable of sustaining life,
and these first three terms

00:04:33.690 --> 00:04:38.140
will give you the average
number of planets--

00:04:38.140 --> 00:04:41.520
or I should say the number,
the total number of planets

00:04:41.520 --> 00:04:45.160
in the galaxy that have been
capable of sustaining life

00:04:45.160 --> 00:04:47.310
at some point in their history.

00:04:47.310 --> 00:04:48.850
Multiply it by this.

00:04:48.850 --> 00:04:52.780
This is the number of planets in
the galaxy that have sustained

00:04:52.780 --> 00:04:54.710
actual life, not just
capability of it.

00:04:54.710 --> 00:04:58.260
They actually had life on them
at some point in their history.

00:04:58.260 --> 00:05:00.680
Multiply it by this,
this is the fraction

00:05:00.680 --> 00:05:03.219
that have developed intelligent
life on these planets.

00:05:03.219 --> 00:05:05.010
The number of planets
with intelligent life

00:05:05.010 --> 00:05:06.810
at some point in their history.

00:05:06.810 --> 00:05:08.700
Multiply it by this fraction.

00:05:08.700 --> 00:05:09.530
All of these terms.

00:05:09.530 --> 00:05:12.380
We have the number of
planets in the galaxy that

00:05:12.380 --> 00:05:16.340
have had intelligent life
that became detectable,

00:05:16.340 --> 00:05:18.685
that started emitting some
type of radio signature.

00:05:18.685 --> 00:05:20.060
We don't know,
some type of thing

00:05:20.060 --> 00:05:23.970
like that at some
point in their history.

00:05:23.970 --> 00:05:27.700
So over here, all of
these first six terms

00:05:27.700 --> 00:05:30.580
tell us the number of
detectable civilizations that

00:05:30.580 --> 00:05:34.020
occurred at some point in
the history of the stars,

00:05:34.020 --> 00:05:36.740
the solar systems, the planets
that are out there right now.

00:05:36.740 --> 00:05:38.900
But we care about the ones
that are detectable now.

00:05:38.900 --> 00:05:40.990
We don't care about the
ones that came and went,

00:05:40.990 --> 00:05:44.850
and their radio signature went
past us while we were still

00:05:44.850 --> 00:05:46.780
living in caves, or we
were hunter gatherers.

00:05:46.780 --> 00:05:49.080
We care about the ones
that the radio signatures

00:05:49.080 --> 00:05:50.890
are receiving us now.

00:05:50.890 --> 00:05:53.740
And that's why we have this
little term right over here.

00:05:53.740 --> 00:05:55.737
So this is the
civilization of-- or I

00:05:55.737 --> 00:05:57.320
guess you could say
this is the length

00:05:57.320 --> 00:05:58.900
of the detectable civilization.

00:05:58.900 --> 00:06:03.080
So while they were actually
releasing a radio signature

00:06:03.080 --> 00:06:06.490
divided by the life of that
planet, or that solar system,

00:06:06.490 --> 00:06:08.080
or that star.

00:06:08.080 --> 00:06:10.050
So for any given
star or planet that

00:06:10.050 --> 00:06:11.740
meets all of these
criterion, what's

00:06:11.740 --> 00:06:15.550
the probability that it's
releasing its-- so at some

00:06:15.550 --> 00:06:19.630
point in the history, there
was a detectable civilization

00:06:19.630 --> 00:06:23.600
or more that was releasing
some type of a radio signature.

00:06:23.600 --> 00:06:26.230
But what's the probability
that it's doing it right now?

00:06:26.230 --> 00:06:28.740
And so that's the
detectable life span

00:06:28.740 --> 00:06:33.870
of that civilization divided by
the life of that solar system,

00:06:33.870 --> 00:06:34.760
or of that planet.

00:06:34.760 --> 00:06:36.880
Because frankly, the
star and the solar system

00:06:36.880 --> 00:06:38.880
and the planet, they're
all going to essentially

00:06:38.880 --> 00:06:41.210
have, give or take, a few
hundreds of thousands of years,

00:06:41.210 --> 00:06:43.157
or even a few millions
of years, because we're

00:06:43.157 --> 00:06:44.740
thinking in the
billions here, they're

00:06:44.740 --> 00:06:47.140
going to have roughly
the same life span.

00:06:47.140 --> 00:06:52.750
And so let's say, and just
to make this a little bit

00:06:52.750 --> 00:06:57.269
more tangible, let's say
that the sun has a lifespan,

00:06:57.269 --> 00:06:59.560
and let's say that with the
Earth and our solar system,

00:06:59.560 --> 00:07:02.980
has a lifespan of
approximately 10 billion years.

00:07:02.980 --> 00:07:05.580
10 billion years.

00:07:05.580 --> 00:07:10.470
And let's say that us as humans,
let me be pretty optimistic

00:07:10.470 --> 00:07:14.790
about it, let's say that we are
detectable as a civilization

00:07:14.790 --> 00:07:17.540
for one million years.

00:07:17.540 --> 00:07:19.890
Our best days are ahead of us.

00:07:19.890 --> 00:07:23.330
We are detectable for
one million years.

00:07:23.330 --> 00:07:26.100
So this term right over
here will be 1 million

00:07:26.100 --> 00:07:27.490
over 10 billion.

00:07:27.490 --> 00:07:31.730
So this will be 1/10,000.

00:07:31.730 --> 00:07:34.000
So even though we might
be around sending out

00:07:34.000 --> 00:07:37.010
detectable signals
for a million years,

00:07:37.010 --> 00:07:41.680
the odds relative to the
entire span of the history of--

00:07:41.680 --> 00:07:43.830
And I'm making some
simplifying assumptions here,

00:07:43.830 --> 00:07:46.420
but relative to the
entire span of the history

00:07:46.420 --> 00:07:49.420
of our planet and our
sun, if someone is just

00:07:49.420 --> 00:07:52.450
randomly sampling
our solar system

00:07:52.450 --> 00:07:55.220
at a random time in its
history, in a random part

00:07:55.220 --> 00:07:57.430
of this 10 billion
years, there's only a 1

00:07:57.430 --> 00:08:02.100
in 10,000 chance that they'll
be sampling us at a time

00:08:02.100 --> 00:08:03.610
that we are releasing signals.

00:08:03.610 --> 00:08:06.670
Assuming that there weren't
any other civilizations on Mars

00:08:06.670 --> 00:08:08.350
or Venus, or whatever
else, that there

00:08:08.350 --> 00:08:11.010
weren't any other civilizations
on Earth hundreds of thousands

00:08:11.010 --> 00:08:13.390
of years ago that
were doing this,

00:08:13.390 --> 00:08:16.170
they'll definitely only
have a 1 in 10,000 chance

00:08:16.170 --> 00:08:18.352
of detecting us, assuming
that they're sampling.

00:08:18.352 --> 00:08:20.060
There could have been
a civilization that

00:08:20.060 --> 00:08:22.400
was around three
million years ago,

00:08:22.400 --> 00:08:25.050
and they did this whole search
for extraterrestrial life.

00:08:25.050 --> 00:08:29.180
Maybe they're 20, or 100,
or 1,000 light years away,

00:08:29.180 --> 00:08:31.910
and they pointed their
radio telescopes at us.

00:08:31.910 --> 00:08:33.570
But a million or two
million years ago,

00:08:33.570 --> 00:08:35.819
they would have pointed at
the direction of our sun,

00:08:35.819 --> 00:08:37.860
and they would have not
gotten any radio signals,

00:08:37.860 --> 00:08:40.610
and they're like, man, when
is extraterrestrial life going

00:08:40.610 --> 00:08:41.360
to show up?

00:08:41.360 --> 00:08:45.420
Even though the sun and Earth
does eventually develop us,

00:08:45.420 --> 00:08:48.290
they weren't able to observe
us because when they sampled

00:08:48.290 --> 00:08:52.190
was outside of that
1 in 10,000 window.


Detectable civilizations in our galaxy 1

WEBVTT
Kind: captions
Language: en

00:00:00.425 --> 00:00:08.636
It's estimated that our galaxy, the Milky Way, has 100 to 400 billion stars,

00:00:08.636 --> 00:00:12.029
and when you hear a number like that the obvious question is:

00:00:12.029 --> 00:00:15.944
Are there civilizations on planets that are orbiting any of these stars

00:00:15.944 --> 00:00:20.241
and maybe even more interesting question is can we detect any of those civilizations?

00:00:20.241 --> 00:00:24.379
Have they gotten to the level of technological progress like us,

00:00:24.379 --> 00:00:27.662
that they are emitting electromagnetic waves into space

00:00:27.662 --> 00:00:30.287
and other civilization like ours can detect and say,

00:00:30.287 --> 00:00:36.954
"Hey, there is someone else out there watching television or using radio or whatever else they might be doing."

00:00:36.954 --> 00:00:39.933
So, what I want to do with this video is not answer that question.

00:00:39.933 --> 00:00:41.938
That's a big open question, we don't know the answer.

00:00:41.938 --> 00:00:45.508
We don't have anywhere near enough information to definitively answer that question,

00:00:45.508 --> 00:00:49.502
but I want to do is come up with a framework for at least thinking about that question,

00:00:49.502 --> 00:00:51.236
a way of actually estimating

00:00:51.236 --> 00:00:56.929
how many detectable civilizations there are in just our galaxy,

00:00:56.929 --> 00:01:00.918
and there is a formula, that you may or may not have heard of,

00:01:00.918 --> 00:01:02.959
called the Drake Equation.

00:01:02.959 --> 00:01:04.379
and what we are going to do is independently

00:01:04.379 --> 00:01:08.000
derive our own version of the Drake Equation.

00:01:08.000 --> 00:01:09.831
It's going to be slightly different,

00:01:09.831 --> 00:01:11.554
but it is the same thought process

00:01:11.554 --> 00:01:14.425
and in the future video, I'm going to maybe reconcile

00:01:14.425 --> 00:01:17.425
what we come up with, with the Drake Equation,

00:01:17.425 --> 00:01:20.338
and just so you know Drake equation is named for Frank Drake,

00:01:20.338 --> 00:01:23.810
who is a Professor at University of California Santa Cruz.

