Intuition for i to the power i
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ii, visualized and explained. Full playlist: https://www.youtube.com/playlist?list...Home page: https://www.3blue1brown.comBrought to you by you: https://3b1b.co/ldm-thanksMistakes: At
, when changing r to equal 0.69*i, I said "this is what we might think of as (2i)x", but that is not correct. It's what we'd think of as [Exp(ln(2)*i)]x for whatever complex number Exp(ln(2)*i) is. Beautiful notes by Ngân Vũ https://twitter.com/ThuyNganVu/status...Video by Matt Parker
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Exponential function for i
Question 
Plug-in imaginary number in exp(x) polynomia
Answer 1 and explanatio
What it really means ii
eit as a position vector
Question 
Audience question from twitte
Answer 
Where you get after traveling π/2 units of time for position vector ei
Question 
Audience tweet
Answer 
Question 
Answer 
How exp(rx) or bx really works
Question 
Audience tweet
Answer 
Visualization of f(x)= exp(r*x) i.e. e(r*x), where r= unique complex numbe
Questions to think abou
Audience tweets
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00:00
welcome back to what is now the second
00:02
to last in the lockdown math series so
00:05
we've been talking a lot about
00:06
exponential functions and complex
00:09
numbers and the interplay of the two of
00:10
them and I thought it might be a very
00:12
nice culminating topic to help bring it
00:13
all together and show really the full
00:16
breadth of what exponential functions
00:17
are and what they can be would be to
00:19
talk about raising I to the power I now
00:23
this is a this is a funny little
00:24
expression because usually if we think
00:26
about powers if we were doing something
00:28
like I to the fourth you know we know
00:30
how to think about it in terms of
00:31
repeated multiplication we're just
00:33
multiplying something by itself four
00:35
different times and in general you'd
00:39
like to think of I to the X as some
00:41
variant of that but as soon as X becomes
00:43
a complex number we should realize that
00:45
it's really nothing like the above so
00:50
it's really nothing like this instead
00:55
the way that we think about it is going
00:57
to have to be very related to the way
00:58
that we think about raising e to various
01:00
different numbers ie being that kind of
01:02
home base for us and in fact if we go
01:05
that's not sure where things are going
01:07
too much here if we go ahead and rewrite
01:10
that base I in terms of e it can help us
01:15
make sense out of this expression but I
01:17
don't want to do that I want you to do
01:19
that for me so if you head on over to 3b
01:21
1 0 / live link is in the description
01:23
they before would you on over to item
01:25
pool which now asks us write a value of
01:30
x so that e to the X is equal to I that
01:33
way we'll be able to replace it in the
01:35
base of our expression and then once we
01:37
do that maybe we can use our knowledge
01:38
of e and Exponential's of Yi in order to
01:42
make some sense out of this expression
01:44
now let me just add as a quick reminder
01:46
while everyone is rolling in with
01:47
answers for that that when we write e to
01:51
the X and complex numbers are in the mix
01:53
we are no longer talking about repeated
01:55
multiplication and instead I think the
01:57
healthiest way to think about it is that
01:59
e to the X is a shorthand for this
02:01
polynomial that I might call X it's the
02:04
it's sort of the main exponential
02:06
function all other exponential functions
02:07
derived from this in a way that I'll
02:09
describe later on in in this episode
02:11
actually and
02:13
it's a lovely polynomial it's one plus X
02:15
plus x squared over 2 plus X cubed over
02:17
6 each term looks like a power of X
02:20
divided by the factorial of that power
02:23
so the reason for doing that is then it
02:26
can actually make sense to plug in
02:27
things like complex values and one of
02:30
the things that we looked at earlier in
02:31
the series was how if you're plugging in
02:33
imaginary values you know let's say I go
02:35
over here and I'm gonna plug in I where
02:37
I is the square root of negative 1 times
02:39
some value theta we can very literally
02:41
read off what it means to exponentiate
02:43
it in that context for example this very
02:46
first term one we might think of as a
02:48
vector pointing from zero to one the
02:50
next term which is some scaled version
02:51
of I well it's a 90 degree rotation of
02:54
that first vector scaled a little bit
02:56
differently then the next vector is a 90
02:58
degree rotation of that scale
02:59
differently and depending on what this
03:02
value theta equals the way that these
03:04
things scale is going to look very
03:06
different and it's in general a
03:08
difficult thing to think about but it's
03:10
at least a sensible thing to think about
03:11
right so that when we go to our
03:13
expression e to the X and we're going to
03:16
start thinking about I and various
03:18
complex numbers it makes sense
03:19
computationally what it means and then
03:21
there's going to be a separate
03:22
illustration that I'll pull up in a
03:23
moment for how it makes sense in terms
03:25
of intuition what it means but first
03:28
let's just see how people are doing on
03:29
our live quiz finding a value of x so
03:32
that e to the X is equal to I now how
03:38
have people done here okay so the top
03:43
answer is in a sense absolutely correct
03:46
you could say by definition the natural
03:48
log is the inverse function of
03:50
exponentiation so whatever whatever the
03:53
answer here is here we should be able to
03:55
write it as natural log of I the problem
03:56
is that's not really helpful it's not
03:58
going to move us forward because if we
04:00
if we tried to use that you know we pop
04:02
over here and we say instead of writing
04:04
I'm gonna write e to the natural log of
04:06
I and then I'm going to be raising that
04:08
to some kind of power it begs the
04:10
question and the only thing I can really
04:12
do from here is simplify that down to BI
04:14
which sort of gets us nowhere so to move
04:16
forward and to think is there a number
04:18
some concrete point on the plane that I
04:20
can plug in for that we'd want to look
04:22
at
04:25
I accidentally skipped ahead of my
04:26
questions here let me go back
04:29
pull up people's answers got a little
04:32
excited a little trigger-happy with some
04:33
of my clicking around here and the
04:37
second most common answer is something
04:39
that's gonna actually help us move
04:40
forward here which was one half of I
04:42
times pi and then that's the same as
04:45
what was written what was categorized as
04:46
the third most common one one half pi
04:48
times I just swapping the two numbers
04:51
there and I think yeah I think the one
04:55
following that I'm guessing people
04:56
either forgot to mention the I but but
04:59
let's just let's just write that out and
05:01
kind of see where this moves us one-half
05:02
I times pi now if you're confused about
05:05
where that came from
05:06
I think this is a good time to remind
05:08
ourselves of Euler's formula and what
05:10
it's really saying it's 1/2 pi times I
05:13
where again computationally this very
05:16
literally means if we plug in PI over 2
05:20
for theta here which would be something
05:22
like one point five seven or so let me
05:25
see if I can I don't think I'll be able
05:27
to get it right on the head but
05:28
something in the 1.57 range if you were
05:31
to play out this whole sum the claim of
05:34
saying that e to the PI halves I equals
05:36
I is that when you play all of this out
05:39
and you add all of the terms together
05:41
you're going to get to the top here but
05:43
usually this is kind of a complicated
05:45
way to think about it people don't want
05:46
to pull up this whole polynomial and
05:47
imagine the spiraling sum I think it's
05:49
helpful just in case you find yourself
05:51
doubting the reality of what we're
05:54
talking about or the fact that it's
