What is Euler's formula actually saying?
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What does it mean to compute e{pi i}? Full playlist: https://www.youtube.com/playlist?list...Home page: https://www.3blue1brown.comBrought to you by you: https://3b1b.co/ldm-thanksBeautiful pictorial summary by @ThuyNganVu:https://twitter.com/ThuyNganVu/status...https://twitter.com/ThuyNganVu/status...------------------ Video Timeline (Thanks to user "Just TIEriffic"
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Ending Animation Previe
Reminders from previous lectur
Q1: Prompt (Relationship with eiθ=…
Q1: Result
WTF, Whats The Functio
Exploring exp(x
Exploring exp(x) in Pytho
Important exp(x) propert
Q2: Prompt (Given f(a+b) = f(a)f(b)…
Ask: Which is more interesting, special cases or the general cas
Q2: Result
Will a zero break Q2
The ex conventio
Q3: Prompt (i2 = -1, in = -1
Ask: Zero does not break Q
Q3: Result
Comparison to Rotatio
Visualizing this relationshi
The special case of 
Periodic nature of this relationshi
Q4: Prompt (e3i
Q4: Result
Explaining the celebrity equatio
Homework / Things to think abou
Ask: Zero does break Q2
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00:00
welcome back to lockdown math today we
00:02
are going to be talking about Euler's
00:04
formula and just to give you a little
00:06
sense of where we're going to be ending
00:07
up with this lesson I'm gonna go ahead
00:09
and show you what we're aiming for at
00:10
the end which is a certain visualization
00:12
so I don't expect you to necessarily
00:15
understand this immediately but the
00:17
point is that this is something we're
00:18
going to walk towards what we're going
00:21
to analyze is an extension of the idea
00:23
of Exponential's in a way that works in
00:26
the complex plane and the illustration
00:28
that you're looking at is showing very
00:30
literally what the claim of Euler's
00:32
formula is because what I want you to
00:34
appreciate is what the actual statement
00:36
says rather than letting it be shrouded
00:38
in a certain mystery or a certain
00:40
question of what the conventions are now
00:42
needless to say this is kind of a
00:44
confusing thing we've got this spiral of
00:46
vectors and if it's not entirely clear
00:48
don't worry about it
00:49
I just want to give you a little sense
00:50
of where we're going to be going with
00:52
this but before any of that let's take a
00:54
step back and remember where were we
00:56
okay back in the end of the last lesson
00:59
when we were talking about complex
01:00
members one of the key types of complex
01:03
numbers that we were looking at were
01:04
those that existed on the unit circle so
01:07
here I have a little complex plane drawn
01:09
we've got the real number line with the
01:11
points 1 and negative 1 indicated we've
01:13
got the imaginary number line I being
01:15
the square root of negative 1 and if you
01:18
remember one of the main points that we
01:20
emphasized last time is that when you
01:21
have a number who's sitting one unit
01:24
away from the origin at some angle theta
01:26
multiplying by this number has the
01:29
effect of rotating things by that angle
01:32
this is incredibly important throughout
01:33
physics throughout electrical
01:34
engineering all throughout math you see
01:36
these numbers everywhere they describe
01:38
wave mechanics they're very important
01:41
for polynomials it's really hard to
01:42
overstate how important numbers that sit
01:44
on this unit circle are now one way that
01:47
you could write them is with the real
01:49
and imaginary parts and based on lecture
01:51
two if we know our trigonometry the x
01:53
coordinate is going to be the cosine of
01:55
that angle and the y coordinate which is
01:57
the imaginary part is going to be I
01:59
times the sine of that angle okay so you
02:03
might think all throughout physics all
02:04
throughout electrical engineering you
02:06
see the expression cosine of theta plus
02:08
I sine of theta
02:10
in fact what you often see is another
02:12
form of this almost always you see this
02:15
written down as e to the power I times
02:18
theta and this relationship is what's
02:22
known as Euler's formula okay now he is
02:25
a special constant of nature and I
02:27
always remember in high school it was
02:28
never crystal clear to me exactly what
02:30
it was it was something that was just
02:31
kind of handed down okay it's 2.71828 on
02:35
and on and we were just taking you know
02:39
we were to take this as a an analogue of
02:41
Pi it's an irrational number that
02:43
evidently the universe side finds
02:45
significant I had one calculus teacher
02:47
who would tell us it was the Andrew
02:48
Jackson number because the president
02:50
Andrew Jackson served two terms he was
02:53
the seventh president and he was elected
02:54
in 1828 so there you go
02:57
I don't know if Andrew Jackson
02:58
appreciated this relationship that you
03:00
had to Euler but it always helped me
03:02
remember that number now just a gauge
03:04
sentiment because I'm not entirely sure
03:06
where the audience is I want to know
03:08
what your current relationship with this
03:11
particular formula is so we're going to
03:13
do a poll to start and this poll is
03:14
mostly going to be helpful to me so
03:17
pulling up some of the warm-up questions
03:18
that we had we're evidently most of you
03:20
if you were to choose one of Euler's
03:21
formulas would have gone with the one
03:23
that we're talking about today which is
03:24
great you know appropriate given where
03:27
we want to head as a poll that's
03:29
actually going to be helpful to me let
03:31
me ask you which of the following best
03:33
describes your relationship with the
03:35
formula e to the power I theta equals
03:39
cosine of theta plus I times the sine of
03:41
theta okay and the options here are that
03:44
you've never seen it before
03:45
totally understandable that you've seen
03:47
it but you're confused by it that you
03:50
still don't understand it but you've
03:51
grown used to it or that you understand
03:54
it well so as you can see answers are
03:56
rolling in please participate this is
03:58
going to be way fun if more of you
04:00
participate you can go to three b-1b
04:02
deco slash live link is also in the
04:04
description that four would do to the
04:06
place where you can answer this poll and
04:08
why all the answers are rolling in I
04:11
just want to remind us of a more famous
04:14
variant of this expression that you
04:16
often see which is basically what
04:18
happens when you plug in PI and so if we
04:21
say e to the I times
04:23
pi we're now we're thinking of pi as an
04:26
angle it's a it's a number of radians a
04:28
distance around the unit circle you
04:31
would plug in cosine of pi plus I times
04:37
the sine of pi okay and the way to think
04:41
about that is to look at the unit circle
04:42
and ask what if you walked around until
04:46
you'd walked a total distance of pi
04:47
you know if it's a lake with the radius
04:49
1 and you walk around the boundary until
04:52