00:01:23.810 --> 00:01:27.021
He first, kind of put some structure around this problem,

00:01:27.021 --> 00:01:29.790
that's why the formula or the equation has his name,

00:01:29.790 --> 00:01:32.364
but the equation is not an equation you can apply

00:01:32.364 --> 00:01:35.523
on a daily basis and get results that you can use to build things,

00:01:35.523 --> 00:01:40.871
but what it's, it's the structures of our thinking around this question of

00:01:40.871 --> 00:01:46.041
How many detectable civilizations are there in our galaxy?

00:01:46.041 --> 00:01:50.067
And to answer this question, I am going to start little bit differently than Frank Drake did

00:01:50.067 --> 00:01:53.498
He starts with the number of new stars that're born each year,

00:01:53.498 --> 00:01:56.282
we will see that our definitions are actually pretty close to each other

00:01:56.282 --> 00:02:00.856
what I want to do, is start with the total number of stars,

00:02:00.856 --> 00:02:03.902
So, we are trying to come up with is,

00:02:03.902 --> 00:02:07.825
I will call it 'N' and this is number of detectable civilizations,

00:02:07.825 --> 00:02:20.256
number of detectable civilizations in the Milky Way, in our galaxy.

00:02:20.256 --> 00:02:25.867
and once again, there could be civilizations, looking back in the star field right over here

00:02:25.867 --> 00:02:30.179
this star right over here, maybe it has a planet that is in the right place

00:02:30.179 --> 00:02:33.579
that has liquid water and maybe there is intelligent life on that planet,

00:02:33.579 --> 00:02:37.000
but they might not be detectable,

00:02:37.000 --> 00:02:42.375
because, they aren't technologically advanced enough that they are using electromagnetic radiation

00:02:42.375 --> 00:02:45.215
or maybe they just figured out some other way to communicate

00:02:45.215 --> 00:02:49.768
or maybe they are beyond using electromagnetic radiation, you know, radio waves and all the rest to communicate

00:02:49.768 --> 00:02:53.415
so we will never be able to detect them.

00:02:53.415 --> 00:02:57.215
We are talking about civilizations like ours that are, to some degree,

00:02:57.215 --> 00:03:00.605
using technology not too different than our own.

00:03:00.605 --> 00:03:03.698
That's what we mean by 'detectable',

00:03:03.698 --> 00:03:06.021
so let's think about that little bit.

00:03:06.021 --> 00:03:11.200
I like to start with just the total number of stars in our solar system.

00:03:11.769 --> 00:03:15.667
So let's just start with, I will call it, N*,

00:03:15.667 --> 00:03:33.000
so this is the number of stars in our galaxy, and our best guess, I said is this's going to be 100-400 billion stars.

00:03:33.000 --> 00:03:35.815
We don't even know how many there are, some of them are undetectable,

00:03:35.815 --> 00:03:38.271
and the center of our galaxy is just a big blur to us

00:03:38.271 --> 00:03:40.646
we don't even know what's on the other side of that,

00:03:40.646 --> 00:03:44.267
e can't even see all the stars that are packed into the center,

00:03:44.267 --> 00:03:49.200
so this is our best guess, 100-400 billion stars.

00:03:49.200 --> 00:03:54.113
Now obviously, there is going to be subset of those stars that even have planets.

00:03:54.113 --> 00:03:56.954
So, let's multiply the times that subset; so lets multiply

00:03:56.954 --> 00:03:59.615
times the frequency of having a planet.

00:03:59.615 --> 00:04:04.067
If you are a star, this is the percent chance or the frequency or

00:04:04.067 --> 00:04:08.010
of the fraction of these stars that have planets (f sub p)

00:04:08.010 --> 00:04:17.485
so, I'll write this way "fractions that have planets",

00:04:17.485 --> 00:04:21.575
So, if this is a hundred billion, let's say I am making a guess here

00:04:21.575 --> 00:04:23.138
and we are learning more about this everyday,

00:04:23.138 --> 00:04:27.387
there are all these discoveries of 'exoplanets'- planets outside our solar systems,

00:04:27.387 --> 00:04:29.098
maybe this is one fourth.

00:04:29.098 --> 00:04:33.102
Then you could say, well that means there are 100 billion times one fourth,

00:04:33.102 --> 00:04:37.148
that means there are 25 billion stars that have planets around them.

00:04:37.148 --> 00:04:40.036
But that's still not enough to go to civilizations.

00:04:40.036 --> 00:04:43.646
We also need to think about planets, there could be a planet like Jupiter,

00:04:43.646 --> 00:04:49.205
and we don't know how life as we know it can survive on a planet like Jupiter or Neptune or Mercury.

00:04:49.205 --> 00:04:52.267
It has to have planets that are good for sustaining life.

00:04:52.267 --> 00:04:55.938
Preferably have a rocky core, liquid water on the outside,

00:04:55.938 --> 00:04:58.538
that's what we think are the ingredients we need for life,

00:04:58.538 --> 00:05:02.021
maybe we are just not being creative enough, that's what we know as life is being.

00:05:02.021 --> 00:05:12.744
So, let's multiply this times the average number of life sustaining or planets that could sustain life on them.

00:05:12.744 --> 00:05:15.129
So we don't necessarily know that they are going to have life,

00:05:15.129 --> 00:05:17.344
but they seem like they are just the right distance from the star,

00:05:17.344 --> 00:05:23.708
not too hot, not too cold. They have the right amount of gravity, water, all the other stuff,

00:05:23.708 --> 00:05:26.762
and we still don't know what this means,

00:05:26.762 --> 00:05:44.662
but this means average number, so given a number of solar systems with planets, what's the average number of planets capable of sustaining life,

00:05:44.662 --> 00:05:46.225
and once again, you don't know this answer.

00:05:46.225 --> 00:05:53.308
maybe it is 0.1, it's probably less than 1.

00:05:53.308 --> 00:05:56.025
Therefore any given solar system that has planets,

00:05:56.025 --> 00:05:59.621
the average number capable of sustaining life maybe its 0.1 maybe it's more than 1, I don't know.

00:05:59.621 --> 00:06:02.667
We don't know the exact answer here, but I will throw an answer in,

00:06:02.667 --> 00:06:05.846
I 'll throw out a guess, maybe it is 0.1.

00:06:05.846 --> 00:06:10.902
And here the fraction that has planets (f sub p), i don't know, i will throw that out as,

00:06:10.902 --> 00:06:15.277
and once again I am making up these numbers, we really don't know the right answers, this is one-fourth.

00:06:15.277 --> 00:06:18.185
But if we were to multiply this out we would have

00:06:18.185 --> 00:06:25.185
the average number of planets in our solar system that are capable of sustaining life

00:06:25.185 --> 00:06:30.067
that are around stars that have planets and these planets are capable of sustaining life.

00:06:30.067 --> 00:06:43.533
Now, n sub p will give us the total number of planets in our galaxy capable of sustaining life.

00:06:43.533 --> 00:06:48.533
Now, just because you have liquid water and right temperature and all the other rest of the ingredients

00:06:48.533 --> 00:06:52.333
doesn't mean that you will actually have life happening on your planet.

00:06:52.333 --> 00:06:58.733
So let's multiply that times the fraction that actually generate life (f sub l)

00:06:58.733 --> 00:07:03.267
So f sub p is the fraction that actually have life

00:07:03.267 --> 00:07:04.600
and this is actually a very , we dont know this answer, so this is the fraction

00:07:04.600 --> 00:07:10.867
that have life on them

00:07:10.867 --> 00:07:13.800
And this is a really big open question

00:07:13.800 --> 00:07:16.067
maybe if you have the ingredients, maybe every planet has life

00:07:16.067 --> 00:07:21.000
maybe it is a frequent thing that's happening in our galaxy and frankly our universe

00:07:21.000 --> 00:07:23.267
or maybe it's a very infrequent thing

00:07:23.267 --> 00:07:26.467
maybe it's just the right kind of freak set of circumstances that just have to happen

00:07:26.467 --> 00:07:30.267
i will throw out a number, just for the sake of having a number

00:07:30.267 --> 00:07:32.933
maybe it is one out of every ten planets

00:07:32.933 --> 00:07:36.600
that have all the right ingredients for life actually do generate life.

00:07:36.600 --> 00:07:39.867
My personal guess is probably higher than that given that

00:07:39.867 --> 00:07:44.400
life seems life seems such a robust and flexible thing we have seen in all sort of weird circumstances.

00:07:44.400 --> 00:07:48.333
Actually let me make it higher number than that so let me make it one half (0.5)

00:07:48.333 --> 00:07:52.800
assuming that we have all of the ingredients.

00:07:52.800 --> 00:07:53.800
So this should tell us essentially

00:07:53.800 --> 00:08:00.384
how many planets in our galaxy have had life on them at some point

00:08:00.384 --> 00:08:04.600
in those planets' lives.

00:08:04.600 --> 00:08:10.200
The life might have come and gone

00:08:10.200 --> 00:08:12.600
maybe destroyed itself through nuclear war.

00:08:12.600 --> 00:08:16.800
But this would tell us the number of planets in our galaxy

00:08:16.800 --> 00:08:20.933
that had life at at least one point in their history.

00:08:20.933 --> 00:08:24.133
Now we care about civilizations, about intelligent life..

00:08:24.133 --> 00:08:33.667
So maybe if the asteroid never hit earth...

00:08:33.667 --> 00:08:38.000
the dinosaurs would have never evolved to the point of generating

00:08:38.000 --> 00:08:41.867
radios and tvs and telephones and all the rest..

00:08:41.867 --> 00:08:44.067
and so it is kind of a freak circumstance

00:08:44.067 --> 00:08:47.800
that because they were destroyed these gaps in the ecosystem developed

00:08:47.800 --> 00:08:49.792
so we could emerge and and be intelligent and

00:08:49.792 --> 00:08:51.018
do all these crazy things

00:08:51.018 --> 00:08:52.410
like make YouTube videos and all the rest.

00:08:52.410 --> 00:08:55.623
So, let's multiply this times the fraction

00:08:55.623 --> 00:08:56.792
if you get all of this, the fraction

00:08:56.792 --> 00:08:59.908
that actually end up having intelligent life

00:08:59.908 --> 00:09:05.942
and maybe this fraction is,

00:09:05.942 --> 00:09:15.338
to a number of 1/10, then probably

00:09:15.338 --> 00:09:17.008
in the next video I'll calculate it all.