05:55
something sensible almost always anytime
05:57
that people are thinking about complex
05:58
Exponential's instead they find
06:00
themselves thinking something like e to
06:04
the I times theta they just immediately
06:07
know of maybe not immediately but you
06:09
come to immediately know of Euler's
06:11
formula that it's telling you you walk a
06:12
distance theta around the unit circle
06:14
which is to say an angle of theta
06:16
radians and whatever point you're
06:19
sitting at on the complex plane that's
06:20
the output so people who knew about this
06:23
would say hmm I as a number is also
06:25
sitting on the unit circle so I just
06:27
have to walk an angle of 90 degrees
06:29
pi-halves radians and from here we can
06:31
actually move forward and get kind of a
06:33
funny-looking answer because that I will
06:35
hop to the inside and we'll be looking
06:37
at e to the
06:38
I haves times I squared I squared is
06:41
negative 1 by definition so this looks
06:43
like e to the negative PI halves that is
06:47
a purely real number you could just plug
06:48
that into your calculator and it's
06:50
actually around 0.2 or so it's about 1/5
06:54
which is weird that you take an
06:57
imaginary number and raise it to an
06:58
imaginary power and that gets you a real
07:01
number maybe it feels a little bit weird
07:03
that a and pi have shown up not that
07:05
much if we're thinking in terms of
07:06
euler's formula and all of that but I
07:10
kind of want to go deeper on this one
07:12
right I think it's one thing to plug in
07:14
some numbers and see what pops out and
07:16
there's a couple great videos who have
07:17
talked about I to the ice from some of
07:19
my favorite channels stand-up math with
07:21
Matt Parker he talked about I to the I
07:22
did a beautiful job black pin red pin
07:25
did another I to the I video truly great
07:27
one thing that I try to you know I tried
07:30
to do is add something into the mix if
07:32
I'm talking about a topic that others
07:33
already have and in this case I think
07:35
one of the most interesting questions
07:36
you can ask before we jump in I mean
07:38
there's lots of interesting things about
07:40
this expression I to the eye but if
07:42
we're seeking intuition thinking about
07:44
how I is a 90 degree rotation right
07:48
that's the way that it acts when we're
07:49
multiplying things on the complex plane
07:50
we think of it very intuitively as the
07:52
thing that upon multiplication turns
07:55
stuff 90 degrees or PI halves radians so
07:58
when you look at this expression that I
07:59
to the power I equals e to the negative
08:01
PI halves it begs the question in what
08:04
sense do two 90 degree rotations combine
08:07
to make this real number this very
08:09
specific real number of e to the
08:11
negative PI halves around zero point two
08:13
zero seven nine on and on and it's like
08:17
those games that sometimes you play with
08:18
arithmetic where someone says oh I'll
08:20
give you the numbers 1 3 4 & 6 see if
08:23
you can combine them in some way with
08:24
addition multiplication division to get
08:26
some other specific number like 24 in
08:29
that case is a particularly fun puzzle
08:30
in this case it's kind of like that
08:32
style of puzzle on steroids how do we
08:34
take the idea of a 90 degree rotation
08:36
and combine it with itself twice in some
08:40
mathematical sense such that we get out
08:42
this number e to the negative PI halves
08:44
where's that intuition coming from and
08:46
the first step I would say
08:49
to remind ourselves of the intuition for
08:52
why it is that oil errs formula works
08:54
why it is that I can take an expression
08:56
like e to the PI halves I and say that
08:59
that's the same thing as I you know I
09:02
just showed what it means in a very
09:03
computational sense what the claim is
09:05
with this spiraling sum but that tells
09:07
us nothing about why why it would be the
09:09
case that this is how things behave now
09:12
I've at this point made you know quite a
09:13
number of videos on variants of this but
09:16
I think the quickest one is to think of
09:17
it dynamically somehow so if we take an
09:19
expression like e to the I times T and
09:21
we're thinking of that as describing a
09:23
dynamic you're gonna have a position
09:25
that changes over time then what I'll do
09:28
is I'm going to start drawing that with
09:30
a blue vector and we know as an initial
09:32
condition that when we raise e to the 0
09:35
that's at 1 so where we are at time T
09:38
equals 0 is equal to 1 but let's say
09:41
that's all we know right we haven't been
09:42
told about Euler's formula or how we
09:45
should be thinking about it with circles
09:46
or anything like that that's just where
09:48
we're starting well one thing we might
09:50
know is that e to the X is its own
09:51
derivative so if we're thinking of
09:53
dynamics here we might ask about the
09:55
velocity of our position taking its
09:56
derivative and because e to the X is its
10:00
own derivative this is quite nice the
10:01
outer expression is gonna stay the same
10:04
and then the inner expression as we're
10:06
applying the chain rule is just I times
10:08
T so we multiply the whole outer thing
10:10
by I and this gives us a literal way to
10:13
read the equation in terms of dynamics
10:17
right what we're saying is take whatever
10:19
your position vector is if you rotate
10:21
that 90 degrees you apply this action of
10:24
I that's going to give you the velocity
10:26
vector so wherever you're standing draw
10:29
a vector from 0 the origin up to where
10:32
you are rotate that vector 90 degrees
10:34
that gives you your velocity and that's
10:36
enough to tell you how to move assuming
10:38
you know that where you're starting and
10:40
of course we have to draw this out in
10:42
the complex plane the fact that we've
10:43
rotated 90 degrees already brings us
10:45
there and I could draw out a bunch of
10:48
different potential position vectors you
10:50
know suggestively maybe putting them on
10:52
a circle and saying really imagine
10:53
yourself sitting at any one of those and
10:55
following the rule for this dynamic and
10:57
what it will tell you is ok I've got a
10:59
sit there rotate my vector 90 degrees
11:02
that does
11:03
my velocity and it doesn't necessarily
11:05
have to be any one of the points on that
11:06
circle any point in the plane if you
11:08
were following this dynamic you say
11:10
rotate that vector 90 degrees which if
11:12
we're drawing a vector field we often
11:13
scale down and it's all well and good
11:15
and here you could phrase this question
11:17
without even talking about Exponential's
11:20
or complex numbers or anything like that
11:21
and in fact I'm going to go back to the
11:23
quiz just to really emphasize that
11:27
Euler's formula and what it's claiming
11:29
and then how we apply it to expressions
11:31
like I to the power I is very it's
11:34
really intuitive as soon as we're
11:35
putting some dynamics into it and we can
11:37
ask questions were moved from the idea
11:38
of Exponential's that actually have the
11:41
same substance in their answers so here
11:44
I want you to imagine starting a walk
11:46
from the point one zero on the
11:48
coordinate plane in such a way that at
11:51
all moments your velocity vector is a
11:53
90-degree counterclockwise rotation of
11:55
the vector drawn from 0 0 the origin up
11:59
to where you are after PI halves units
12:02
of time where will you be on the plane
12:04
okay so kind of think about that think
12:06
about that dynamic and where you'll end
12:08
up after a little bit of time while you
12:20
answer let me see if we have any
12:21
questions that have popped in from the
12:23
audience no questions at the moment at
12:26
least that I see so if ever you do want
12:28
to ask go to Twitter use the hashtag
12:30
lockdown math those will be forwarded to
12:33
me and then sometimes they'll be on the
12:35
screen that I can pull up here and that
12:37
always makes for a fun time for example
12:40
it seems like now we actually have one