you've got a distance of pi kind of by
04:54
the definition of what pi is that takes
04:55
you halfway around the circle so the X
04:57
component cosine is going to be negative
05:00
1 and then the Y component sine is
05:03
actually 0 so there's no imaginary part
05:05
it's just negative 1 and this gets us
05:08
you know what might be the most
05:09
celebrity equation in all of math e to
05:12
the I times pi is equal to negative 1 ok
05:15
but it's baffling genuinely baffling
05:18
because you've got this idea of raising
05:20
a constant which is a little wishy-washy
05:22
to start with what is e but you're
05:24
raising it to an imaginary power and if
05:26
you don't understand what that means
05:27
you're in good company I think very few
05:29
people in the world actually do now
05:32
turning back to our poll where we've got
05:33
a good number of responses and a pretty
05:35
good spread too
05:36
I'm genuinely curious to know who am I
05:40
talking to right now okay cuz it looks
05:42
like we have a pretty even distribution
05:43
among all camps all right so the number
05:46
one most common answer is C which is
05:48
those who still don't understand it but
05:50
they've grown used to it this I'm gonna
05:52
guess includes people like math majors
05:56
or engineering majors or people who have
05:58
gone into a technical field where it
06:00
comes up a lot so you have to grow used
06:02
to it this is also revealing the fact
06:04
that even though these lectures are
06:05
targeted at high school students that
06:07
this channel has a certain base
06:09
demographic to start with so sometimes
06:11
those sitting in the audience it's more
06:13
like adults who have walk in it walked
06:15
in and are sitting in the back of the
06:16
class very happy to see that D I
06:20
understand it well is a second most
06:23
common answer I'm curious
06:25
among those who answered e if by the end
06:28
of the lecture they would say that what
06:29
they understood about this is the same
06:31
as what I'm going to teach because for
06:32
example I would make the claim that this
06:35
formula actually has nothing to
06:37
do with the number II okay that the
06:39
number II doesn't actually play a role
06:40
in what this formula is computationally
06:43
saying and I'd be curious if those who
06:45
claim that they understand it well would
06:46
agree with that statement after that we
06:48
have those who've seen it but are
06:49
confused by it and at the very end is
06:51
the actual target demographic of those
06:54
who've never seen it before I don't know
06:56
if that's it's fair to call that the
06:57
target demographic because I think
06:59
sometimes you see it in a non
07:00
circumstance here there but definitely
07:02
those who answered a or B you're my
07:03
people you're the ones that I want to
07:05
talk to about this okay but let's say
07:07
you see this can you see this weird
07:09
formula e to the PI I or the
07:11
generalization e to the I times some
07:12
angle I think the healthy reaction to
07:16
have to this okay if you're just seeing
07:17
this for the first time the healthy
07:19
question to ask is w.t.f
07:23
okay what is the function at play and
07:30
how is it defined okay what's the
07:33
function because in this case the
07:35
function is e to the X and we're
07:37
plugging in certain imaginary inputs and
07:40
I think a lot of people think that that
07:42
refers to taking a number E and
07:44
multiplying by itself some number of
07:47
times and that X describes how often
07:50
you're multiplying by itself and that
07:52
yeah there's some notion of extending
07:54
that to things like 1/2 or negative 1 or
07:56
any kind of real number but that it's
07:58
based in this idea of repeated
08:00
multiplication now the thing that makes
08:03
this equation misleading is that that's
08:06
not the function that is not what each
08:08
of the X is referring to let's emphasize
08:09
that very heavily this is not what e to
08:12
the X means this is not its convention
08:15
instead what has emerged in math is that
08:19
we use e to the X to be a shorthand for
08:21
another function a function which I'm
08:25
going to give the name X this is how you
08:27
often see it in literature and it's
08:29
defined to be a certain polynomial it's
08:32
1 plus X plus x squared divided by 2
08:36
plus X cubed divided by 6 plus X
08:42
to the power 4 divided by 24 and in fact
08:45
it's not a polynomial it's an infinite
08:48
polynomial we add infinitely many terms
08:50
each of which look like X to the N
08:52
divided by n factorial the proper term
08:56
for this is a series in practice if
08:58
you're actually computing this because
09:00
that denominator grows so quickly you
09:01
can chop off the series pretty early to
09:03
get an approximate value for what this
09:06
actually is now right away we can see
09:09
that this is going to work for lots of
09:11
kinds of X that we could plug in
09:13
anything where we know how to raise X to
09:15
a power just a whole number power
09:16
multiply it by itself and where we know
09:18
how to divide by a factorial and add
09:20
those together we can come up with a
09:22
nice meaning for what this X function
09:25
means but Before we jump into things
09:27
like complex numbers and throwing that
09:29
in I think it would be very unsatisfying
09:32
to do that if we didn't first draw the
09:34
connection to the number E and the idea
09:37
of repeated multiplication because on
09:39
the surface this infinite polynomial
09:41
seems very different from the idea of
09:44
some special constant of nature E and
09:46
raising it to a power so let's build up
09:48
a little familiarity we're going to take
09:50
you know a couple minutes to do this and
09:52
it's very it's a very healthy exercise I
09:53
think to become friends with this
09:55
function by starting to plug in a couple
09:57
different values one of the first you
09:59
might plug in is the number one so that
10:03
way we get one and then X is 1 so that's
10:05
one x squared is still 1 so that's one
10:09
half and then one-sixth and then one
10:14
twenty-fourth
10:16
and in general we're adding 1 divided by
10:19
n factorial in math we say that this is
10:23
a series that converges in the sense
10:25
that as we add more and more terms it
10:28
approaches a certain value and it gets
10:29
closer and closer to some specific value
10:31
and that value ends up being around
10:35
2.71828 the Andrew Jackson number but at
10:41
this point just think of it as it was
10:42
going to be something X but one was
10:44
going to be something this is what X but
10:46
one happens to be at the moment there
10:48
isn't necessarily anything special about
10:49
that but let's say you wanted to plug in
10:52
other values ok just to get a little
10:54
practice with what
10:55
that would look like if I wanted Expo -
10:57
it would look like one plus two plus two
11:01
squared over two plus two cubed over six
11:06
and in principle if you were on a desert
11:08
island and you just needed to compute
11:10
Expo - you could work this all out by
11:13
hand if you were comfortable with long