00:09:17.008 --> 00:09:19.227
Now this is very important to realise

00:09:19.227 --> 00:09:22.374
because once again, you could have life

00:09:22.374 --> 00:09:25.637
these are all examples of life right over here

00:09:25.637 --> 00:09:27.797
This is actually life on Earth, on our planet

00:09:27.797 --> 00:09:29.377
even though this looks quite alien

00:09:29.377 --> 00:09:32.236
this is a weeble, kind of looked at from very close-up,

00:09:32.236 --> 00:09:35.226
but there is all sorts of forms of life

00:09:35.226 --> 00:09:37.963
many of which we probably can't even begin to imagine

00:09:37.963 --> 00:09:41.975
but what we care is that if intelligent life

00:09:41.975 --> 00:09:44.158
starts to emerge on the planet

00:09:44.158 --> 00:09:46.526
because only intelligent life has a chance,

00:09:46.526 --> 00:09:49.943
we believe of being able to eventually communicate in ways

00:09:49.943 --> 00:09:53.342
that are detectable by us.

00:09:53.342 --> 00:09:55.152
Now I said intelligent life, but

00:09:55.152 --> 00:09:57.577
maybe not all intelligent life will eventually

00:09:57.577 --> 00:10:00.704
get to the technological sophistication, where

00:10:00.704 --> 00:10:05.719
they will be using radio waves, electro-magnetic radiation to communicate with each other

00:10:05.719 --> 00:10:08.542
maybe we might have stagnated at this stage

00:10:08.542 --> 00:10:11.253
if nothing of the right things didn't happen..

00:10:11.253 --> 00:10:15.385
So what we need to do now, is multiply this

00:10:15.385 --> 00:10:19.600
so right here, we would have the number of planets

00:10:19.600 --> 00:10:25.376
in our galaxy that have had intelligent life on them at some point in their history

00:10:25.376 --> 00:10:28.943
maybe not at a time that coincides with ours,

00:10:28.943 --> 00:10:34.235
but what we wanna do is bring it down even more to the percentage that get to the point

00:10:34.235 --> 00:10:38.527
that they can develop technology that allows us to detect them.

00:10:38.527 --> 00:10:47.044
So, let me multiply times the fraction that are, I'll put a 'C' here,

00:10:47.044 --> 00:10:49.976
'C' for, maybe they're using communication; 'C' for communications

00:10:49.976 --> 00:10:53.627
that allow us to detect them.

00:10:56.703 --> 00:10:59.405
So this is detectable, the fraction that are detectable.

00:10:57.734 --> 00:11:02.840
Now you might think that we're done, this will give you the Total Number of Civilizations, or life-forms

00:11:02.840 --> 00:11:11.125
in our Galaxies or the planets that have life forms that developed detectable technologies at some point

00:11:11.125 --> 00:11:19.206
in their history.. Now it would be nice if civilizations be not born and die

00:11:19.206 --> 00:11:22.895
but the reality is, they do die. They might destroy themselves or whatever,

00:11:22.895 --> 00:11:26.960
and they might exist for only a small period of time for the history of that planet

00:11:26.960 --> 00:11:28.891
or the history of that Solar System.

00:11:28.891 --> 00:11:35.659
So in order to make it the number of civilizations that are in existance now

00:11:35.659 --> 00:11:39.902
and I'll clarify what "now" means in the next video because really if we are

00:11:39.902 --> 00:11:43.192
detecting something from a star that's 10,000 lightyears away

00:11:43.192 --> 00:11:45.892
our "now" means we are just receiving their signals

00:11:45.892 --> 00:11:50.527
so, they released the signals 10,000 years ago.

00:11:50.527 --> 00:11:54.453
But what I want to do is get to know what is the fraction of these whose signals

00:11:54.453 --> 00:11:57.658
are reaching us right now.

00:11:57.658 --> 00:12:02.688
and here, I'll say what's the average lifespan of a civilization

00:12:02.688 --> 00:12:06.342
I'll put that "L", who knows what is, maybe 10,000 years

00:12:06.342 --> 00:12:15.022
so civilization lifespan as going to be that over the life of the star

00:12:15.022 --> 00:12:22.145
So, I'll put a 'T' here, for the average lifespan for the star,

00:12:22.145 --> 00:12:24.410
and I could say the average lifespan for the planet,

00:12:24.410 --> 00:12:26.703
or whatever, but we're assuming once our star

00:12:26.703 --> 00:12:29.888
supernovas, you're not going to have a chance for

00:12:29.888 --> 00:12:33.258
life on Earth to develop anymore.

00:12:33.258 --> 00:12:35.542
So, maybe this thing appears ten thousand years

00:12:35.542 --> 00:12:39.219
and this down here is maybe 10 billion years.

00:12:39.219 --> 00:12:42.043
and if you were to multilply this out, you should get

00:12:42.043 --> 00:12:46.676
the number of detectable civilizations in our galaxy right now.

00:12:46.676 --> 00:12:49.109
I'll leave you there, and next we'll discuss it a little bit

00:12:49.109 --> 00:12:54.911
more, and reconcile it with the more famous version of Drake's equation,

00:12:54.911 --> 00:12:58.145
and I'll also try and talk about this little piece of bit,

00:12:58.145 --> 00:13:00.222
because I think it might be a little confusing, and

00:13:00.222 --> 00:13:03.000
I'll try to diagram that out a little bit more..


Scale of the small

WEBVTT
Kind: captions
Language: en

00:00:00.394 --> 00:00:01.810
What I want to do
in this video is

00:00:01.810 --> 00:00:04.740
explore what happens when we
get to really, really, really

00:00:04.740 --> 00:00:05.660
small scales.

00:00:05.660 --> 00:00:07.290
And before we even
think about it,

00:00:07.290 --> 00:00:11.160
I want to familiarize
ourselves with the units here.

00:00:11.160 --> 00:00:15.160
So, we're all familiar with
what a meter looks like.

00:00:15.160 --> 00:00:18.360
The average adult male is
a little under two meters.

00:00:18.360 --> 00:00:21.155
If you were to divide a
meter into 1,000 units,

00:00:21.155 --> 00:00:22.280
you would get a millimeter.

00:00:22.280 --> 00:00:24.446
And I think we probably
know what a millimeter is.

00:00:24.446 --> 00:00:26.070
If you've ever looked
at a meter stick,

00:00:26.070 --> 00:00:29.650
it's the smallest measurement
on that meter stick.

00:00:29.650 --> 00:00:32.170
So it's already pretty
hard to look at.

00:00:32.170 --> 00:00:34.440
Now, if you were to divide
each of those millimeters

00:00:34.440 --> 00:00:37.745
into 1,000 sections,
you'd get a micrometer.

00:00:37.745 --> 00:00:39.495
Or another way to think
about a micrometer

00:00:39.495 --> 00:00:42.880
is, it's one
millionth of a meter.

00:00:42.880 --> 00:00:44.550
So this is kind of
beyond what we're

00:00:44.550 --> 00:00:46.710
capable of really perceiving.

00:00:46.710 --> 00:00:48.750
If you were to take each
of those micrometers

00:00:48.750 --> 00:00:51.230
and divide them
into 1,000 sections,

00:00:51.230 --> 00:00:52.900
you would get a nanometer.

00:00:52.900 --> 00:00:55.600
So now we're at one
billionth of a meter.

00:00:55.600 --> 00:01:00.490
You divide that by 1,000,
you get a picometer.

00:01:00.490 --> 00:01:06.390
So a picometer is 1,000
billionth of a meter,

00:01:06.390 --> 00:01:08.660
or you could say a
trillionth of a meter.

00:01:08.660 --> 00:01:11.080
You divide one of
those by 1,000,

00:01:11.080 --> 00:01:12.760
and you would get a femtometer.

00:01:12.760 --> 00:01:15.904
So these are unimaginably
small things.

00:01:15.904 --> 00:01:17.570
Now once you're
familiar with the units,

00:01:17.570 --> 00:01:19.890
let's explore what
types of things

00:01:19.890 --> 00:01:22.600
we can expect to find at
these different scales.

00:01:22.600 --> 00:01:23.830
And I'll start over here.

00:01:23.830 --> 00:01:25.900
And I've written them
on the left as well,

00:01:25.900 --> 00:01:28.350
but it's more compelling
when you see the pictures.

00:01:28.350 --> 00:01:32.130
We'll start over
here with the bee.

00:01:32.130 --> 00:01:34.530
And I've arbitrarily picked
something of this scale.

00:01:34.530 --> 00:01:37.010
There's many, many, many,
almost an infinite number

00:01:37.010 --> 00:01:39.260
of things I could have
picked at this scale.

00:01:39.260 --> 00:01:42.220
But the average bee is
about two centimeters long.

00:01:42.220 --> 00:01:43.300
This bee right over here.

00:01:43.300 --> 00:01:48.410
It's about, give or take, it's
about one hundredth the length

00:01:48.410 --> 00:01:51.720
of the average
adult human being.

00:01:51.720 --> 00:01:53.137
But once again,
the honey bee not

00:01:53.137 --> 00:01:54.970
too exciting, although
it is pretty exciting

00:01:54.970 --> 00:01:56.570
to see it zoomed in like this.

00:01:56.570 --> 00:01:58.720
But a honey bee is something
that we can relate to.

00:01:58.720 --> 00:02:00.960
We've all seen honey bees.

00:02:00.960 --> 00:02:03.280
Now, what I want
to do is zoom in,

00:02:03.280 --> 00:02:07.680
or look at something that's 50
times smaller than a honey bee.

00:02:07.680 --> 00:02:11.490
So something that if I
were to show how big it

00:02:11.490 --> 00:02:15.670
is relative to this honey
bee, it would look something

00:02:15.670 --> 00:02:16.460
like this.

00:02:16.460 --> 00:02:17.940
I'm doing it very rough.

00:02:17.940 --> 00:02:20.190
And that is a dust mite.

00:02:20.190 --> 00:02:24.750
And this right here, these are
both pictures of dust mites.