12:41
which is how about other solutions to
12:44
the expression Expo x equals I such as
12:46
five PI halves oh I'm so glad you asked
12:48
I'm so very glad you asked we will
12:50
certainly be talking about that and if
12:53
you just if you just hold on for like
12:54
one little moment that's exactly what
12:57
we're going to get to so I'll keep that
12:59
on file for us to pull up in the little
13:02
moment but right now while we're just
13:04
trying to get intuition around the first
13:06
you know there will be multiple answers
13:08
the first answer we saw was e to the
13:10
negative PI halves let's consider this
13:13
lake question so it seems like most of
13:17
you correctly said that this would get
13:19
you at the coordinates 0 1 it's
13:21
essentially walking you a quarter of the
13:23
way around I said late question I'm sort
13:25
of picturing in my head is a big
13:26
circular lake that you're going for a
13:28
walk around because when you have this
13:30
rule for dynamics where your position
13:32
vector rotated by 90 degrees gives you
13:35
your velocity vector that is circular
13:38
motion because the tangent line to a
13:40
circle is always perpendicular to its
13:43
radius so this rule of motion
13:46
corresponds with walking around a circle
13:48
and that's kind of the intuition for
13:49
Euler's formula right that you keep
13:52
increasing that value of T and this
13:54
dynamical rule if e to the I T is going
13:57
to behave according to the derivatives
13:59
that we expect of a function like e to
14:00
the t necessarily walks you around a
14:02
circle so in particular when we're
14:04
wondering how long does it take to get
14:06
to I you would basically say just wait
14:09
for an amount of time equal to well
14:12
whatever the angle to get up there is PI
14:13
halves ok so in this case we're thinking
14:15
PI halves let me see if I can get it
14:17
exactly on the dot ok 1.57 we're
14:19
thinking PI halves is kind of an amount
14:20
of time for these dynamics to get you on
14:23
the number I and that gives us the first
14:26
half of our intuition if we want to go
14:28
back to our formula and we're asking in
14:30
what sense do two 90-degree rotations
14:31
combined to make e to the negative PI
14:33
halves once we're thinking of our
14:38
dynamics here as each one of your
14:41
velocity vectors is a 90-degree rotation
14:43
of your position this base I the way
14:47
that we're kind of thinking about it
14:48
reading it off is something with meaning
14:49
rather than just a number is that as a
14:51
point on the plane it's where you get
14:53
after traveling for PI halves units of
14:56
time according to the dynamics of this
14:58
expression according to the idea that
15:00
your velocity is always a 90 degree
15:02
rotation of your position okay so that's
15:05
awfully nice and in fact there's already
15:06
sort of two different eyes at play here
15:08
which or I guess there's three different
15:10
eyes at play we have this base which is
15:12
describing the 90 degrees that we walk
15:14
around the circle we have the eye that's
15:16
sitting here which is describing the
15:18
rule of rotating your velocity vector 90
15:21
degrees but now we're going to introduce
15:23
another eye which essentially has this a
15:26
of changing what your dynamics are
15:28
because as we go from e to the I times T
15:32
and instead we you know raise things to
15:35
another power of I which I'll just write
15:37
with a little caret hi if we take e to
15:40
the I T and we alter what that
15:42
expression is by raising it to the I
15:43
what we get is e to the negative T okay
15:48
so we have e to the negative T and if we
15:51
try to interpret that with the same sort
15:53
of dynamics that we had above in terms
15:54
of a velocity and a position what that's
15:57
telling us is that the derivative of our
16:02
new dynamic e to the negative T is equal
16:05
to well now the constant sitting in
16:09
front of T is negative one so our chain
16:11
rule will have it be negative one times
16:14
itself times e to the negative T so
16:18
whatever your position is now your
16:19
velocity is negative one times itself so
16:22
the effect of raising to another power I
16:24
it's kind of like we took the dynamics
16:26
we looked at every velocity vector and
16:28
we said rotate another 90 degrees so
16:30
that in this context it would actually
16:32
be in the beginning a velocity vector
16:35
pointing backwards with one unit so if
16:38
you're starting off at the number one
16:40
your initial velocity is to walk
16:42
straight toward zero and as you walk
16:44
even lower if you were sitting at one
16:46
half then you would still be walking
16:47
toward zero but now your velocity vector
16:50
would be negative one times where you
16:52
are which is negative one half and what
16:55
this means for the actual motion that
16:56
this would imply you might kind of
16:58
imagine it
16:58
I haven't animated this nicely yet or
17:01
anything but if you look at this whole
17:02
vector field and you ask each one of
17:04
those vectors take wherever you are and
17:06
rotate another 90 degrees
17:07
counterclockwise they would all be
17:09
pointed in towards the origin so if you
17:12
were to have a little dot move in such a
17:14
way that it's velocity always matches
17:15
whatever victor it's sitting on top of
17:17
what you would end up with is something
17:21
where with each time step it kind of
17:23
takes a step toward zero and it just
17:25
with each of your time steps is walking
17:27
toward zero and each step has a size
17:30
that gets smaller and smaller as you
17:32
actually approach zero and of course in
17:35
practice this would be a bunch of
17:38
phantasmal steps rather than very
17:40
concrete sizing you might be very exact
17:42
about it if you wanted and say what
17:44
we're looking at is scaling down by some
17:47
number just less than one and we're
17:49
doing this in different times and then
17:52
we're gonna multiply that by however
17:54
much time we're waiting and you take
17:57
this as a limiting expression that's
18:00
just a what we talked about in the
18:02
compound interest lecture if you're
18:04
curious and it's kind of the standard
18:04
way to talk about e to powers is this
18:06
sort of limit you could also have that
18:08
te living on the inside here instead if
18:10
you wanted but now if we think about our
18:15
original point I our base what that
18:17
meant it was saying look at our dynamics
18:19
and it's the point you get to when you
18:22
wait PI halves units of time so the
18:25
effect of raising to the eye changes our
18:26
dynamics in such a way that instead of
18:28
walking around a circle we're doing this
18:30
kind of exponential decay we're moving
18:33
toward zero at a slowing and slowing
18:34
rate and the place that you end up after
18:38
PI halves units of time will be e to the
18:41
negative PI halves around zero point two
18:44
zero seven nine so you know that's a
18:48
little bit of intuition for in what
18:50
sense do two 90-degree rotations combine
18:52
to get you this very specific value one
18:54
of them changes your dynamics and the
18:56
original one came from the idea of
18:58
walking PI halves radians so I think
19:00
that's kind of satisfying in its own way
19:02
because otherwise when you look at the
19:03
expression I to the I equals e to the
19:05
negative PI halves it's like what does
19:07
any of this mean this doesn't correspond
19:08
to any you know thing that might see in
19:10
the real world but thinking through the
19:12
intuitions actually have you thinking
19:14
through things like what are the
19:15
dynamics of circular motion what are the
19:17
dynamics were exponential decay things
19:18
that come up in the real world all the
19:21
time but there is a potentially more
19:23
pressing question which was raised and
19:26
let's see if it's still up there well
19:28
whoever asked it a little bit earlier
19:29
what about other solutions to e to the x
19:32
equals I I think that's that's actually
19:35
spot-on and in fact let me let me ask
19:37