11:14
division but of course we live in the
11:17
modern world we have computers we have
11:19
programming so I think to make this
11:21
especially concrete let's go ahead and
11:23
actually implement this function so that
11:25
we can calculate a couple values and so
11:27
that it's not a black box it's not just
11:29
a calculator that's been handed to us we
11:31
implement it so we know exactly what
11:33
it's doing so for that let's go ahead
11:35
and pull up a little Python this is
11:37
desmos we might look at that a little
11:39
bit later
11:39
right now let's be a little bit more
11:41
programmatic let's pull up some Python
11:43
let's import some math because that's a
11:46
sign you're always going to have some
11:47
fun then I'm going to define the X
11:50
function that's going to take in some
11:52
number X and what I want is to return
11:54
something that looks like 1 plus X plus
11:56
x squared over 2 where in Python this
12:00
double what do you call it an asterisk
12:02
this double asterisk sign is how we do
12:04
exponentiation and then X cubed over 6
12:07
and we kind of want to add that up a
12:09
whole bunch of course instead of typing
12:12
all that out we can use a little special
12:15
syntax where I'm going to say I want you
12:17
to return a sum of a bunch of terms each
12:22
term is going to look like raising X to
12:24
some power and it knows what X is
12:26
because that's what was handed to it X
12:28
to some power divided by the factorial
12:30
factorial of N and that's built into
12:33
this math package that we imported and
12:36
I'm just going to do this for values of
12:37
n that start at 0 0 factorial is defined
12:42
to be 1 if you're curious we could talk
12:44
all about factorials of weird values at
12:46
a later date but that's all you need to
12:48
know here and I'm just going to have it
12:50
range up to 100 because a hundred
12:52
factorial is going to be huge so that's
12:54
going to be plenty big enough for a
12:56
denominator to be small enough that
12:58
those later terms don't contribute a lot
13:00
okay so even if you don't know Python I
13:02
hope that this is a reasonably clear way
13:05
to turn
13:05
math into something that our computer
13:08
can chew on and crunch through the
13:09
numbers so that we don't have to so for
13:12
example if we type X above 1 we get the
13:15
Andrew Jackson number 2.71828 on and on
13:18
and I could type in expo 2 and it looks
13:22
like that's around seven point three
13:24
eight nine so maybe we even write that
13:26
in our notes we go over here and say
13:28
interesting X book two was about seven
13:31
point three eight nine okay and if you
13:38
spend a long time just kind of playing
13:40
around with this okay I think it's not
13:44
obvious that you might find this but if
13:46
you were just plugging in a couple
13:47
values X above three X before one
13:51
important fact you might stumble across
13:53
is that if I add two numbers in the
13:56
input okay so in this case they get 1096
14:00
when I plug in seven which is three
14:01
times three plus four that actually ends
14:04
up being the same as if I plug in X but
14:07
three times x before okay so adding the
14:12
input corresponds to multiplying in the
14:15
output and three and four weren't
14:17
special here I could have done you know
14:18
five point five and three point two and
14:23
that would have gotten me some value and
14:24
if instead I had added those together
14:26
okay it gets the same value that is not
14:30
at all obvious okay so I think that
14:33
would be a genuine discovery to have
14:35
with respect to this function with this
14:37
polynomial if that was something that
14:38
you found and it's it's important enough
14:41
that I want to write it down X of a plus
14:46
B is actually the same thing as X of a
14:52
times X of B
14:58
now if you just look at the polynomial
14:59
this is not clear okay this is not
15:01
something that like you look at it and
15:03
say oh yes of course it couldn't have
15:05
been any other way at the end of today's
15:07
lesson though I'm gonna give you guys
15:09
some homework and yes homework will make
15:11
you learn better and what we're going to
15:12
do is go through a couple problems that
15:14
are gonna have you show that this fact
15:16
is true that simply from the polynomial
15:18
and the fact that it includes these
15:20
factorial terms excuse me that's going
15:24
to be enough to show this very special
15:26
property now why am i calling it a very
15:28
special property well I want to ask you
15:30
a certain question that will hopefully
15:32
hopefully make this clear and it really
15:35
is important that you think this through
15:36
yourself because I think if we if I just
15:40
kind of tell you the implications here
15:42
it's not going to sink in to the same
15:43
extent as if you really noodle with it
15:45
yourself so I'm going to pull up another
15:47
question and I'm going to give you some
15:50
time to think on this one what the
15:53
question asks us is to suppose that we
15:55
have some function f of X that has this
15:58
special property we're adding two
16:00
numbers in the input F of a plus B gives
16:02
the same result as multiplying in the
16:04
output F of a times F of B and that this
16:07
is true for any real number a and B okay
16:10
suppose you have some function that
16:12
satisfies this property who cares how
16:13
its defined whether it's through this
16:15
polynomial through other means which of
16:17
the following is true one of the options
16:20
is that F of 5 is equal to F of 1 raised
16:22
to the power 5 another is that F of 1/2
16:26
is equal to the square root of F of 1
16:28
and the other is F of negative 1 equals
16:31
1 divided by F of 1 now very important
16:33
I'm not asking does there exist some
16:36
function f where these 3 things will be
16:38
true I'm asking which of these
16:39
necessarily has to be true only from
16:42
that property so you shouldn't be able
16:43
to contrive some sort of adversarial
16:45
function that doesn't satisfy it ok and
16:49
then you have various options for which
16:51
collection of these three things is true
16:55
I'm going to give you some time to think
16:57
about this because I really do think
16:58
it's important so I'm going to turn up
17:00
our pause and ponder music to get us in
17:02
the mood
17:04
and take a desperately-needed drink of
17:07
water
17:11
you
17:24
while you're noodling on that I'm gonna
17:26
go ahead and take a question from the
17:27
audience where it looks like surfed asks
17:30
what do you think is more interesting to
17:33
someone who is a newcomer to higher
17:34
level math special case theorems like e
17:37
to the I PI or more general cases like e
17:40
to the IX oh that is such a great
17:42
question by the way anyone who wants to
17:44
ask questions go to Twitter and just use
17:45
the hashtag lockdown math those will be
17:47
forwarded to me I think the best way to
17:50
learn is to have specific examples and
17:52
really let yourself understand the
17:54
patterns represented by those specific
17:56