00:02:24.750 --> 00:02:28.430
Now dust mites look like these
strange and alien creatures,

00:02:28.430 --> 00:02:31.280
but what's amazing about them
is that they are everywhere.

00:02:31.280 --> 00:02:32.320
They're all around us.

00:02:32.320 --> 00:02:35.780
You probably have many of them
lying on your skin or wherever

00:02:35.780 --> 00:02:39.310
right now, which is
kind of a creepy idea.

00:02:39.310 --> 00:02:40.940
But we're talking
about scale here,

00:02:40.940 --> 00:02:44.895
and the average
dust mite-- so we

00:02:44.895 --> 00:02:47.120
were talking about
centimeters before,

00:02:47.120 --> 00:02:49.450
now we'll talk about
millimeters-- the average dust

00:02:49.450 --> 00:02:52.830
mite is less than
1/2 of a millimeter.

00:02:52.830 --> 00:02:55.380
Or if you want to
talk in micrometers,

00:02:55.380 --> 00:02:58.630
it's about 400 micrometers long.

00:02:58.630 --> 00:03:03.525
So this length right over
here is about 400 micrometers,

00:03:03.525 --> 00:03:07.469
so about 1/50th the length--
remember, this huge thing

00:03:07.469 --> 00:03:09.510
that I'm showing right
here, this is a honey bee.

00:03:09.510 --> 00:03:12.200
It's about 1/50th of the
length of the honey bee.

00:03:12.200 --> 00:03:13.970
Or maybe to put
it in other terms

00:03:13.970 --> 00:03:16.290
that you might be
familiar with, this

00:03:16.290 --> 00:03:19.212
is a zoomed-in
picture of human hair.

00:03:19.212 --> 00:03:21.670
And you might say, oh my god,
this person has horrible hair,

00:03:21.670 --> 00:03:22.470
but no.

00:03:22.470 --> 00:03:25.280
If you looked at your own hair
under an electron microscope,

00:03:25.280 --> 00:03:26.870
you'd be lucky if
it looked this good.

00:03:26.870 --> 00:03:30.600
This person, actually I've seen
pictures of more damaged hair

00:03:30.600 --> 00:03:31.100
than this.

00:03:31.100 --> 00:03:34.020
This is probably smooth
and silky hair right here.

00:03:34.020 --> 00:03:36.380
But the diameter of human
hair, and this is on average,

00:03:36.380 --> 00:03:39.390
it depends on whose hair
you're talking about,

00:03:39.390 --> 00:03:43.910
the diameter of human
hair is about 100--

00:03:43.910 --> 00:03:48.150
you can't see it when
I write in that color.

00:03:48.150 --> 00:03:51.160
It's about 100
micrometers thick.

00:03:51.160 --> 00:03:52.460
That's the diameter.

00:03:52.460 --> 00:03:57.330
So it's about a fourth
the length of a dust mite.

00:03:57.330 --> 00:04:00.370
Or if I were to draw some human
hair relative to this honey

00:04:00.370 --> 00:04:02.850
bee, it would look
something like this.

00:04:02.850 --> 00:04:08.390
It would be about-- and I'm
drawing the whole hair--

00:04:08.390 --> 00:04:11.070
so its width would be the width
of this thing that I just drew.

00:04:11.070 --> 00:04:13.240
Now remember, we're looking
at a honey bee here.

00:04:13.240 --> 00:04:16.610
It looks like some type of
giant, but it is a honeybee.

00:04:16.610 --> 00:04:19.200
Let's zoom in even more.

00:04:19.200 --> 00:04:20.980
So, we started
with the honey bee.

00:04:20.980 --> 00:04:23.220
We zoomed in by 50
to get the dust mite.

00:04:23.220 --> 00:04:25.450
We zoomed in by
another factor of 4

00:04:25.450 --> 00:04:28.350
to get the width of human hair.

00:04:28.350 --> 00:04:31.050
If we zoom in, we're in
the micrometer range now.

00:04:31.050 --> 00:04:34.900
If we zoom in by another,
roughly, another factor of 10,

00:04:34.900 --> 00:04:38.590
we get to the scale of cells.

00:04:38.590 --> 00:04:40.890
And this right here
is a red blood cell.

00:04:40.890 --> 00:04:43.430
I think this is a white
blood cell right over here.

00:04:43.430 --> 00:04:45.920
About 6 to 8 micrometers.

00:04:45.920 --> 00:04:48.210
So once again, if I
were to draw a cell

00:04:48.210 --> 00:04:52.080
relative to this human
hair, it would probably

00:04:52.080 --> 00:04:55.970
look something like this.

00:04:55.970 --> 00:04:58.050
Something on a similar
scale that we can still

00:04:58.050 --> 00:05:00.630
kind of relate to, is
the width of spider silk.

00:05:00.630 --> 00:05:03.450
It's about 3 to 8 micrometers.

00:05:03.450 --> 00:05:07.390
So if I were to draw some
spider silk on the same diagram,

00:05:07.390 --> 00:05:10.400
it would look
something like this.

00:05:10.400 --> 00:05:12.190
This is an actual
image of spider silk.

00:05:12.190 --> 00:05:14.690
So, once again, something
that we can kind of perceive.

00:05:14.690 --> 00:05:16.700
You can bump into it, you
can touch spider silk,

00:05:16.700 --> 00:05:19.237
you can see it if the sun
is reflecting just right,

00:05:19.237 --> 00:05:21.070
or if it has a little
bit of moisture on it.

00:05:21.070 --> 00:05:24.870
But it's about the thinnest
thing that humans can perceive.

00:05:24.870 --> 00:05:28.440
And this is in the ones
of micrometer range.

00:05:28.440 --> 00:05:30.220
At that same range,
you start to have

00:05:30.220 --> 00:05:32.220
some of your larger bacteria.

00:05:32.220 --> 00:05:34.260
Bacteria can be
anywhere from-- and I'm

00:05:34.260 --> 00:05:38.140
speaking very roughly--
1 to 10 micrometers.

00:05:38.140 --> 00:05:41.160
So in general, they're
smaller than cells.

00:05:41.160 --> 00:05:43.610
Most bacteria are
smaller than most cells.

00:05:43.610 --> 00:05:47.022
And just to figure out
where we sit on our scale,

00:05:47.022 --> 00:05:48.864
I have it over here.

00:05:48.864 --> 00:05:50.280
So we started off--
I want to keep

00:05:50.280 --> 00:05:52.220
reminding ourselves-- humans.

00:05:52.220 --> 00:05:54.630
You divide by 100,
you get to the bee.

00:05:54.630 --> 00:05:57.870
So each of these slashes
right here are dividing by 10.

00:05:57.870 --> 00:05:59.710
So this is divide by 10.

00:05:59.710 --> 00:06:02.990
Divide by 10 again, you're
divided in size by 100.

00:06:02.990 --> 00:06:06.160
Divide by 10 again,
you get to millimeter.

00:06:06.160 --> 00:06:07.510
You've divided by 1,000.

00:06:07.510 --> 00:06:09.930
Divide by 10 again,
you are doing

00:06:09.930 --> 00:06:12.520
tenths of millimeters,
which is about the size

00:06:12.520 --> 00:06:13.740
of the human hair.

00:06:13.740 --> 00:06:17.270
You divide again by 10, you're
going to tens of micrometers.

00:06:17.270 --> 00:06:20.080
By 10 again, you get into
the micrometer range.

00:06:20.080 --> 00:06:24.587
So now we're talking about
human hair-- not human hair.

00:06:24.587 --> 00:06:25.670
Human hair we did up here.

00:06:25.670 --> 00:06:26.810
We're talking about cells.

00:06:26.810 --> 00:06:30.420
We're talking about bacteria.

00:06:30.420 --> 00:06:32.920
Now things are going
to get really crazy.

00:06:32.920 --> 00:06:36.160
Now they're going to get
really, really, really crazy.

00:06:36.160 --> 00:06:38.900
This was in the ones
of micrometer range.

00:06:38.900 --> 00:06:41.671
Now we're going to start getting
into the hundreds of nanometer

00:06:41.671 --> 00:06:42.170
range.

00:06:42.170 --> 00:06:44.370
And just to get a
sense of things--

00:06:44.370 --> 00:06:49.270
So remember, a nanometer is
a thousandth of a micrometer,

00:06:49.270 --> 00:06:56.590
or 100 nanometers would be
a tenth of a micrometer.

00:06:56.590 --> 00:07:00.660
And this picture right here,
this big enormous planet

00:07:00.660 --> 00:07:04.190
or asteroid looking thing,
this is a white blood cell.

00:07:08.090 --> 00:07:10.682
The enormous blue
thing in this picture.

00:07:10.682 --> 00:07:13.140
And so if I were to zoom out,
it would might look something

00:07:13.140 --> 00:07:14.710
like this right over here.

00:07:14.710 --> 00:07:16.890
But what's really fascinating
about this picture

00:07:16.890 --> 00:07:19.570
for multiple reasons are
these little green things

00:07:19.570 --> 00:07:23.570
that are emerging after
essentially reproducing,

00:07:23.570 --> 00:07:27.600
emerging from the surface
of this white blood cell.

00:07:27.600 --> 00:07:30.885
And these things right here,
these are AIDS viruses.

00:07:37.210 --> 00:07:41.960
So now if we zoom in roughly
another factor of, you know,

00:07:41.960 --> 00:07:46.020
about 100 to 1,000 from
the size of a cell,

00:07:46.020 --> 00:07:49.210
you are now getting to
the size of a virus.

00:07:49.210 --> 00:07:52.560
And all of the genetic material
necessary to replicate that

00:07:52.560 --> 00:07:55.900
virus is right inside each
of these little capsids.

00:07:55.900 --> 00:07:59.440
It's right inside each of
these little green containers.

00:07:59.440 --> 00:08:01.810
So now, going back
to our scale--

00:08:01.810 --> 00:08:04.570
let me get my scale
right over here--

00:08:04.570 --> 00:08:07.200
we are down to the
scale of a virus.

00:08:07.200 --> 00:08:10.440
So we're in the hundreds
of nanometer range.