you other people to contribute some
19:38
solutions here so let's let's go to our
19:41
quiz let me go to question number three
19:45
which is going to be very similar to a
19:48
question that was already seen which is
19:50
to write a value of
19:52
other than the one we just saw x equals
19:54
PI halves times I so that e to the X
19:57
equals I and feel free to be as creative
20:00
as you want to try to get an answer that
20:02
other people haven't written right just
20:05
to really emphasize that there are
20:06
actually quite a few different options
20:08
here and then in a moment we'll talk
20:10
about why there are quite a few options
20:11
if it's not already clear but I'll give
20:13
you a moment to think of which of the
20:15
many possible values of X you could
20:17
choose is the one that you want to put
20:19
your fingerprint on that you want to be
20:20
contributing to the live stats page that
20:22
we're seeing right now I'll give you a
20:24
little moment
20:42
and as you're answering that we've got a
20:45
question coming in that says wouldn't it
20:47
be more accurate to say that I to the
20:48
power I is I different 90-degree
20:51
rotations not two 90-degree rotations
20:53
after all it's multiplying by I that's
20:56
the rotation so if we stretch the
20:58
meaning of words too thin aren't we
21:00
multiplying by I I times I mean I think
21:02
the most honest answer here is just no
21:04
not at all
21:05
no offense meant but that's just not
21:07
what exponentiation actually means as
21:09
soon as we're extending to the idea of
21:10
complex numbers i I get the intent you
21:15
kind of want to stretch the meaning of
21:16
taking I to the I it's as if you're
21:18
multiplying by itself I different times
21:20
but I I cannot think of a way that that
21:23
actually like satisfactorily makes sense
21:25
in my mind what I do think makes some
21:28
sense is to try to think of the function
21:31
I to the X say what properties do we
21:36
want this function to have and in the
21:38
context of counting numbers you know if
21:39
N and K are just things like 3 & 5 we
21:42
know that it should satisfy this idea of
21:46
multiplying the outputs correspond to
21:48
adding the inputs so we want to say if
21:50
this is any kind of function and you
21:52
know I've said this in many different
21:54
forms in many different places we want
21:56
it to satisfy the property that when you
21:58
add the inputs you multiply the outputs
22:00
that to me is kind of what
22:03
exponentiation is moreso than trying to
22:05
stretch the idea of repeated
22:07
multiplication so it like if it I mean
22:12
correct me if I'm wrong if you can
22:13
definitely if you can find some way
22:15
where if you would read this off in a
22:17
stretching language context saying we're
22:20
taking a 90 degree rotation and applying
22:22
that I different times
22:24
that's not nonsense and that somehow
22:26
like intuitively gets you to the answer
22:28
e to the negative PI halves I'm all ears
22:30
but I think the healthier relationship
22:32
to have is to say we have this central
22:35
property for exponentiation
22:36
which definitely holds in the context of
22:39
repeated multiplication but to extend
22:41
beyond the places where repeated
22:43
multiplication makes any sense because
22:44
it just it just doesn't make sense when
22:47
we're talking about something that's not
22:48
a counting number like I or other crazy
22:50
things that we exponentiate
22:52
like matrices focus focus on the
22:55
property moreso than what some people
22:58
think of as the origins of that property
22:59
or the original intuition behind it
23:02
an interesting question will be you know
23:05
is is there just one such function that
23:07
feels reasonable to write for this
23:08
because you know if we're gonna write it
23:11
as I to the X not only should it satisfy
23:12
this it should also satisfy you know
23:14
when we plug in the number one we get I
23:18
presumably I to the power one however
23:20
we're thinking of this function should
23:21
be I is there just one such function or
23:25
are there multiple such functions that's
23:27
what the quiz question is starting to
23:29
get at and we've got lots of different
23:30
answers which always makes me happy so
23:32
let's see some of the variety now that
23:34
people have thrown in here so we've got
23:36
five PI halves great that absolutely is
23:40
another value that we can plug in for X
23:43
here and just to spell out that a little
23:46
bit more visually if we were to look
23:48
back at our circle here where we've at
23:50
the moment walked for an amount of time
23:53
equal to PI halves which is one point
23:56
five seven what if instead we took
23:57
another full turn and we go another PI
24:00
halves to get us to pi which you know we
24:02
might kind of record that's where the e
24:03
to the PI I value is we walk another PI
24:06
halves we walk another pi halves which
24:08
at this point we will be gone a full
24:10
circle getting us back to one and then
24:12
we walk four five PI halves which
24:14
numerically is about seven point eight
24:17
five yeah that absolutely is another
24:20
number that gets us on top of I and if
24:22
we were to go through the whole
24:23
rigmarole of re expressing I to the
24:28
power I by first writing e to the five
24:33
PI halves I to the power I those eyes
24:38
multiply to become negative and we'd be
24:40
looking at e to the negative 5 PI halves
24:42
which is a very different number right
24:44
we can actually calculate this I'm not
24:47
sure off the top of my head but let's
24:48
take a look at a des most maybe go over
24:51
here let's ask it what is e to the
24:54
negative 5 PI halves zero point zero
24:58
zero zero three eight eight okay zero
25:01
zero zero three eight eight much smaller
25:04
number
25:05
zero zero zero three eight eight which
25:09
begs the question of okay eye to the eye
25:11
what are you right are you about a fifth
25:13
like we saw before around 0.2 or are you
25:17
this much smaller number and in terms of
25:19
our intuition it's basically a a
25:22
question of how we're interpreting that
25:24
base eye right
25:25
are we thinking of it as you wait about
25:27
one point five seven units of time and
25:29
then you translate your dynamics into
25:31
something that looks like decay rather
25:32
than spinning and see where you decay to
25:34
after that amount of time which is 0.2
25:36
or do you wait even longer for about
25:39
seven point eight five units of time
25:41
which is five PI halves and what see
25:43
what happens when you decay for that
25:44
long which gets you to a much smaller
25:46
number but that's not the only answer
25:49
that we could enter right we have other
25:50
people coming in here with negative
25:52
three-halves times i pi which you know
25:56
in terms of a unit circle we could think
25:59
of as saying hey if i want to get to i
26:02
rather than walking ninety degrees PI
26:06
halves radians that way what if I walk
26:09
270 degrees the other way three PI
26:12
halves radians which maybe I'll think of
26:14
as negative because the convention is
26:15
usually the counterclockwise is positive
26:17
that absolutely is another way to
26:19
express it and that would give us a
26:20
different answer if we had e to the
26:22
negative three PI halves I all for the
26:26
power I we go through the same game now
26:28
the I squared cancels with a negative
26:31
that's already there and we have a
26:32
positive three PI halves and numerically
26:37
this gets us an even different looking
26:39
answer from what we had before which if
26:41
we go over and we say hey what is e to
26:44
the three PI not three oh three PI
26:47
halves a hundred and eleven point three
26:50
one very different kind of number than
26:52
what we saw before one hundred and
26:56
eleven point what was it one hundred
26:58
eleven point three one gate had an
27:01
eleven point three one or so and again
27:04
in terms of the intuition well you might
27:06
be asking there is suppose we have this
27:08
rotating dynamic but we move backwards
27:10
in time we see how long ago in time