examples and then generalize them for
17:58
the specific one that you have here I
18:00
think if you just if you just see e to
18:03
the I pi that doesn't really count as a
18:05
good specific example that explains the
18:07
the generality it's more that you're
18:09
plugging in one particular number into a
18:11
formula so in this case e to the i-x
18:15
as you'll see it has everything to do
18:17
with circular motion and between this
18:19
lecture and the next one we're going to
18:21
talk about why that relation might be
18:22
there so the form of having specific
18:24
examples that aids your understanding
18:26
wouldn't be you know a number like e to
18:29
the I pi it would be building a
18:31
relationship with circular motions in
18:32
other circumstances like studying
18:34
physics where you have a tetherball
18:36
with some kind of centripetal force and
18:37
it's orbiting or like orbital mechanics
18:40
anything where you're you're really
18:41
understanding the nature of circular
18:43
motion that actually prepares you for
18:45
understanding e to the i-x as a
18:47
generality a little bit better so with
18:51
all of that answers are still rolling in
18:53
and I don't want you to feel rushed so
18:56
I'm going to give a little bit more time
18:58
here actually because we remember you
19:03
need to really be sure that if you're
19:05
saying that something like 2 or 3 or 1
19:07
is included in your answer that any
19:10
function with this special property
19:12
necessarily follows that ok
19:17
oh let's let's see if we can grade it
19:20
when the on the top answer is the year
19:23
right it's a 2017 that's a little bit in
19:25
the past 2013 we're losing the top
19:27
answer interesting people are changing
19:29
their mind it's like we're going back in
19:30
time so it looks like a couple people
19:33
are oh now we're in the future 2023 yeah
19:37
so we're actually seeing people think
19:39
about it and thinking hang on you know
19:40
is it the case that option 3 here is
19:42
necessarily true and they're really
19:45
being critical about that so that's a
19:46
good sign if you want to keep thinking
19:54
about it please do but for the sake of
19:56
continuing with the lesson I'm going to
19:58
go ahead and lock things in here and see
20:00
how people ended up answering so it
20:02
looks like 20-30 of you the decade in
20:05
the future believed that all three of
20:07
these are necessarily true second most
20:10
common answer was B they only believed
20:13
that the first one is true and then
20:15
after that people who either included
20:17
number two or number three and the
20:19
correct answer is that all of them are
20:21
and it's very interesting to walk
20:24
through why I do think it's very
20:25
elucidating so let's let's go ahead and
20:27
do this the first one which it seems
20:29
like a majority of you believe is that
20:32
if we plug in something like five to a
20:36
function with this special property that
20:38
it'll be the same as taking F of one
20:40
raised to the fifth
20:42
now the reason is that we could also
20:44
write five as one plus one plus one plus
20:47
one plus one and this property of
20:52
addition in the input becoming
20:54
multiplication in the output means
20:55
that's the same thing as writing X both
20:58
one multiplied by itself five times okay
21:03
it lets you rewrite the whole thing in
21:05
terms of X both one which I'm going to
21:08
say five times this is the same as
21:14
taking X both 1 raised to the power 5
21:17
and here when I'm writing Exponential's
21:19
anytime you see a natural number a whole
21:20
number it can literally mean multiplying
21:23
by itself that number of times that's
21:24
not the same as the fact that e to the X
21:26
is a shorthand for this crazy polynomial
21:28
so that can sometimes get a little
21:30
confusing the exponent
21:31
playing two different roles and we might
21:33
give X both one a special name a
21:35
shorthand let's just call it e for short
21:38
that's not why we call this number e by
21:40
the way it it's also not because this is
21:42
what Euler's name starts with it's just
21:45
because whenever Euler was using this
21:46
the first time in a particular book he
21:49
was partial to vowels and the vowel a
21:51
had already been used
21:52
so he was just his arbitrary letter okay
21:55
so simply by virtue of this property we
21:57
can see that expo five has to be written
21:59
in terms of X both one well a little bit
22:02
trickier was the question about plugging
22:04
in one half okay and the key to solving
22:08
this it's a little bit tricky is to
22:09
think about what happens when we
22:11
multiply that by itself X but 1/2 times
22:15
X but 1/2 because of this property has
22:18
to satisfy or I should say has to equal
22:21
X both 1/2 plus X but 1/2 which is of
22:25
course X both 1 which let's say we're
22:28
using the shorthand and we call it e so
22:32
what does that mean X but 1/2 has to be
22:34
a number such that multiplying it by
22:36
itself equals e well that's what we mean
22:39
when we write square roots okay and the
22:43
last one was talking about negative
22:46
inputs so just as an example if we
22:48
inputted something like negative 1 so
22:51
the key here is to ask about multiplying
22:53
it by the value when you plug in 1 by
22:59
this rule that addition in the input
23:01
turns into multiplication in the output
23:03
that has to be the same as X of 0 now
23:08
what is expo zero oh you know what I'm
23:11
realizing hmm I think I might have
23:16
actually made the enter the wrong answer
23:18
there because in our case in our case
23:20
Expo zero actually does come out to be
23:23
the number one okay so if we plug in
23:26
zero for X then all of these you add
23:29
them together the only term that matters
23:31
is one so X book negative 1 times X plus
23:33
1 is equal to X plus 0 which is 1 so
23:37
this is true in the case of X I
23:41
I can't actually think to myself right
23:44
now if that's necessarily true you know
23:46
if we look if we look at our question
23:47
we're saying all we know about F is that
23:50
F of a plus B is f of a times F of B if
23:54
that necessarily implies that F of zero
23:55
is going to be one and I I don't think
24:00
it does I think we could construct a
24:02
function yeah because we could just
24:03
scale e to the X by some other amount
24:05
okay yeah so actually what's graded here
24:08
is is not entirely correct the correct
24:09
answer would be only one and two
24:12
I think someone correct me on Twitter if
24:15
I if I'm wrong about that
24:17
but very interesting so if we go back to
24:21
our paper let's see in the case of X the
24:24
this this value at 0 would be 1 what
24:27
does that imply well if we call X both
24:29
1/e that means we're asking what is the
24:33
number which when you multiply by e
24:34
equals 1 and it would be 1 divided by e
24:37
but yeah and for the general case but we
24:40
need the added condition that when you
24:41
plug in 0 you do get one now the point
24:45
of all of this right the reason that I'm
24:47
saying this is to emphasize why it's