00:08:10.440 --> 00:08:12.590
If we divide by 10
and then divide by 10,

00:08:12.590 --> 00:08:14.350
you get to the nanometer range.

00:08:14.350 --> 00:08:16.640
And right in the ones
of nanometer range,

00:08:16.640 --> 00:08:22.020
you get to the width of the
double helix of a DNA molecule.

00:08:22.020 --> 00:08:24.550
So this right here is,
if you were to zoom in,

00:08:24.550 --> 00:08:26.890
and this is an artist's
depiction of it, obviously.

00:08:26.890 --> 00:08:30.850
Well, this is not a picture,
so to speak, of a DNA molecule.

00:08:30.850 --> 00:08:35.245
But the width of this double
helix is about 2 nanometers.

00:08:35.245 --> 00:08:36.620
Or another way to
think about it,

00:08:36.620 --> 00:08:40.616
about 1/60th the diameter of
one of these viral capsids.

00:08:40.616 --> 00:08:42.240
Which it would have
to be, because it's

00:08:42.240 --> 00:08:43.740
going to have to
get all wound up

00:08:43.740 --> 00:08:47.010
and fit into one of
these viral capsids.

00:08:47.010 --> 00:08:49.640
And DNA, just to make it clear,
this is just the width of DNA.

00:08:49.640 --> 00:08:52.670
It's much, much, much, much,
much, much, much, much longer.

00:08:52.670 --> 00:08:54.880
And we can talk about
that in future videos.

00:08:54.880 --> 00:08:57.540
So once again, we're at
a very, very small scale.

00:08:57.540 --> 00:08:59.480
If you want to think of
it in terms of meters,

00:08:59.480 --> 00:09:03.510
we're at two
billionths of a meter.

00:09:03.510 --> 00:09:06.550
You could put 500
million of these side

00:09:06.550 --> 00:09:10.620
by side to get to a meter.

00:09:10.620 --> 00:09:12.400
Or you could even
think of it this way,

00:09:12.400 --> 00:09:15.620
this is two millionths
of a millimeter.

00:09:15.620 --> 00:09:17.390
So once again, super small.

00:09:17.390 --> 00:09:20.090
You could put these side by
side, one DNA, and another DNA,

00:09:20.090 --> 00:09:24.200
and if you made them touch,
you could put 500,000

00:09:24.200 --> 00:09:27.350
next to each other
in a millimeter.

00:09:27.350 --> 00:09:30.290
So this is unbelievably
small amount of space.

00:09:30.290 --> 00:09:32.840
And now I'll introduce
you to another unit that's

00:09:32.840 --> 00:09:35.040
not kind of in the
conventional, you know,

00:09:35.040 --> 00:09:36.930
prefix followed by meters.

00:09:36.930 --> 00:09:38.220
And this is an angstrom.

00:09:38.220 --> 00:09:41.330
And 10 angstroms
equal one nanometer.

00:09:41.330 --> 00:09:45.185
So the width of this
DNA double helix,

00:09:45.185 --> 00:09:49.940
it would be two nanometers
or 20 angstroms.

00:09:49.940 --> 00:09:54.260
Now, if we were to
divide again by 10,

00:09:54.260 --> 00:09:57.770
you get to something that's
2 angstroms or 0.2 nanometers

00:09:57.770 --> 00:09:59.530
wide, and that is
a water molecule.

00:10:02.150 --> 00:10:04.010
Maybe instead of
using red, I should

00:10:04.010 --> 00:10:05.830
have used blue or something.

00:10:05.830 --> 00:10:10.050
But this right
here is the oxygen,

00:10:10.050 --> 00:10:16.162
and it is bonded to the 2
hydrogens right over here.

00:10:16.162 --> 00:10:17.620
So we're getting,
you know, this is

00:10:17.620 --> 00:10:20.030
beyond, frankly, human
perception, I mean.

00:10:20.030 --> 00:10:22.870
Or even really, stuff
that we can conceptualize.

00:10:22.870 --> 00:10:24.550
Not to even speak
of perception, I

00:10:24.550 --> 00:10:26.970
have trouble imagining
how small we're

00:10:26.970 --> 00:10:28.460
dealing with right over here.

00:10:28.460 --> 00:10:30.160
We're essentially
dealing, remember,

00:10:30.160 --> 00:10:35.010
we're dealing with less, 1/5
of a billionth of a meter,

00:10:35.010 --> 00:10:38.039
or 1/5 of a millionth
of a millimeter.

00:10:38.039 --> 00:10:39.580
Something that I
really can't fathom.

00:10:39.580 --> 00:10:42.050
But we're going to get
even smaller than that.

00:10:42.050 --> 00:10:45.270
If we were to zoom in on one
of these hydrogen atoms--

00:10:45.270 --> 00:10:47.290
and now things start to
get kind of abstract,

00:10:47.290 --> 00:10:49.040
and we start dealing
in the quantum realm.

00:10:49.040 --> 00:10:51.760
And it's hard to define where
one thing ends and one thing

00:10:51.760 --> 00:10:52.280
begins.

00:10:52.280 --> 00:10:53.130
And what is real?

00:10:53.130 --> 00:10:54.110
And what is not real?

00:10:54.110 --> 00:10:55.410
And all of that silliness.

00:10:55.410 --> 00:10:57.730
But if we try our best to
do it, if we were to zoom in

00:10:57.730 --> 00:11:00.760
and we sort of put some
boundary on a hydrogen atom--

00:11:00.760 --> 00:11:03.300
because electrons actually
could jump around anywhere--

00:11:03.300 --> 00:11:06.190
but if we set some boundary of
where the electrons are most

00:11:06.190 --> 00:11:10.230
likely to be found, the
diameter of a hydrogen atom

00:11:10.230 --> 00:11:13.642
is roughly 1 angstrom.

00:11:13.642 --> 00:11:15.350
Which makes sense from
this diagram, too.

00:11:15.350 --> 00:11:19.310
It's about 1/2 of the diameter
of this water molecule.

00:11:19.310 --> 00:11:22.990
What's extra crazy
is one, this atom

00:11:22.990 --> 00:11:24.464
is super, super duper small.

00:11:24.464 --> 00:11:25.880
Something that we
can't, you know,

00:11:25.880 --> 00:11:30.550
this is one ten
billionth of a meter,

00:11:30.550 --> 00:11:32.890
or one ten millionth
of a millimeter.

00:11:32.890 --> 00:11:35.430
So something we really,
really can't fathom.

00:11:35.430 --> 00:11:38.549
But what's crazier than that,
is that it's mostly free space.

00:11:38.549 --> 00:11:40.090
We've gotten this
small, we're trying

00:11:40.090 --> 00:11:41.830
to get to these
fundamental units,

00:11:41.830 --> 00:11:44.860
and this thing right here
is mostly free space.

00:11:44.860 --> 00:11:46.875
And that's because if
you look at an electron,

00:11:46.875 --> 00:11:49.400
and when we say
radius here, it's

00:11:49.400 --> 00:11:51.502
really hard to define
where it starts and ends.

00:11:51.502 --> 00:11:53.710
And you have to do some
things related to the charge.

00:11:53.710 --> 00:11:56.085
And we're not even thinking
about quantum effects and all

00:11:56.085 --> 00:11:56.832
of that.

00:11:56.832 --> 00:12:00.530
An electron has a
radius of 3 times 10

00:12:00.530 --> 00:12:02.540
to the negative 1/5 angstroms.

00:12:02.540 --> 00:12:04.510
And the nucleus of
a hydrogen atom,

00:12:04.510 --> 00:12:07.130
which is really just a proton,
has a radius a little bit--

00:12:07.130 --> 00:12:09.830
and you don't even worry
about this number right here.

00:12:09.830 --> 00:12:12.380
The general idea is, it's
the same order of magnitude.

00:12:12.380 --> 00:12:15.460
It's about 1/10,000th
of an angstrom.

00:12:15.460 --> 00:12:17.740
And just to give a
sense of what it's like,

00:12:17.740 --> 00:12:21.170
if you have the entire, if you
view the entire atomic radius

00:12:21.170 --> 00:12:23.470
to be about an
angstrom, kind of,

00:12:23.470 --> 00:12:26.900
just have a conception
for scale of the atom

00:12:26.900 --> 00:12:29.580
and how much free space
there is in an atom,

00:12:29.580 --> 00:12:33.140
if we even want to think
what is free space.

00:12:33.140 --> 00:12:36.270
Imagine a nucleus
being maybe a marble

00:12:36.270 --> 00:12:38.270
at the center of a
football stadium,

00:12:38.270 --> 00:12:40.200
of a domed football stadium.

00:12:40.200 --> 00:12:44.940
And imagine an electron being a
honey bee just randomly jumping

00:12:44.940 --> 00:12:47.520
around random parts
of that entire volume

00:12:47.520 --> 00:12:49.600
inside of that football stadium.

00:12:49.600 --> 00:12:51.729
And obviously, it's
a quantum honey bee,

00:12:51.729 --> 00:12:53.395
so it can jump around
from spot to spot,

00:12:53.395 --> 00:12:54.860
and it's not easy
to predict where

00:12:54.860 --> 00:12:56.651
it's going to go next,
and all of the rest.

00:12:56.651 --> 00:12:58.940
But that will give you
a sense of the scale

00:12:58.940 --> 00:13:02.730
of the electron and the
proton relative to the atom

00:13:02.730 --> 00:13:03.370
as a whole.

00:13:03.370 --> 00:13:04.870
But even more
crazy, it gives you

00:13:04.870 --> 00:13:14.010
a sense for how empty atoms,
and really all matter really is.


Introduction to limits

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.540
In this video, I want
to familiarize you

00:00:02.540 --> 00:00:07.060
with the idea of a limit, which
is a super important idea.

00:00:07.060 --> 00:00:10.610
It's really the idea that all
of calculus is based upon.

00:00:10.610 --> 00:00:13.157
But despite being
so super important,

00:00:13.157 --> 00:00:15.740
it's actually a really, really,
really, really, really, really

00:00:15.740 --> 00:00:17.350
simple idea.

00:00:17.350 --> 00:00:19.540
So let me draw a
function here, actually,

00:00:19.540 --> 00:00:23.690
let me define a function here,
a kind of a simple function.