27:12
would I have to be such that if I played
27:14
things forward from there
27:16
I would land on the
27:17
number one my initial condition and you
27:19
have to go back in time three PI halves
27:21
units and then if you were to translate
27:23
to the decay dynamics which is what
27:25
raising to the AI is doing in this
27:27
context you say if I'm starting at the
27:29
number one but I want to move backwards
27:30
in time and say where should I have
27:33
started if I want to decay down such
27:35
that I end up at the number one after
27:37
three PI halves units of time the answer
27:40
is evidently starting at around a
27:41
hundred and eleven for that kind of
27:43
exponential decay and you can see where
27:45
this is going where there's actually
27:46
infinitely many different values that we
27:48
could plug in for X if we're thinking of
27:51
e to the X as being I and people have
27:53
entered a lot more here
27:55
excuse me throwing my pin onto the
27:58
ground as one does classic for third
28:01
place nine PI halves great choice
28:03
17:29 PI halves y'all are my favorite
28:06
lots and lots of different options
28:08
infinitely many different values which
28:10
feels a little disconcerting at first
28:12
right because we look at an expression
28:13
that seems like you know there's just
28:16
gonna be some computation I just plug
28:17
that into my calculator and see what
28:19
pops out and we've got multiple
28:21
different values for it so what's going
28:23
on here right what's going on and I
28:26
think this really cuts to the idea of
28:28
how we think about Exponential's in
28:29
general right but before that I do want
28:32
to emphasize that this isn't the only
28:33
time in math where we come across a kind
28:35
of ambiguity for how to interpret
28:37
something because if I say something
28:39
like what is the square root of 25 you
28:43
know I think a lot of us say well it's 5
28:46
but if we are saying you know what
28:48
should the square root be it should be
28:50
some number X such that when you square
28:53
it you get 25 well there's two different
28:56
answers to that who's to say that our
28:58
conventions should be that the square
28:59
root function is positive gives us a
29:01
positive number rather than negative 5
29:04
so we have one expression that seems
29:05
like it wants to have multiple different
29:07
values right and this could actually
29:09
happen in another context where instead
29:12
of square roots what if I was asking for
29:14
the fourth root of something like 16
29:17
usually we would think of this as
29:19
positive number two right 2 is a number
29:22
such that when you multiply by itself
29:23
four times you get 16 seems like a
29:25
decent answer to a fourth root but if
29:27
we're thinking of this as answering the
29:29
question what number
29:31
the fourth equals 16 there there is
29:34
another answer to this we could also say
29:37
negative 2 that's a number that when you
29:39
multiply by itself four times you get 16
29:41
but there's another answer you could say
29:44
two times I that seems valid but there's
29:46
another answer you could think negative
29:49
two times I all four of these numbers
29:51
satisfy that property so who's to say
29:53
that the fourth root of 16 should be two
29:57
and the answer ends up being well we
30:00
adopted convention when there's multiple
30:02
options like this when you have a multi
30:04
valued function we often just choose one
30:07
of those values to be what we mean when
30:09
we want to treat it as a function as
30:10
something with a single input and a
30:11
single output in fancier lingo
30:15
this comes up all the time when we're
30:16
dealing with complex numbers the idea of
30:17
something as an operation kind of
30:19
wanting to have multiple values you'll
30:21
sometimes hear the phrase branch where
30:23
you choose a branch of the square root
30:25
function which is to say you choose a
30:27
certain convention and real numbers it's
30:30
nice and easy sometimes because you say
30:32
just choose the positive one but there's
30:33
no notion of which complex members are
30:35
like the positive complex numbers when
30:37
we want to take square roots just to
30:39
give one example let's say we wanted to
30:41
take the square root of I and we want to
30:44
know what should that be because there's
30:46
multiple different answers you know we
30:49
think of I again as this 90 90 degree
30:51
rotation and if we were thinking of it
30:53
as a 90 degree rotation it feels like
30:55
the square root should be you know
30:58
something sitting at a 45 degree angle
31:01
maybe that's the square root of I which
31:03
we could write out very explicitly as
31:04
root 2 over 2 root 2 over 2 I that's
31:09
just using trigonometry but if we were
31:11
thinking of I instead as being a
31:14
negative 270 degree rotation it feels
31:17
like half of that doing half of that
31:19
operation should actually get us on the
31:21
other side maybe the number sitting down
31:24
here should be the square root of I and
31:25
that's actually just the negative of
31:27
what we saw before negative root 2 over
31:31
2 minus root 2 over 2 times I now in the
31:36
context of real valued functions we can
31:37
say yeah just choose the square root to
31:39
be whatever the positive answer is but
31:41
which of these do you consider the
31:42
positive answer you know maybe it feels
31:44
like
31:44
we should can consider this upper one
31:46
because it's coordinates have positive
31:48
numbers but however you try to define
31:50
positive in a nice way here that's going
31:52
to be consistent you know for example
31:54
two positive numbers should always
31:55
multiply to make a positive number
31:56
you're not really going to be able to do
31:58
it the way that you can for real numbers
32:00
and in fact this phenomenon here where
32:02
we're taking roots it's actually the
32:04
same as the phenomenon we were just
32:06
looking at when we were talking about
32:07
multiple values for I raised to the
32:10
power of I because forget I raised to
32:12
the power of I let me ask what much
32:14
might look like it's a much simpler
32:15
question of taking two to the power
32:18
one-half okay
32:20
what's 2 to the one-half yeah I think
32:22
you say well we know what this is we
32:24
kind of define it to be the square root
32:25
of 2 all is well and good but what if I
32:27
said let's approach this the same way
32:29
that we were approaching our eye to the
32:32
eye expression I want to first express
32:34
things as e to the something right and
32:38
then I'm going to raise that to the 1/2
32:40
by multiplying the 1/2 into the exponent
32:42
and I'd say okay I can I guess I can do
32:45
that e to the what is equal to 2 well
32:48
that's the natural log of 2 it's a
32:50
constant which is around zero point six
32:52
nine or so if we raise e to that power
32:56
we'll get to so we could be thinking of
32:59
this as e to the natural log of 2 times
33:02
one-half and if you wanted to if you
33:05
were thinking of e to the X you know
33:09
this might be kind of overkill in the
33:11
context of real numbers but if you were
33:12
thinking of e to the X as shorthand for
33:14
this X function you could plug in the
33:16
value zero point six nine times one half
33:18
which I guess would be around zero point
33:20
three four five is something like that
33:24
you plug in that very concrete value
33:27
into your polynomial see what it outputs
33:28
and it will output around one point four
33:31
one for a nice real number it's square
33:34
root of two what you would expect but if
33:36
we do the same thing we were just doing
33:38
with I and acknowledging that there's
33:39
actually multiple different answers when
33:41
we want to write something as e to a
33:43
power we could also write this this
33:46
might seem funny but we could write it
33:47
as e to the natural log of 2 plus 2 pi I
33:53
that whole thing raised to the one-half
33:57
right after all this value will come to
34:00
equal to you could break it down as it's
34:03
e to the natural log of 2 multiplied by