24:49
reasonable that we use the shorthand
24:51
that when we say X both X for real
24:54
numbers X Y it would be very reasonable
24:56
to write this as e to the power x
24:59
because basically this special property
25:01
means that because you can use the
25:03
number 1 to access pretty much all the
25:06
real numbers by adding it to itself or
25:08
dividing as needed or negating and then
25:11
if you have certain continuity
25:13
restrictions that will let you extend
25:14
from rational numbers to all the reals
25:16
but that's a technical point you don't
25:18
need to you don't need to fuss over for
25:21
what we're doing right now
25:23
it lets you basically express everything
25:25
that this polynomial can output in terms
25:27
of what it outputs at the number 1 so
25:30
you might read e to the X as just or
25:32
whatever this polynomial is at the input
25:34
1 we can start exponentiating in terms
25:37
of that so that's the connection right
25:39
we could we could if we want to define
25:41
the number e to simply be where is this
25:43
polynomial at the number 1 and then all
25:46
of this would follow but I really want
25:48
to emphasize that only makes sense for
25:50
real numbers because as soon as we start
25:51
introducing things like complex numbers
25:53
you can't just add
25:54
to itself or subtract and divide and get
25:57
the number I and at math you often do
25:59
even crazier things where you plug in
26:00
things like matrices into this
26:02
polynomial which seems weird but you
26:05
know you can take a matrix and square it
26:07
you can divide it by two you can add all
26:09
of those together and that's actually a
26:11
very useful thing it's very useful for a
26:13
field called differential equations
26:15
which in turn is useful for physics in
26:18
quantum mechanics you often plug in
26:20
these things called operators which are
26:22
kind of like the mature older brother of
26:24
matrices and all of that looks totally
26:26
nonsensical if we're talking about a
26:28
constant raised to some kind of power
26:30
but what you have to understand is that
26:32
it's being plugged into this polynomial
26:34
okay so I understand why we have the
26:37
convention of writing this as e to the X
26:39
it makes sense for real numbers but I
26:42
think that's actually a bad convention
26:43
as soon as we start extending it and I
26:45
think that causes a lot of undo
26:46
confusion so with all of that said let's
26:49
finally have some fun and plug in some
26:52
complex values okay and before we do
26:54
that just as kind of a warm-up I want to
26:56
make sure that we're comfortable with
26:57
powers of I because that's what's going
26:59
to be important here yeah so let's pull
27:01
up our quiz and let's go ahead and ask
27:05
one more question and then I'll take a
27:07
question from the audience too while you
27:08
guys think about this remember that I is
27:12
defined to be a value that satisfies I
27:15
squared equals negative one okay it's
27:17
not it's defined for which values of n
27:20
does I to the power n equal negative I
27:23
okay so the spirit of this question is
27:26
to have people thinking deeply about
27:28
powers of I and I kind of I wanted it to
27:31
be a question that's not totally obvious
27:33
so one that you do have to put a little
27:36
mental energy into so let's go ahead and
27:40
take a question from the audience while
27:43
I take a drink of water
27:45
[Music]
27:47
f of X equals zero doesn't work with F
27:51
of negative 1 equals 1 over F of 1 yeah
27:57
f of X equals zero oh interesting yeah
28:00
okay so so they're saying that if f of X
28:04
ever outputs zero actually I'm not
28:07
entirely sure what say I'm saying f of X
28:08
equals zero doesn't work with F of
28:10
negative 1 equals 1 over F of 1 so I
28:17
guess that fact that F of negative 1 is
28:20
1 over F of 1 doesn't imply that the
28:23
output will ever be 0 so I could ask for
28:26
some clarification but I think that
28:30
might be giving at the general point of
28:31
the mistake of the question which was to
28:33
assume to extend the idea of X a little
28:36
bit too much where the fact that expose
28:38
0 equals 1 gives us this other property
28:40
and then Crispin Simmons says 3 is
28:45
included because f of X plus 0 oh ok
28:48
yeah ok great
28:48
great yeah someone thought this through
28:50
a little bit more deeply than I can
28:52
while live and on camera and under some
28:54
pressure f of X plus 0 is the same as f
28:57
of X times F of 0 which implies f of X
29:00
equals 1 awesome crispin you saved me he
29:02
saved me on this one that's a beautiful
29:04
way of thinking about it yeah all of
29:06
these these questions where you have
29:07
like a property of a function and you're
29:09
supposed to deduce facts they can be
29:11
very subtle and so in this case okay the
29:14
question was graded correctly the
29:17
property simply that addition in the
29:19
input becomes multiplication in the
29:21
output is enough to imply F of 0 equals
29:23
1 which in turn locks in the value for F
29:26
of negative 1 oh it's so pleasing one
29:28
little property can just lock in basic
29:30
not the whole real number line because
29:32
it didn't assume continuity but it locks
29:34
in pretty much any value you want
29:35
awesome excellent questions I love live
29:38
stuff even if I make errors it does feel
29:40
a little more interactive than the usual
29:42
videos so let's see how we're doing on
29:44
the quiz looks like we've still got some
29:46
answers rolling in we're trying to
29:48
understand which powers of I are equal
29:53
to negative I and the options include
29:56
multiples of 3 or multiples of 3 as long
29:59
as they're positive
30:00
or integers one below a multiple of four
30:03
or those where you're restricted to them
30:05
just being positive and I'm going to go
30:09
ahead and grade this question and as
30:12
always if you feel like you weren't done
30:14
by the time I'm locking it in know that
30:16
that's just for the sake of letting the
30:17
lesson progress forward if you wanted to
30:19
pause and think about it further
30:20
yourself especially if you're watching
30:22
this later that would be that would be
30:24
totally in the spirit of these lessons
30:26
okay so the majority or the plurality I
30:30
guess of you answered that it's all
30:33
integers one below a multiple of four
30:35
which is correct and the second most
30:37
common answer was basically the same but
30:40
restricted to positives great so let's
30:44
think through what powers of I actually
30:46
look like because this is going to be
30:47
the only thing we really need to plug in
30:49
complex values into this X function and
30:52
the way this can work if I pull out my
30:55
infinite supply of unit circles is to
30:58
remember that multiplication in the
31:01
complex plane includes a rotation
31:02
component and because I has a magnitude
31:05
of one there's only rotation so
31:07
multiplying by I has the effect of
31:09
rotating 90 degrees if you multiply by I
31:12
twice okay I squared you end up 180
31:16