00:00:23.690 --> 00:00:27.460
So let's define f of x,
let's say that f of x

00:00:27.460 --> 00:00:32.846
is going to be x minus
1 over x minus 1.

00:00:32.846 --> 00:00:34.220
And you might say,
hey, Sal look,

00:00:34.220 --> 00:00:36.511
I have the same thing in the
numerator and denominator.

00:00:36.511 --> 00:00:38.480
If I have something
divided by itself,

00:00:38.480 --> 00:00:39.730
that would just be equal to 1.

00:00:39.730 --> 00:00:42.200
Can't I just simplify
this to f of x equals 1?

00:00:42.200 --> 00:00:44.750
And I would say, well,
you're almost true,

00:00:44.750 --> 00:00:46.860
the difference between
f of x equals 1

00:00:46.860 --> 00:00:50.300
and this thing right over here,
is that this thing can never

00:00:50.300 --> 00:00:54.200
equal-- this thing is
undefined when x is equal to 1.

00:00:54.200 --> 00:00:56.560
Because if you set,
let me define it.

00:00:56.560 --> 00:00:59.040
Let me write it over
here, if you have f of,

00:00:59.040 --> 00:01:02.650
sorry not f of 0, if you
have f of 1, what happens.

00:01:02.650 --> 00:01:04.599
In the numerator,
we get 1 minus 1,

00:01:04.599 --> 00:01:07.770
which is, let me just write
it down, in the numerator,

00:01:07.770 --> 00:01:08.910
you get 0.

00:01:08.910 --> 00:01:13.590
And in the denominator, you
get 1 minus 1, which is also 0.

00:01:13.590 --> 00:01:16.800
And so anything divided by
0, including 0 divided by 0,

00:01:16.800 --> 00:01:17.550
this is undefined.

00:01:21.350 --> 00:01:23.520
So you can make
the simplification.

00:01:23.520 --> 00:01:28.980
You can say that this is you the
same thing as f of x is equal

00:01:28.980 --> 00:01:35.080
to 1, but you would have to add
the constraint that x cannot be

00:01:35.080 --> 00:01:36.340
equal to 1.

00:01:36.340 --> 00:01:38.190
Now this and this
are equivalent,

00:01:38.190 --> 00:01:40.470
both of these are
going to be equal to 1

00:01:40.470 --> 00:01:44.820
for all other X's other
than one, but at x equals 1,

00:01:44.820 --> 00:01:46.070
it becomes undefined.

00:01:46.070 --> 00:01:48.770
This is undefined and
this one's undefined.

00:01:48.770 --> 00:01:51.540
So how would I
graph this function.

00:01:51.540 --> 00:01:54.610
So let me graph it.

00:01:54.610 --> 00:01:59.130
So that, is my y is
equal to f of x axis,

00:01:59.130 --> 00:02:04.330
y is equal to f of x axis,
and then this over here

00:02:04.330 --> 00:02:05.666
is my x-axis.

00:02:09.680 --> 00:02:12.940
And then let's say this is
the point x is equal to 1.

00:02:12.940 --> 00:02:18.320
This over here would be
x is equal to negative 1.

00:02:18.320 --> 00:02:22.140
This is y is equal to 1, right
up there I could do negative 1.

00:02:22.140 --> 00:02:25.430
but that matter much relative to
this function right over here.

00:02:25.430 --> 00:02:26.890
And let me graph it.

00:02:26.890 --> 00:02:31.180
So it's essentially for
any x other than 1 f

00:02:31.180 --> 00:02:33.390
of x is going to be equal to 1.

00:02:33.390 --> 00:02:35.390
So it's going to
be, look like this.

00:02:35.390 --> 00:02:40.250
It's going to look
like this, except at 1.

00:02:40.250 --> 00:02:42.430
At 1 f of x is undefined.

00:02:42.430 --> 00:02:44.590
So I'm going to put a
little bit of a gap right

00:02:44.590 --> 00:02:48.310
over here, the circle to signify
that this function is not

00:02:48.310 --> 00:02:48.810
defined.

00:02:48.810 --> 00:02:51.210
We don't know what this
function equals at 1.

00:02:51.210 --> 00:02:52.840
We never defined it.

00:02:52.840 --> 00:02:55.010
This definition of the
function doesn't tell us

00:02:55.010 --> 00:02:55.760
what to do with 1.

00:02:55.760 --> 00:02:58.730
It's literally undefined,
literally undefined

00:02:58.730 --> 00:03:00.740
when x is equal to 1.

00:03:00.740 --> 00:03:04.120
So this is the function
right over here.

00:03:04.120 --> 00:03:07.850
And so once again, if someone
were to ask you what is f of 1,

00:03:07.850 --> 00:03:10.340
you go, and let's say that
even though this was a function

00:03:10.340 --> 00:03:12.126
definition, you'd go,
OK x is equal to 1,

00:03:12.126 --> 00:03:15.100
oh wait there's a gap in
my function over here.

00:03:15.100 --> 00:03:17.320
It is undefined.

00:03:17.320 --> 00:03:19.440
So let me write it again.

00:03:19.440 --> 00:03:22.875
It's kind of redundant, but I'll
rewrite it f of 1 is undefined.

00:03:25.820 --> 00:03:27.770
But what if I were
to ask you, what

00:03:27.770 --> 00:03:32.320
is the function
approaching as x equals 1.

00:03:32.320 --> 00:03:35.630
And now this is starting to
touch on the idea of a limit.

00:03:35.630 --> 00:03:38.080
So as x gets closer
and closer to 1.

00:03:38.080 --> 00:03:40.190
So as we get closer
and closer x is

00:03:40.190 --> 00:03:47.490
to 1, what is the
function approaching.

00:03:47.490 --> 00:03:49.960
Well, this entire
time, the function,

00:03:49.960 --> 00:03:51.760
what's a getting
closer and closer to.

00:03:51.760 --> 00:03:53.830
On the left hand side,
no matter how close

00:03:53.830 --> 00:03:56.080
you get to 1, as long
as you're not at 1,

00:03:56.080 --> 00:03:58.880
you're actually at f
of x is equal to 1.

00:03:58.880 --> 00:04:02.130
Over here from the right hand
side, you get the same thing.

00:04:02.130 --> 00:04:05.370
So you could say, and
we'll get more and more

00:04:05.370 --> 00:04:08.530
familiar with this idea
as we do more examples,

00:04:08.530 --> 00:04:13.710
that the limit as x and
L-I-M, short for limit,

00:04:13.710 --> 00:04:24.590
as x approaches 1 of f of x
is equal to, as we get closer,

00:04:24.590 --> 00:04:27.610
we can get unbelievably, we
can get infinitely close to 1,

00:04:27.610 --> 00:04:29.240
as long as we're not at 1.

00:04:29.240 --> 00:04:31.202
And our function is
going to be equal to 1,

00:04:31.202 --> 00:04:33.160
it's getting closer and
closer and closer to 1.

00:04:33.160 --> 00:04:35.490
It's actually at
1 the entire time.

00:04:35.490 --> 00:04:38.030
So in this case, we
could say the limit

00:04:38.030 --> 00:04:41.214
as x approaches
1 of f of x is 1.

00:04:41.214 --> 00:04:43.880
So once again, it has very fancy
notation, but it's just saying,

00:04:43.880 --> 00:04:45.380
look what is a
function approaching

00:04:45.380 --> 00:04:48.140
as x gets closer
and closer to 1.

00:04:48.140 --> 00:04:51.040
Let me do another example where
we're dealing with a curve,

00:04:51.040 --> 00:04:54.680
just so that you have
the general idea.

00:04:54.680 --> 00:04:58.230
So let's say that
I have the function

00:04:58.230 --> 00:05:01.230
f of x, let me just for
the sake of variety,

00:05:01.230 --> 00:05:02.970
let me call it g of x.

00:05:02.970 --> 00:05:05.940
Let's say that we have
g of x is equal to,

00:05:05.940 --> 00:05:11.710
I could define it this way, we
could define it as x squared,

00:05:11.710 --> 00:05:16.930
when x does not equal, I don't
know when x does not equal 2.

00:05:16.930 --> 00:05:24.565
And let's say that when x
equals 2 it is equal to 1.

00:05:24.565 --> 00:05:26.190
So once again, a kind
of an interesting

00:05:26.190 --> 00:05:30.110
function that, as you'll
see, is not fully continuous,

00:05:30.110 --> 00:05:32.110
it has a discontinuity.

00:05:32.110 --> 00:05:33.880
Let me graph it.

00:05:33.880 --> 00:05:37.870
So this is my y
equals f of x axis,

00:05:37.870 --> 00:05:42.180
this is my x-axis
right over here.

00:05:42.180 --> 00:05:46.350
Let me draw x equals 2, x,
let's say this is x equals 1,

00:05:46.350 --> 00:05:53.750
this is x equals 2, this is
negative 1, this is negative 2.

00:05:53.750 --> 00:05:57.650
And then let me draw, so
everywhere except x equals 2,

00:05:57.650 --> 00:06:00.400
it's equal to x squared.

00:06:00.400 --> 00:06:01.870
So let me draw it like this.

00:06:01.870 --> 00:06:08.670
So it's going to be a parabola,
looks something like this,

00:06:08.670 --> 00:06:10.880
let me draw a better
version of the parabola.

00:06:10.880 --> 00:06:13.110
So it'll look
something like this.

00:06:16.970 --> 00:06:19.370
Not the most beautifully
drawn parabola

00:06:19.370 --> 00:06:21.570
in the history of
drawing parabolas,

00:06:21.570 --> 00:06:23.720
but I think it'll
give you the idea.

00:06:23.720 --> 00:06:26.750
I think you know what a
parabola looks like, hopefully.

00:06:26.750 --> 00:06:28.674
It should be symmetric,
let me redraw it

00:06:28.674 --> 00:06:29.840
because that's kind of ugly.

00:06:32.950 --> 00:06:34.860
And that's looking better.

00:06:34.860 --> 00:06:37.740
OK, all right, there you go.

00:06:37.740 --> 00:06:42.210
All right, now, this would be
the graph of just x squared.

00:06:42.210 --> 00:06:43.020
But this can't be.