34:05
e to the 2 pi this one just has the
34:09
effect of rotating things 360 degrees so
34:11
it's just going to equal one so we're
34:13
looking at two times one great that
34:15
feels like a valid substitution and yet
34:18
when we play the same game of taking
34:21
this and raising it to a power and
34:22
treating that by multiplying the power
34:24
into the exponent look at what happens
34:26
we have e to the natural log of 2 times
34:30
1/2 plus well what's 2 pi I times 1/2
34:34
well that will be pi times I now this
34:38
first part e to the natural log of 2
34:40
times 1/2 that will end up being the
34:42
familiar square root of 2 that's all
34:45
well and good but we're going to be
34:47
multiplying that by e to the PI I right
34:50
and quite famously e to the PI I is
34:52
negative 1 so in this case it seems to
34:56
be suggesting that if we are solving
34:58
this expression 2 to the 1/2 by playing
35:01
around with the different answers we
35:03
could plug in for something like e to
35:05
the X equaling 1/2 what we end up with
35:09
is another answer what we might
35:11
traditionally write as this negative
35:13
square root of 2 and here I mean it's a
35:16
little funny for it to have multiple
35:17
values to look at 2 to the 1/2 and say
35:19
that's not equaling one thing but based
35:21
on choices we make it could equal
35:23
multiple different things but the two
35:25
things that it could seem quite
35:27
reasonable if there's going to be
35:28
anything that 2 to the one-half is it
35:30
seems like it should either be the
35:32
positive square root that we're familiar
35:33
with or the negative variant of that
35:34
that doesn't actually seem like such a
35:36
problem and in fact we could um we could
35:39
play this game even further where let me
35:41
ask you for even more creative answers
35:43
to this expression because maybe we can
35:46
find other funny powers of something
35:48
like 2 to the power X as we start
35:52
plugging in various different values of
35:53
X based on what substitution we make if
35:55
we're abiding by the same rules that we
35:57
were using in evaluating I to the power
35:59
I so this time the question asks or it
36:03
specifies that one solution of the
36:05
equation e to the x equals 2 is the real
36:06
number natural log of 2 u ok
36:08
that one we know it it's not boring but
36:10
it's boring in comparison to what else
36:12
we can do can you think of another one
36:14
and can you write some other answer to
36:17
the question e to the x equals 2 and
36:20
again creativity is welcomed so I will
36:23
give you another little moment for that
36:51
all right I will go ahead and walk in
36:53
some answers here if that's all right
36:55
with you I'm not sure how much time it
36:58
necessarily takes to do the math entry
37:00
depending on what device you're looking
37:01
at but don't be too stressed if it's
37:05
before you got the chance to into the
37:07
question that you want into the answer
37:08
that you want it to answer so it looks
37:13
like 131 of you have entered the variant
37:16
where we take Ln of 2 and we add 2 hyah
37:19
and I guess I am in writing this
37:21
question mistakenly like marked one of
37:23
the answers as being correct when in
37:25
fact there's quite a few different
37:26
correct ones
37:27
so that's on me for the fact that I
37:28
don't know if it looks to any of you
37:30
like oh it's red you got it wrong when
37:31
you entered Ln of 2 plus 42 I pi which
37:34
is of course a great choice but you
37:37
could also have something like 4 pi plus
37:39
the natural log of 2 or 6 pi I or really
37:42
any integer multiple of 2 pi I if you
37:45
add that it doesn't affect e to the X
37:48
because it just has the effect of
37:50
multiplying by e to the 2 pi which is
37:52
the effect of multiplying by 1 and again
37:54
this has kind of a funny consequence
37:56
where it seems to output kind of
37:58
reasonable results when we do it as
38:01
another example it looks like the second
38:03
most common entered expression there was
38:05
that we might replace 2 so let's think
38:08
we're thinking 2 to the power of one
38:09
fourth okay there was a suggestion that
38:12
we replaced 2 with e to the natural log
38:16
of 2 plus 4 PI I okay plus 4 PI and we
38:24
raise all of that to the 1/4 all right
38:27
well if you were to play the same game
38:28
you would get e to the natural log of 2
38:32
times 1/4 and we'd be multiplying by e
38:35
to the PI I now the first part of that
38:38
is going to be the usual positive 4th
38:41
root of 2 the thing we mean when you
38:43
plug in an expression like 4th root of 2
38:45
into a calculator a nice small positive
38:47
number but then this second part is
38:49
negative 1 so it seems to be saying you
38:53
know if we returned her for 2 in this
38:55
different way raising it to the 1/4 you
38:57
know it's not the usual answer that we
38:59
get but it's a reasonable answer it's
39:00
another number that when you raise it to
39:02
the fourth power you get 2
39:04
and if we had done this with even
39:07
different values if instead we had been
39:08
using two pi you know it's kind of fun
39:12
to think about how that would have
39:13
changed things if instead of four pi we
39:15
had been adding two pi
39:17
well then over here we would have been
39:19
looking at PI halves times I and instead
39:22
of multiplying by negative one we would
39:25
have instead been multiplying by I which
39:27
again is a valid answer that seems like
39:30
a reasonable output for something like
39:31
two to the one-fourth so when you're
39:33
looking at the fact that I to the power
39:35
I seems to have multiple different
39:36
values for it right we have this funny
39:40
phenomenon where we can plug in e to the
39:43
five PI halves I negative three PI
39:44
halves I and we get what seemed like
39:46
wildly different answers something super
39:48
small something super big all very
39:50
different from the 1/5 approximately 1/5
39:53
answer that we found before up here it's
39:55
exactly the same phenomenon as when
39:57
you're asking something like what's 2 to
39:58
the 1/4 and acknowledging that there's
40:00
actually multiple different solutions to
40:02
the expression X to the 4th equals 2 for
40:05
different solutions in fact and what
40:07
you're looking at is the fact that
40:08
there's multiple different solutions to
40:10
the expression e to the x equals some
40:12
kind of base whether that base is I
40:14
whether that base is 2 whatever it might
40:17
be and one way that we might think about
40:20
this is that we're dealing with real
40:22
numbers
40:23
things are just lovely things are nice
40:25
there's one-to-one relationships you've
40:27
got positives it's great where if we
40:31
want to think about exponential
40:32
functions let me just cover some of this
40:35
stuff up
40:38
we have this nice back and forth where
40:40
you can choose to express any
40:42
exponential as a base to X like 2 to the
40:44
X or you could express that same
40:46
exponential as X of R times X which you
40:49
know that is the polynomial that we
40:52
refer to whenever implicitly referred to
40:55
whenever we write something like e to
40:56
the X and these are lovely back and
40:58
forth because you can just take a
40:59
natural logarithm of B and it gives you
41:01
one answer assuming that B is a positive
41:03
number and that's the same thing as
41:06
saying that X bar is equal to B so one
41:09
way that I've talked about this earlier
41:10
in the series is that if you were
41:12
looking at the family of all possible
41:14
Exponential's
41:15
and we could write them as X both are
41:17
times X and change what R is and this is
41:20
exactly the same thing as writing e to
41:22
the R times X if that's something you're
41:23
more comfortable with so e to the R
41:26
times X X pivara times X those are the
41:28
same thing we could think about changing
41:29
what that is but on the other hand if
41:32
you were to think about all possible
41:34
Exponential's as some base let me do
41:37
base for the power of X and we're going
41:39