degrees around reassuring it's supposed
31:19
to have a square of negative one and
31:20
this geometry lines up with that fact if
31:22
we multiply by itself three times to get
31:26
I cubed it ends up being a vector
31:29
pointing straight down multiplying by
31:32
itself a fourth time 90 degrees takes us
31:36
back to one so I to the fourth is 1 and
31:39
then after that they all start repeating
31:40
so basically all you need to know is
31:43
where's that exponent with respect to
31:45
powers of 4 if it's a power of 4 you're
31:48
at 1 if it's 1 above a power 4 you're at
31:51
I now the the trickiness of the question
31:53
was to also include the nuance of
31:55
whether this is only for positive powers
31:57
or negative powers but what a negative
31:59
power means if we write something like I
32:01
to the negative 1 that's defined to be 1
32:04
over I the number such that when you
32:07
multiply by it you get 1 and you can see
32:10
that that's actually the same as I cubed
32:13
this is the same as I cubed because
32:15
that's a number where when we multiply
32:17
it by I we get one negative one that's
32:21
also one below a multiple of four and in
32:23
general anything that's one below all
32:25
the negative multiples of four will also
32:27
satisfy this property all right so it's
32:29
a little tricky question but it's just
32:30
because I wanted you to think deeply
32:32
about this idea that powers of I of
32:35
whole numbers just have us ticking
32:37
forward by 90 degrees and finally we can
32:39
have our fun okay we can take our crazy
32:41
X X function and we could plug it into
32:46
Python if we wanted to and see how
32:48
things play there but first let me go
32:50
ahead and visualize it for you so I'm
32:53
going to let's see that circle is
32:56
already suggestively drawn but I don't
32:58
want that to be there I just want you to
33:00
think about this polynomial where I'm
33:01
gonna plug in I scaled by some kind of
33:03
constant which very suggestively I'm
33:06
going to give the name theta and then
33:07
what that means is taking 1 plus I times
33:10
theta plus I theta squared over 2 on and
33:13
on and just as an example let's crank
33:15
theta on up until it equals 1 and let's
33:18
try to think pretty deeply about what
33:20
this actually means computationally the
33:22
first term is 1 which we might draw with
33:24
this green vector on the bottom pointed
33:27
one unit to the right the next term is
33:29
going to be I times 1 so just I it's
33:32
pointed straight up with magnitude 1 now
33:35
think about what the next term is it's
33:40
going to be I squared or I theta squared
33:42
but theta is 1 so it's I squared which
33:44
is negative 1 over 2 so that's why this
33:47
next vector is pointed to the left and
33:49
it has length 1/2 okay then after that I
33:53
cubed is pointed straight down and we
33:55
divide it by 6 so now we just have an
33:57
itty-bitty vector that is of magnitude
34:00
1/6 and then they just get smaller from
34:02
there the next one is pointed to the
34:03
right but it has magnitude 1 over 24 and
34:06
they just shrink a whole bunch and so we
34:08
get something it's clear that this is a
34:11
computation that makes sense to do it's
34:13
not clear how we we know where it ends
34:15
up that feels like it's we're going to
34:16
require real math to have some kind of
34:18
theory behind it we could be
34:20
computational and pull up Python and
34:23
remember that it actually has built-in
34:24
support for complex numbers if
34:26
write something like complex of to three
34:28
it gives me a complex number with real
34:30
part two and imaginary part three and
34:32
what's cool is it it already knows the
34:35
rules for multiplication so I could
34:36
multiply this by other things it knows
34:38
how to take their product very cool now
34:41
given that our definition of this
34:42
exponential function if you'll remember
34:43
was only ever in terms of raising
34:46
something to a power and scaling it by a
34:48
factorial and then adding all those
34:49
terms up well it should make sense to
34:52
plug in something like complex zero one
34:55
which is how we write I in Python and
34:58
see what it pops out and indeed it
35:00
crunches through the numbers and it gets
35:02
for us something whose real part is
35:03
evidently around point five four who
35:06
knew and whose imaginary part is around
35:09
point eight four and in terms of our
35:11
visual you know that checks out it looks
35:13
like the real part is a little above 0.5
35:16
so point five five four I believe that
35:18
and the imaginary part is it claimed
35:21
what point eight four yeah that seems
35:23
about right
35:23
so at this point we can actually think
35:26
what is the claim of Euler's formula
35:29
okay what is it telling us about the
35:31
behavior here and my hope is that
35:33
Whitley's formula might have started off
35:35
mysterious but mysterious in a bad way
35:38
okay because when you initially see this
35:41
expression e to the I theta is equal to
35:44
cosine theta plus I sine theta telling
35:46
you that oh I guess imaginary exponents
35:48
get us on the unit circle it doesn't
35:51
leave you wondering about a real pattern
35:52
it leaves you wondering what the
35:54
convention is right it leaves you
35:56
wondering WTF
35:58
however right now what we can see is
36:00
what this is saying is if I plug in I
36:03
times theta evidently that's the same as
36:05
walking around a unit circle now that
36:08
feels very substantive there's a lot of
36:09
content to be had in that expression
36:11
because if we go to our visual what it's
36:13
telling us is that if I if I crank up
36:17
this value of theta the place that this
36:19
sum of vectors converges always sits on
36:22
a circle that's one of radius one that's
36:26
not obvious if you just look at this
36:28
expression and you think through what
36:29
it's saying very literally that is
36:31
definitely not something that just pops
36:32
right out of it but it becomes a lot
36:34
more beautiful in my opinion because
36:36
you're no longer asking questions of
36:37
convention you're asking questions of
36:40
the universe of actual patterns so just
36:43
to take the most famous input that there
36:44
is plugging in PI around 3.14 let's
36:48
think about what this is actually saying
36:50
it's saying okay the first term is 1 so
36:53
we've got our vector pointed 1 unit to
36:54
the right the next term is I times pi
36:57
okay so that's gonna be a vector that's
37:00
pointed up because of the I with the
37:02
length of pie around 3.14 which it looks
37:05
like it is the next one is going to be I
37:08
squared times pi squared over 2 so that
37:11
I squared means that it's negative okay
37:13
it's pointing to the left and I guess
37:15
the magnitude is pi squared over 2 we
37:18
could do a quick gut check if we wanted
37:19
and just ask ourselves you know what is
37:21
what is pi squared it's around 9.8 same
37:25
as the gravity on earth in terms of
37:27
meters per second squared which is
37:28
always fun and dividing that by 2
37:31