00:06:43.020 --> 00:06:45.600
It's not x squared
when x is equal to 2.

00:06:45.600 --> 00:06:49.290
So once again, when
x is equal to 2,

00:06:49.290 --> 00:06:53.200
we should have a little bit
of a discontinuity here.

00:06:53.200 --> 00:06:56.440
So I'll draw a gap right over
there, because when x equals 2

00:06:56.440 --> 00:06:58.610
the function is equal to 1.

00:06:58.610 --> 00:07:00.560
When x is equal to
2, so let's say that,

00:07:00.560 --> 00:07:03.200
and I'm not doing them on the
same scale, but let's say that.

00:07:03.200 --> 00:07:06.970
So this, on the graph of f
of x is equal to x squared,

00:07:06.970 --> 00:07:11.020
this would be 4, this would
be 2, this would be 1,

00:07:11.020 --> 00:07:12.850
this would be 3.

00:07:12.850 --> 00:07:17.755
So when x is equal to 2,
our function is equal to 1.

00:07:23.720 --> 00:07:25.480
So this is a bit of
a bizarre function,

00:07:25.480 --> 00:07:26.890
but we can define it this way.

00:07:26.890 --> 00:07:29.280
You can define a function
however you like to define it.

00:07:29.280 --> 00:07:33.490
And so notice, it's
just like the graph of f

00:07:33.490 --> 00:07:36.800
of x is equal to x squared,
except when you get to 2,

00:07:36.800 --> 00:07:39.410
it has this gap,
because you don't

00:07:39.410 --> 00:07:43.830
use the f of x is equal to x
squared when x is equal to 2.

00:07:43.830 --> 00:07:46.170
You use f of x--
or I should say g

00:07:46.170 --> 00:07:48.497
of x-- you use g
of x is equal to 1.

00:07:48.497 --> 00:07:49.580
Have I been saying f of x?

00:07:49.580 --> 00:07:50.940
I apologize for that.

00:07:50.940 --> 00:07:52.670
You use g of x is equal to 1.

00:07:52.670 --> 00:07:56.470
So then then at 2, just
at 2, just exactly at 2,

00:07:56.470 --> 00:07:57.830
it drops down to 1.

00:07:57.830 --> 00:08:00.250
And then it keeps going
along the function

00:08:00.250 --> 00:08:03.310
g of x is equal to, or I
should say, along the function

00:08:03.310 --> 00:08:05.240
x squared.

00:08:05.240 --> 00:08:06.542
So my question to you.

00:08:06.542 --> 00:08:08.000
So there's a couple
of things, if I

00:08:08.000 --> 00:08:10.860
were to just evaluate
the function g of 2.

00:08:10.860 --> 00:08:12.840
Well, you'd look
at this definition,

00:08:12.840 --> 00:08:16.260
OK, when x equals 2, I use
this situation right over here.

00:08:16.260 --> 00:08:18.950
And it tells me, it's
going to be equal to 1.

00:08:18.950 --> 00:08:20.940
Let me ask a more
interesting question.

00:08:20.940 --> 00:08:23.150
Or perhaps a more
interesting question.

00:08:23.150 --> 00:08:30.370
What is the limit as x
approaches 2 of g of x.

00:08:30.370 --> 00:08:33.159
Once again, fancy notation,
but it's asking something

00:08:33.159 --> 00:08:35.220
pretty, pretty, pretty simple.

00:08:35.220 --> 00:08:38.758
It's saying as x gets closer and
closer to 2, as you get closer

00:08:38.758 --> 00:08:40.799
and closer, and this isn't
a rigorous definition,

00:08:40.799 --> 00:08:42.419
we'll do that in future videos.

00:08:42.419 --> 00:08:48.050
As x gets closer and closer to
2, what is g of x approaching?

00:08:48.050 --> 00:08:53.210
So if you get to 1.9, and
then 1.999, and then 1.999999,

00:08:53.210 --> 00:08:57.960
and then 1.9999999, what
is g of x approaching.

00:08:57.960 --> 00:09:00.450
Or if you were to go from
the positive direction.

00:09:00.450 --> 00:09:02.680
If you were to say
2.1, what's g of 2.1,

00:09:02.680 --> 00:09:08.050
what's g of 2.01, what's g of
2.001, what is that approaching

00:09:08.050 --> 00:09:10.100
as we get closer
and closer to it.

00:09:10.100 --> 00:09:12.510
And you can see it visually
just by drawing the graph.

00:09:12.510 --> 00:09:15.610
As g gets closer
and closer to 2,

00:09:15.610 --> 00:09:18.350
and if we were to
follow along the graph,

00:09:18.350 --> 00:09:20.560
we see that we
are approaching 4.

00:09:20.560 --> 00:09:22.930
Even though that's not
where the function is,

00:09:22.930 --> 00:09:24.840
the function drops down to 1.

00:09:24.840 --> 00:09:30.940
The limit of g of x as x
approaches 2 is equal to 4.

00:09:30.940 --> 00:09:33.540
And you could even do this
numerically using a calculator,

00:09:33.540 --> 00:09:38.170
and let me do that, because I
think that will be interesting.

00:09:38.170 --> 00:09:39.795
So let me get the
calculator out,

00:09:39.795 --> 00:09:43.230
let me get my trusty TI-85 out.

00:09:45.980 --> 00:09:48.820
So here is my calculator,
and you could numerically

00:09:48.820 --> 00:09:50.770
say, OK, what's it
going to approach

00:09:50.770 --> 00:09:53.770
as you approach x equals 2.

00:09:53.770 --> 00:09:57.310
So let's try 1.94,
for x is equal to 1.9,

00:09:57.310 --> 00:09:59.550
you would use this top
clause right over here.

00:09:59.550 --> 00:10:04.070
So you'd have 1.9 squared.

00:10:04.070 --> 00:10:07.280
And so you get 3.61, well what
if you get even closer to 2,

00:10:07.280 --> 00:10:12.730
so 1.99, and once again,
let me square that.

00:10:12.730 --> 00:10:14.410
Well now I'm at 3.96.

00:10:14.410 --> 00:10:18.715
What if I do 1.999,
and I square that?

00:10:21.840 --> 00:10:23.760
I'm going to have 3.996.

00:10:23.760 --> 00:10:25.520
Notice I'm going
closer, and closer,

00:10:25.520 --> 00:10:27.930
and closer to our point.

00:10:27.930 --> 00:10:29.930
And if I did, if I
got really close,

00:10:29.930 --> 00:10:36.290
1.9999999999 squared,
what am I going to get to.

00:10:36.290 --> 00:10:38.290
It's not actually
going to be exactly 4,

00:10:38.290 --> 00:10:40.150
this calculator just
rounded things up,

00:10:40.150 --> 00:10:42.650
but going to get to a number
really, really, really, really,

00:10:42.650 --> 00:10:44.917
really, really, really,
really, really close to 4.

00:10:44.917 --> 00:10:47.250
And we can do something from
the positive direction too.

00:10:47.250 --> 00:10:48.958
And it actually has
to be the same number

00:10:48.958 --> 00:10:52.280
when we approach from the below
what we're trying to approach,

00:10:52.280 --> 00:10:54.700
and above what we're
trying to approach.

00:10:54.700 --> 00:11:00.020
So if we try to 2.1
squared, we get 4.4.

00:11:00.020 --> 00:11:00.980
If we do 2.

00:11:00.980 --> 00:11:02.870
let me go a couple
of steps ahead,

00:11:02.870 --> 00:11:07.920
2.01, so this is much
closer to 2 now, squared.

00:11:07.920 --> 00:11:09.420
Now we are getting
much closer to 4.

00:11:09.420 --> 00:11:11.710
So the closer we
get to 2, the closer

00:11:11.710 --> 00:11:13.230
it seems like
we're getting to 4.

00:11:13.230 --> 00:11:15.180
So once again,
that's a numeric way

00:11:15.180 --> 00:11:19.110
of saying that the
limit, as x approaches 2

00:11:19.110 --> 00:11:22.620
from either direction of g
of x, even though right at 2,

00:11:22.620 --> 00:11:25.230
the function is equal to 1,
because it's discontinuous.

00:11:25.230 --> 00:11:26.980
The limit as we're
approaching 2,

00:11:26.980 --> 00:11:30.803
we're getting closer, and
closer, and closer to 4.


Visualizing Taylor polynomial approximations

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:05.890
Let's say we've got the function
f of x is equal to e to the x.

00:00:05.890 --> 00:00:08.245
And just to get a sense
of what that looks like,

00:00:08.245 --> 00:00:13.140
let me do a rough drawing of
f of x is equal to e to the x.

00:00:13.140 --> 00:00:15.275
It would look
something like this.

00:00:18.520 --> 00:00:20.690
So that is e to the x.

00:00:20.690 --> 00:00:23.380
And what I want to do is
I want to approximate f

00:00:23.380 --> 00:00:28.010
of x is equal to e to the
x using a Taylor series

00:00:28.010 --> 00:00:30.930
approximation, or a
Taylor series expansion.

00:00:30.930 --> 00:00:33.440
And I want to do it not
around x is equal to 0.

00:00:33.440 --> 00:00:37.050
I want to do it around
x is equal to 3,

00:00:37.050 --> 00:00:39.562
just to pick another
arbitrary value.

00:00:39.562 --> 00:00:41.520
So we're going to do it
around x is equal to 3.

00:00:41.520 --> 00:00:43.570
This is x is equal to 3.

00:00:43.570 --> 00:00:44.860
This right there.

00:00:44.860 --> 00:00:46.610
That is f of 3.

00:00:46.610 --> 00:00:48.780
f of 3 is e to the third power.

00:00:48.780 --> 00:00:52.080
So this is e to the third
power right over there.

00:00:52.080 --> 00:00:54.520
So when we take the
Taylor series expansion,

00:00:54.520 --> 00:00:59.120
if we have a 0 degree
polynomial approximating it,

00:00:59.120 --> 00:01:02.630
the best we could probably do
is have a constant function

00:01:02.630 --> 00:01:05.379
going straight through
e to the third.

00:01:05.379 --> 00:01:09.920
If we do a first
order approximation,

00:01:09.920 --> 00:01:11.750
so we have a first
degree term, then

00:01:11.750 --> 00:01:14.610
it will be the tangent line.