to change what that base is at first it
41:41
feels like that's a different kind of
41:43
expression to manipulate but it's just
41:44
another way of expressing the same
41:46
family and a way that you might think
41:49
about this for how do we think about
41:51
what base does it correspond to if we're
41:54
thinking a little bit more abstractly as
41:55
X part times X and there's a reason I'm
41:58
doing this because we're about to apply
41:59
this to complex numbers where it's going
42:00
to look weird er so follow through with
42:02
me here if instead of looking at that
42:03
base one thing I could do is say what is
42:06
the value of x both are right which is
42:11
basically this function but when we plug
42:13
in one so X bar times one if you prefer
42:16
to think of it that way and that is
42:19
being represented by our Green Line and
42:21
what you can see is okay if I get R that
42:24
factor in front of X in my exponential
42:27
function X bar X to be zero point six
42:30
nine which I know is around the natural
42:33
log of two what this means is that X
42:36
both 1 is about 2 and so this
42:39
corresponds with the function that we
42:40
would usually write as 2 to the power X
42:42
right okay and basically as I change
42:46
around my our you know I could try to
42:48
change it to something so that it looks
42:50
like 3 so around 1.1 that exponential
42:53
looks like 3 which we would usually
42:55
write as 3 to the power X I would like
42:57
to argue that it's a little bit
42:59
healthier to think about varying this
43:01
value R rather than varying the base and
43:03
the main reason is that as soon as we
43:05
get to complex contacts and we're
43:07
thinking of exponentiation you have this
43:12
overloading that goes on where if you
43:14
change around what sits in front of the
43:16
X that's all well and good I could have
43:18
X above R times X where maybe are
43:20
something like zero point 6 9 but I
43:23
could shift that down by 2 pi I and that
43:26
doesn't change the base that it would
43:27
correspond
43:28
that would still correspond to two or it
43:30
could shift it up by 2 pi that doesn't
43:32
change the base that it corresponds to
43:33
because in all of those cases when we
43:35
plug in x equals 1 we get the same thing
43:37
however all of these for different
43:40
values of X are distinct functions this
43:42
is why we saw multiple different values
43:44
for I to the power I because I to the X
43:47
is an ambiguous function in that context
43:49
it would be unambiguous if we decided
43:52
which value of R such that what we're
43:55
representing is X of R times X which
43:58
value of R do we choose as soon as we
44:00
choose one it's an unambiguous function
44:02
but at that point it just feels like
44:04
maybe what we want is a to stop thinking
44:06
about things in terms of some base
44:09
raised to the power X maybe as soon as
44:11
we're in the context of complex numbers
44:12
we should just write them all as X of
44:15
some constant times X if for no other
44:17
reason it makes crystal clear how we
44:19
actually plug in numbers if we want to
44:21
do a computation or just to do math on
44:23
top of it we've got this nice infinite
44:25
polynomial that we plug them into and
44:27
I'll make another case for you that this
44:30
is this is maybe the the correct way to
44:32
think about Exponential's as soon as
44:35
we're extending into other domains
44:36
things like complex numbers and for that
44:38
let's just let's just back up go back to
44:41
Oh doorbell something's arrived go back
44:44
to the original way that we extend the
44:47
idea of exponentiation and just think of
44:49
like what is 2 to the X we know how to
44:54
think about this for natural numbers you
44:56
have something like 2 to the 3 repeated
44:59
multiplication how is it that you're
45:01
first taught to think about something
45:02
like 2 to the X for fractional amounts
45:04
or for negative amounts and things like
45:07
that
45:07
well you're usually taught that 2 to the
45:10
1/2 should be something where you know
45:14
if I multiply it by itself and this
45:17
follows the usual rules that
45:19
Exponential's do with counting numbers
45:20
where we're able to add things in that
45:21
exponent I should get 2 to the 1 so it
45:25
should be some number that when I
45:26
multiply it by itself I get 2 and you
45:29
know at that point you have a choice
45:30
maybe it's positive maybe it's negative
45:32
but if you always decide to make the
45:35
positive choice you're going to be able
45:36
to get a nice continuous function out of
45:37
this same deal if we asked about
45:39
negative numbers what should 2 to the
45:41
name
45:41
one be well that should be something
45:43
where when I multiply it by two to the
45:44
one it gets me two to the zero and
45:47
that's kind of the justification for our
45:49
convention that negative exponents look
45:51
like one half but what's really going on
45:54
here is we're saying whatever this is it
45:56
should be some kind of function that
45:58
satisfies this property f of a plus B
46:02
equals F of a times F of B and moreover
46:07
the fact that the base is 2 is basically
46:09
telling us that it's not just any such
46:11
function it's a function where when we
46:13
plug in 1 we get 2 and just as a little
46:17
you know sanity check style question to
46:20
see if you're following along with some
46:21
of the implications here I want to ask
46:23
you what is I'm gonna call it like a
46:27
softball but this is this isn't meant to
46:29
be like an incredibly deep question
46:32
necessarily it's just more of a check if
46:34
you're following along with the idea of
46:36
abstractly starting with properties of a
46:39
function and then kind of deducing ways
46:40
that we might want to write it down
46:42
based on those properties if f of X
46:44
satisfies this exponential property f of
46:47
a plus B equals F of a times f of B for
46:49
all inputs and it also satisfies F of 1
46:52
equals 2 which of the following is true
46:54
which is to say which of the following
46:56
is necessarily true no matter which such
47:00
function you're starting with and those
47:02
of you who remember which lecture was it
47:04
it's whichever one we were talking about
47:05
how to enter what Euler's formula is
47:07
really saying I asked a question of this
47:09
style where I neglected a single
47:12
condition you know I didn't write down
47:13
the fact that we want to make sure f of
47:16
X is nonzero everywhere and then that
47:17
caused some amount of confident which is
47:20
cool could come follow mint on screen
47:21
that happens to all of us but the the
47:24
intent of it was to basically show that
47:25
this abstract property have something
47:28
that turns addition into multiplication
47:29
is
47:32
it is enough to basically make you want
47:35
to write the function as whatever it
47:37
equals is 1 raised to some kind of power
47:39
this is the the spirit of the question
47:44
now we've got a couple questions
47:47
actually about power towers that seem to
47:49
have popped up here which is great
47:50
connected to last time let's let's hold
47:54
off on the power tower question for just
47:56
a moment so that we first get like a
47:58
deeper feel of like what exponentiation
48:01
should mean here because because we can
48:04
be what I want to claim is we can answer
48:05
it in like multiple different ways so if
48:08
you give me just one we'll talk about
48:09
power towers and then just as a number
48:12
line can be represented in a logarithmic
48:13
scale can the same done be done for a
48:15
complex plane maybe mapping the complex
48:17
plane onto an infinite cylinder in the
48:19
logarithmic sense yeah yeah in fact
48:23
there's a visualization that I'm going
48:24
to get to in just a moment here where we
48:27
do something quite similar to that
48:28
because what we'll do is play around
48:31
with different exponential functions X
48:33
bar times X but we're going to change
48:35
that value of R which is going to be
48:37
represented by a little yellow dot so
48:39
we'll kind of talk through this it's not