it looks like it's 4.9 3 ok does that
37:34
check out yeah I buy it that the the
37:37
length of this vector is just less than
37:39
5 so 4 point 9 3 and then the next one
37:42
because of the I cubed is pointed down
37:44
and the magnitude is PI cubed over 6
37:47
evidently that's around the magnitude of
37:49
PI cubed over 6 and so far they've the
37:52
vectors have been getting bigger but
37:53
it's at this point they start turning
37:54
around because basically each time on
37:57
the for each new term the numerator gets
38:00
an extra pie thrown in but the
38:02
denominator now gets an extra 4 thrown
38:04
in like going from 6 to 24 you're
38:06
multiplying by a 4 and then it's going
38:08
to get an extra 5 and then a 6 so the
38:11
denominator is going to start growing
38:12
much more than the numerator which is
38:14
why the vectors shrink and shrink and
38:16
start to converge to a point so the I in
38:19
there is what's letting these vectors
38:20
all turn 90 degrees each time and give
38:22
us this very pretty spiral sum and the
38:24
claim of Euler's formula is this very
38:27
fascinating fact that evidently when the
38:29
vectors are rotating according to powers
38:31
of I and when their lengths are changing
38:34
according to powers of pi divided by n
38:36
factorial they all conspire in just the
38:39
right way that they land at the point
38:42
negative 1 that's interesting still
38:45
mysterious but mysterious in a good way
38:47
and it's making an even stronger claim
38:50
which is that any other angle that we
38:52
might put in
38:53
it puts us on a unit circle at that
38:56
angle now this is especially mysterious
38:59
when we consider the fact that as we let
39:01
this value just continue increasing it's
39:04
not even clear that the function would
39:05
be periodic right because as theta
39:08
increases more and more of the vectors
39:11
become relevant because it takes longer
39:12
and longer for the denominator of our
39:14
terms to win out and yet they stayed
39:18
very nicely constrained on to the unit
39:20
circle I think that's interesting
39:22
so just to just to do a last little
39:24
concept check here I'm gonna ask you one
39:27
final question to basically see if
39:29
you've been paying attention see if the
39:31
the claims-made stand to reason so
39:33
getting rid of our powers of I the last
39:37
question is a a wind down question of
39:39
sorts for the evening which of the
39:44
following values is closest to e to the
39:47
3 I okay so as usual this is the link
39:52
you can go to which of the following
39:54
values is closest to e to the 3 I
40:03
[Music]
40:14
[Music]
40:17
you
40:19
now as you're answering let me just
40:22
remind you that the main takeaway I want
40:24
you to have from this lesson is that the
40:26
way that you read this is not to think
40:28
about a constant e raised to an
40:31
imaginary power because that doesn't
40:33
really make sense when we're thinking of
40:35
powers as repeated exponential repeated
40:38
multiplication however what does let's
40:44
see it seems like we have some
40:47
controversy over the last question we'll
40:48
address that in a moment but what you
40:52
should think about here is the fact that
40:53
e to the X is shorthand for this general
40:55
polynomial okay and this general
40:57
polynomial it turns out to be very
40:59
relevant in things like calculus in a
41:01
future lecture I'd like to start talking
41:03
about why this idea of X to the N over N
41:06
factorial and adding them up why is that
41:08
useful why would anyone care why would
41:11
it give rotation but if you know that
41:14
that's how to read it hopefully this
41:15
isn't a nonsense question and it's not
41:17
one that just asks you to blindly
41:19
believe convention it kind of asks you
41:22
to blindly believe a claim about the
41:23
fact that this follows a circle but
41:25
we'll address that in the next lecture
41:26
and I think that's enough time to go
41:29
ahead and grade this and the correct
41:31
answer is B which around 3800 W nicely
41:36
got well done well it done indeed and
41:38
it's around negative 0.99 plus 0.1 4i
41:42
and so the expectation isn't that you're
41:44
a machine who can calculate that in your
41:46
head it's instead the idea that if we go
41:48
over to our visual where we're plugging
41:51
in values of theta and you ask what
41:53
happens when you plug in 3 that the
41:56
claim of Euler's formula is that we walk
41:58
three units around a unit circle knowing
42:02
that pi units takes us all the way
42:04
around three units should take us so
42:06
that the real part is really quite close
42:07
to negative one and the imaginary part
42:10
is something positive and of the options
42:12
you were given those were the only ones
42:13
in there now the very last thing that I
42:16
would like to emphasize here is that
42:18
nothing about this actual expression has
42:21
to do with the number E has to do with
42:24
the number 2.71828 on and on if we were
42:27
to go to the computation taking place if
42:29
I plug in something like X of
42:32
a complex number whose real part is zero
42:34
and whose imaginary part is pi okay
42:36
which at first it might look like it's
42:39
outputting something other than negative
42:40
one but it's saying the real part is
42:42
negative one and then some numerical
42:44
error because partly because of how we
42:46
defined it you know only doing so many
42:47
terms and partly because computers can't
42:50
can do infinite precision the imaginary
42:53
part it might look like it's saying
42:55
three point four five but really it's
42:57
saying times 10 to the negative 16
42:59
that's what that means
43:00
so it's basically zero if you wanted to
43:02
see this we could maybe import numpy and
43:07
do something like round it so I can
43:10
round this thing to I don't know eight
43:12
decimal places and we see that okay it's
43:15
basically negative one so this is the
43:17
function that we wrote showing us that e
43:19
to the PI is negative one
43:21
but what I'd like to emphasize is that
43:22
nowhere in that function did we include
43:25
the value 2.718 in fact nowhere in the
43:28
computer's memory during the execution
43:31
of this computation would it encounter
43:33
the value 2.718 I would submit to you
43:35
that that value is not relevant to the
43:38
equation e to the PI I equals negative
43:40
one which is kind of funny all right
43:44
let's at least find out what the
43:46
relationship is because you know I'm not
43:49
going to say there's no connection
43:50
between e and pi there is one but I
43:52
think when you write the very famous
43:54
celebrity equation e to the PI I equals
43:56
negative one it's it heavily overstates
44:00
the connection that there is between
44:02
this e and that pi what's really going
44:05
on is that we have this function X that
44:10
we've defined and that when you plug in
44:12
real values okay
44:14
when you plug in real numbers for the
44:16
reasons we talked about earlier it makes
44:18
sense to write x of x in terms of
44:21
whatever its value at one happens to be
44:24