00:01:14.610 --> 00:01:16.360
And as we add more and
more degrees to it,

00:01:16.360 --> 00:01:19.090
we should hopefully be
able to kind of contour

00:01:19.090 --> 00:01:21.912
or converge with the curve
better and better and better.

00:01:21.912 --> 00:01:23.870
And in the future, we'll
talk a little bit more

00:01:23.870 --> 00:01:25.950
about how we can
test for convergences

00:01:25.950 --> 00:01:28.630
and how well are we converging
and all that type of thing.

00:01:28.630 --> 00:01:30.762
But with that said, let's
just apply the formula

00:01:30.762 --> 00:01:32.220
that hopefully we
got the intuition

00:01:32.220 --> 00:01:34.810
for in the last video.

00:01:34.810 --> 00:01:38.170
So the Taylor series
expansion for f

00:01:38.170 --> 00:01:42.560
of x is equal to e to the
x will be the polynomial.

00:01:42.560 --> 00:01:44.590
So what's f of c?

00:01:44.590 --> 00:01:46.410
Well, if x is equal
to 3, we're saying

00:01:46.410 --> 00:01:49.230
that c is 3 in this situation.

00:01:49.230 --> 00:01:53.360
So if c is 3, f of 3 is
e to the third power.

00:01:53.360 --> 00:01:57.620
So it's e to the third power
plus-- what's f prime of c?

00:01:57.620 --> 00:02:00.984
Well f prime of x is also
going to be e to the x.

00:02:00.984 --> 00:02:03.400
You take the derivative of e
to the x, you get e to the x.

00:02:03.400 --> 00:02:06.470
That's one of the super cool
things about e to the x.

00:02:06.470 --> 00:02:08.460
So this is also f prime of x.

00:02:08.460 --> 00:02:12.142
Frankly, this is the same thing
as f the nth derivative of x.

00:02:12.142 --> 00:02:13.850
You could just keep
taking the derivative

00:02:13.850 --> 00:02:15.840
of this and you'll
get e to the x.

00:02:15.840 --> 00:02:18.390
So f prime of x is e to the x.

00:02:18.390 --> 00:02:22.250
You evaluate that at 3, you
get e to the third power

00:02:22.250 --> 00:02:29.700
again times x minus 3, c is
3, plus the second derivative

00:02:29.700 --> 00:02:31.140
our function is
still e to the x,

00:02:31.140 --> 00:02:34.910
evaluate that at 3, you get
e to the third power over 2

00:02:34.910 --> 00:02:39.830
factorial times x minus
3 to the second power.

00:02:39.830 --> 00:02:41.100
And then we could keep going.

00:02:41.100 --> 00:02:43.090
The third derivative
is still e to the x.

00:02:43.090 --> 00:02:46.330
Evaluate that at 3. c
is 3 in this situation.

00:02:46.330 --> 00:02:50.290
So you get e to the third
power over 3 factorial times

00:02:50.290 --> 00:02:52.560
x minus 3 to the third power.

00:02:52.560 --> 00:02:54.170
And we can keep going
with this, but I

00:02:54.170 --> 00:02:55.840
think you get the general idea.

00:02:55.840 --> 00:02:58.590
But what's even more interesting
than just kind of going

00:02:58.590 --> 00:03:01.180
through the mechanics of
finding the expansion,

00:03:01.180 --> 00:03:04.800
is seeing how as we add
more and more terms,

00:03:04.800 --> 00:03:07.740
it starts to approximate e
to the x better and better

00:03:07.740 --> 00:03:08.240
and better.

00:03:08.240 --> 00:03:11.560
And our approximation gets
good further and further away

00:03:11.560 --> 00:03:13.900
from x is equal to 3.

00:03:13.900 --> 00:03:16.520
And to do that, I
used WolframAlpha,

00:03:16.520 --> 00:03:19.700
available at wolframalpha.com.

00:03:19.700 --> 00:03:24.270
And I think I typed in
Taylor series expansion e

00:03:24.270 --> 00:03:26.700
to the x and x equals 3.

00:03:26.700 --> 00:03:28.640
And it just knew what
I wanted and gave me

00:03:28.640 --> 00:03:30.290
all of this business
right over here.

00:03:30.290 --> 00:03:31.735
And it actually
calculated the expansion.

00:03:31.735 --> 00:03:33.370
And you can see it's
the exact same thing

00:03:33.370 --> 00:03:36.100
that we have over here, e to the
third plus e to the third times

00:03:36.100 --> 00:03:38.030
x minus 3.

00:03:38.030 --> 00:03:40.660
We have e to the third plus e
to the third times x minus 3

00:03:40.660 --> 00:03:41.780
plus 1/2.

00:03:41.780 --> 00:03:43.669
They actually expanded
out the factorial.

00:03:43.669 --> 00:03:45.835
So instead of 3 factorial,
they wrote a 6 over here.

00:03:45.835 --> 00:03:47.660
And they did a bunch
of terms up here.

00:03:47.660 --> 00:03:49.910
But what's even more interesting
is that they actually

00:03:49.910 --> 00:03:55.600
graph each of these polynomials
with more and more terms.

00:03:55.600 --> 00:03:58.050
So in orange, we
have e to the x.

00:03:58.050 --> 00:04:01.390
We have f of x is
equal to e to the x.

00:04:01.390 --> 00:04:05.070
And then they tell us,
"order n approximation

00:04:05.070 --> 00:04:07.330
shown with n dots."

00:04:07.330 --> 00:04:10.140
So the order one
approximation, so that

00:04:10.140 --> 00:04:14.160
should be the situation where we
have a first degree polynomial,

00:04:14.160 --> 00:04:16.740
so that's literally-- a
first degree polynomial

00:04:16.740 --> 00:04:18.860
would be these two
terms right over here.

00:04:18.860 --> 00:04:21.430
Because this is a 0-th degree,
this is a first degree.

00:04:21.430 --> 00:04:25.260
We just have x to the
first power involved here.

00:04:25.260 --> 00:04:28.030
If we just were to plot this--
if this was our polynomial,

00:04:28.030 --> 00:04:29.900
that is plotted with 1 dot.

00:04:29.900 --> 00:04:33.580
And that is this one right
over here, with one dot,

00:04:33.580 --> 00:04:36.440
and they plot it
right over here.

00:04:36.440 --> 00:04:38.770
And we can see that
it's just a tangent line

00:04:38.770 --> 00:04:41.520
at x is equal to 3.

00:04:41.520 --> 00:04:43.420
That is x is equal to
3 right over there.

00:04:43.420 --> 00:04:45.240
And so this is the tangent line.

00:04:45.240 --> 00:04:49.140
If we add a term, now we're
getting to a second degree

00:04:49.140 --> 00:04:52.070
polynomial, because we're
adding an x squared.

00:04:52.070 --> 00:04:54.330
If you expand this out,
you'll have an x squared term,

00:04:54.330 --> 00:04:55.830
and then you'll
have another x term,

00:04:55.830 --> 00:04:58.840
but the degree of the polynomial
will now be a second degree.

00:04:58.840 --> 00:05:00.330
So let's look for two dots.

00:05:00.330 --> 00:05:02.890
So that's this one
right over here.

00:05:02.890 --> 00:05:04.750
So let's see, two dots.

00:05:04.750 --> 00:05:05.890
Two dots coming in.

00:05:05.890 --> 00:05:08.190
See, you'll notice
one, two dots.

00:05:08.190 --> 00:05:12.140
So you have two dots,
and it comes in.

00:05:12.140 --> 00:05:13.620
And this is a parabola.

00:05:13.620 --> 00:05:15.550
It's a second degree
polynomial, and then it

00:05:15.550 --> 00:05:16.950
comes back like this.

00:05:16.950 --> 00:05:21.410
But notice it does a better job,
especially around x equals 3,

00:05:21.410 --> 00:05:23.170
of approximating e to the x.

00:05:23.170 --> 00:05:25.860
It stays with the curve
a little bit longer.

00:05:25.860 --> 00:05:31.030
You add another term-- let me
do this in a new color, a color

00:05:31.030 --> 00:05:32.640
that I have not used.

00:05:32.640 --> 00:05:33.680
You add another term.

00:05:33.680 --> 00:05:35.644
Now you have a third
degree polynomial.

00:05:35.644 --> 00:05:37.060
If you have all
of these combined,

00:05:37.060 --> 00:05:40.030
if this is your polynomial, and
you were to graph that-- and so

00:05:40.030 --> 00:05:42.320
let's look for the three
dots right over here.

00:05:42.320 --> 00:05:43.790
So one, two, three.

00:05:43.790 --> 00:05:45.520
So it's this curve.

00:05:45.520 --> 00:05:50.240
Third degree polynomial is
this curve right over here.

00:05:50.240 --> 00:05:51.970
And notice, it
starts contouring e

00:05:51.970 --> 00:05:54.750
to the x a little bit sooner
than the second degree version.

00:05:54.750 --> 00:05:56.925
And it stays with it
a little bit longer.

00:06:00.820 --> 00:06:03.090
And so you have
it just like that.

00:06:03.090 --> 00:06:05.960
You add another term to it,
you add the fourth degree term

00:06:05.960 --> 00:06:06.780
to it.

00:06:06.780 --> 00:06:09.642
So now we have all of
this plus all of this.

00:06:09.642 --> 00:06:11.100
If this is your
polynomial, now you

00:06:11.100 --> 00:06:13.810
have this curve right over here.

00:06:13.810 --> 00:06:15.600
Notice every time
you add a term,

00:06:15.600 --> 00:06:18.000
it's getting better and
better at approximating

00:06:18.000 --> 00:06:22.260
e to the x further and further
away from x is equal to 3.

00:06:22.260 --> 00:06:25.206
And then if you add another
term, you get this one up here.

00:06:25.206 --> 00:06:26.580
But hopefully that
satisfies you,

00:06:26.580 --> 00:06:29.440
that we are getting closer and
closer, the more terms we add.

00:06:29.440 --> 00:06:32.180
So you can imagine it's a
pretty darn good approximation

00:06:32.180 --> 00:06:37.750
as we approach adding an
infinite number of terms.