x but one raised to the X and that we
44:27
often call that value X of 1e okay this
44:32
is a property of the function another
44:35
fact about this is that when we plug in
44:37
imaginary numbers the claim of Euler's
44:40
formula which I haven't shown why it's
44:42
true but the the claim that it's making
44:45
is that it's periodic basically that it
44:49
walks around a circle and it's periodic
44:51
with a period of 2pi
44:58
I basically when you increase that input
45:00
by 2pi you get back to where you started
45:02
and even more specifically that you do
45:04
so by walking along a circle so it's
45:07
saying that each of these constants is a
45:08
child of the function X of X right but
45:12
the way that they're related kind of
45:15
happens along different dimensions that
45:16
I think is the healthy way to understand
45:18
this formula now in the next lecture we
45:21
are going to be talking about why this
45:23
thing walks around in circles and that
45:26
misses that necessarily involves a
45:28
little bit of calculus so we're going to
45:31
get a little bit of calculus into the
45:32
mix here but as minimal as I can and
45:35
before then I want to leave you with
45:37
some homework because if you remember
45:39
the core property that related this
45:42
strange polynomial with factorials to
45:45
the number E was that addition in the
45:47
input is the same as multiplication in
45:49
the output so before next time I won't
45:53
grade this this is just you know to do
45:55
to do on your own time in your own way
45:59
question number one I want you to show
46:02
that when you fully expand X but X times
46:06
X of Y and you know you might have
46:08
written down for yourself what this
46:09
function is it's all those factorial
46:11
things so that when you expand that out
46:13
each term has the form X to the K times
46:17
y to the M divided by K factorial M
46:20
factorial for certain whole numbers K
46:22
and M and remember zero factorial is
46:25
defined to be 1 and question number to
46:29
show that when you expand Expo of X plus
46:32
y so addition in the input each term is
46:36
going to look like 1 over N factorial
46:38
times n choose K and times X to the K
46:43
times y to the N minus K if you're not
46:45
familiar with this term n choose K take
46:48
a look at the binomial formula I'm sure
46:49
there's plenty of excellent videos on
46:51
YouTube about it I'll leave links in the
46:52
description as I find them but show that
46:55
that's true question 3 compare the two
46:58
results
46:59
you just found to explain this key
47:01
property right the thing that relates
47:04
this function to some special constant
47:06
and the idea of raising it to powers and
47:08
then if you're up for the the bonus
47:10
question this is kind of the this is for
47:13
all of the extra credit in the world
47:15
justify see if you can say will this
47:17
result still be true if x and y are
47:19
complex numbers and will it still be
47:21
true if they're matrices yeah that
47:24
actually I think is a very important
47:25
exercise to go through to understand
47:27
when this is true for what kinds of
47:30
inputs because like I said as you get
47:32
deeper and deeper into math this X
47:34
function remains very relevant and you
47:36
start plugging in things that seem
47:37
wilder and wilder it's useful it is
47:40
genuinely useful and it's important to
47:42
know what properties carry over and
47:43
which ones don't
47:44
no I was talking a little about this
47:47
lecture with a friend before hand and I
47:50
mentioned the spirit of the lockdown
47:51
math series being like high school
47:52
classes and you know he looks at me
47:54
quizzically and is like is this is that
47:57
a high school topic and the honest
47:59
answer is I don't know what I can say is
48:02
that it is the case that in high school
48:04
we came across the formula e to the I
48:06
theta I think the first time that I ever
48:08
saw it was in you know learning about
48:10
complex numbers it was just hot as the
48:13
polar representation of complex numbers
48:15
to emphasize this idea that when you
48:16
multiply two things you you multiply
48:20
their magnitudes and you add their
48:21
angles but it was just this strange
48:23
thing handed from on high the usual way
48:26
that we teach a relationship between e
48:28
to the X in this polynomial comes from a
48:31
piece of calculus called Taylor series
48:32
and you start with the function e to the
48:34
X always defined in a little bit of a
48:36
wishy-washy way depending on the course
48:39
that you're taking and then you show
48:40
that it's connected to this polynomial
48:42
but I think there's no reason we can't
48:43
go the other way around
48:44
and I actually wish that I had seen this
48:46
I wish that I had known from the
48:48
beginning that through math especially
48:51
through deeper math that this is what
48:53
the expression e to the X actually means
48:55
so with that I'm going to call an end to
48:58
the proper lecture and just take a
49:00
couple more questions from the audience
49:01
to finish things off maintain the
49:04
interactivity that is the spirit of our
49:06
of our elections
49:07
all right great great we have more
49:09
discussion I love this Crispin's proof
49:12
is wrong the constant function f of X
49:14
equals 0 ah wonderful
49:16
ah excellent counter example I think
49:18
this is maybe niggling in the back of my
49:20
mind it's like I don't think this is
49:21
unique
49:22
great so you have the case of a function
49:25
that is constantly equal to zero and I
49:29
think ok whoever was asking a question
49:30
earlier that I didn't quite understand I
49:32
think this must be what you were getting
49:33
at and I didn't I just wasn't reading it
49:35
properly but yeah that would be a
49:38
function where f of 0 is not necessarily
49:41
1 beautiful alright so that question
49:43
that I said was wrong and then I said it
49:44
wasn't wrong it actually was wrong so
49:46
thank you very much Erik moss and to
49:48
everyone who can think more clearly
49:49
about things than I can why alive on
49:52
camera and then another person points
49:54
out exactly the same thing I love this
49:57
this is just like a real class I'm up
50:00
here I get to make mistakes you're down
50:01
there you get to correct them and I
50:03
think it makes it better for everyone
50:04
watching there's a legitimate discussion
50:06
to be had what's relevant in the case of
50:08
the X function is just that it's not
50:11
constantly 0 that it does in fact equal
50:13
1 at the input x equals 0 but I love
50:18
this discussion and I think I can
50:19
imagine no better place to end things
50:21
than right there so as always thank you
50:24
for joining I hope this helped provide a
50:26
little bit of a different perspective
50:27
than what you usually see in other
50:29
classes on what's a very popular topic
50:31
on the Internet at least as math goes
50:33
and next time we're going to talk about
50:35
why it's true where the circle would
50:37
come from I hope to see you then
50:48
[Music]
51:02
you
51:07
[Music]