48:41
going to map the whole plane but just a
48:42
couple sample points from the real axis
48:44
and the imaginary axis but the idea is
48:46
that as we move around what that
48:47
constant is we're going to be able to
48:49
kind of visualize the different things
48:51
that it does to the plane and
48:53
effectively it's like it's turning the x
48:56
axis into a logarithmic scale and then
48:58
wrapping the imaginary axis along a
49:00
circle and then as soon as that value of
49:01
R becomes imaginary it swaps the role of
49:04
those real numbers get put on the circle
49:06
and imaginary numbers get put on a
49:08
logarithmic scale positive axis so great
49:11
question all three of which I guess are
49:13
sort of jumping the gun ahead for where
49:15
I want to go but nice to see that's
49:16
where people are thinking so on this one
49:18
let's go ahead and just create it the
49:20
idea is that this this property of f of
49:25
a plus B ends up letting you express a
49:27
lot of different things purely in terms
49:28
of what F of 1 is and just to spell that
49:32
out very explicitly something like f of
49:36
5 is the same thing as F of 1 plus 1
49:41
plus 1 plus 1 plus 1
49:43
which is the same thing as f of 1
49:45
multiplied by itself 5 times because of
49:48
this property which if F of 1 is 2 is
49:53
the same as 2 to the power 5 and then
49:55
something like F of negative 5 it should
49:58
be the case that when we multiply it by
50:00
F of 5 we get whatever F of 0 is and
50:05
it's not immediately clear what F of 0
50:06
is but we could say that F of 1 plus 0
50:10
is equal to whatever F of 1 is times
50:14
what F of 0 is but F of 1 is equal to 2
50:17
and so this is also equal to 2 so we're
50:20
saying 2 is equal to 2 times something
50:22
well that something has to be a 1 so in
50:24
this context this guarantees that F of
50:26
negative 5 is 2 to the negative 5 it's 1
50:30
over 2 to the fifth so we can explicitly
50:32
write this as 2 to the negative 5 which
50:35
is all to say these two properties
50:36
together make us really want to write
50:39
the function as 2 to the X because any
50:43
counting number that we put in it's
50:45
going to satisfy it's it's going to look
50:48
like 2 multiplied by itself that number
50:50
of times any fractional number we put in
50:52
it's going to satisfy these properties
50:54
that we wanted and you might wonder is
50:56
that unique and in the context of real
50:58
valued functions it actually would be
51:00
but in the context of complex valued
51:03
functions there would be multiple such
51:04
functions f that we could write for this
51:06
one of which is what we were looking at
51:08
before where we could have a function
51:10
defined to be X of the natural log of 2
51:16
plus 2 pi all of that times X ok forgive
51:25
the sloppiness here I just get excited
51:27
writing about this and this is actually
51:29
a different function as evidenced by
51:31
what happens if you plug in x equals 1/2
51:33
we saw a little bit earlier how when you
51:36
plug in 1/2 what you get is the negative
51:37
square root of 2 and then if you plug in
51:39
1/4 you get not the fourth root of 2 but
51:43
I times the fourth root of 2 so it is a
51:45
different function but it still
51:46
satisfies these properties and it kind
51:48
of makes us want to write it as 2 to the
51:50
X and it makes it suggests that maybe 2
51:52
to the X is an ambiguous bit of notation
51:54
and we should just write everything
51:56
in terms of X of our times something but
51:59
you might wonder well you know maybe
52:01
we're just not being creative enough
52:03
with all of the functions that satisfy
52:04
this property maybe there's an ambiguity
52:06
when we write X bar times something and
52:08
there's different values of R that could
52:09
come into play but I'm just going to put
52:12
down a little claim and then maybe give
52:14
like a sketch of what the proof would
52:15
look like if you want which is that
52:17
let's say you have some complex function
52:19
f and it satisfies the following
52:20
properties first you're able to take a
52:23
derivative of it it's differentiable
52:24
which just keeps it from being some you
52:27
know totally messy discontinuous thing
52:29
that's like taking on some random values
52:31
depending on you know the span of
52:34
whatever vector space over
52:36
I don't know fractional amounts you
52:38
might want to think of in crazy ways
52:39
it's a nice function that's
52:40
differentiable it's not equal to zero
52:43
everywhere so the condition that sort of
52:46
slipped my mind and I forget which
52:48
lecture lecture four or something like
52:49
that
52:50
and then it has this central property
52:51
that it turns addition into
52:53
multiplication if you have such a
52:55
function I claim that there is a unique
52:58
maybe have to really specify there exist
53:00
a unique complex number R so that you
53:08
could write f of X as basically being
53:09
this exponential function of R times
53:12
that value X which is you know basically
53:15
saying that if you have expose a
53:17
function this infinite polynomial with
53:20
nice derivative properties and all of
53:21
that if you have this you have every
53:24
exponential that you want in a very like
53:25
abstract generic sense of the word
53:27
exponential just based on a property
53:29
that we could want from it and the
53:31
sketch of the proof would look something
53:32
like this if you want to first look at
53:35
what is the derivative of this value
53:37
which we're assuming exists everywhere
53:38
and and you explicitly write out what
53:41
the limit of that is I'll just talk to
53:43
her very quickly here for those who want
53:44
to like pause and think through the
53:45
details feel free to the central
53:48
property that we have lets us expand out
53:51
the f of X plus h term so we're thinking
53:54
you know a change slight change to the
53:56
output over the change to the input that
53:57
cost it that's what DF DX unwraps to and
54:01
because we can factor that out we can
54:03
factor f of X out of the expression
54:05
entirely and the whole limit is
54:08
only in terms of H which if you think
54:11
about what it means in the context of
54:13
derivatives and the fact that F of 0
54:14
necessarily equals 1
54:16
this whole limiting expression is just
54:20
some constant but more specifically it's
54:22
whatever the derivative of our function
54:23
at 0 is so you have this funny thing
54:25
where if you know it's derivative at 0
54:27
that determines what its derivative is
54:29
everywhere and in the context of
54:31
exponential functions this is hopefully
54:32
quite familiar because all with all that
54:34
we're really saying is the derivative of
54:36
an exponential function is proportional
54:38
to itself and that proportionality
54:40
constant is equal to whatever the
54:42
derivative at 0 is this is all very
54:44
abstractly phrased and such but but the
54:46
purpose of it is to emphasize that it's
54:49
not necessarily just functions that we
54:51
already think of as a to the power X but
54:53
it is a potentially much more broad
54:55
class of functions that to satisfy this
54:57
abstract property of turning addition
54:59
into multiplication but if you have that
55:02
it actually guarantees that you also
55:03
have a second derivative and for that
55:07
matter a third derivative and such
55:08
because the derivative function is just
55:10
proportional to itself so in order to
55:12
take the nth derivative you just look at
55:15
that proportionality constant and raise
55:17
it to the power N and then from here you
55:21
could do a Taylor series expansion and I
55:23
might leave that as sort of the advanced
55:25
homework for those of you who are
55:26
comfortable with Taylor series and that
55:27
idea especially if you want to intermix
55:30
the idea of any differentiable function
55:34
that's differentiable in a sense of
55:35
complex numbers which is sort of a
55:37
definitely college topic you know you
55:41
you you you could intermix the reasoning
55:42
there as you want but fuzzy reasoning is
55:44
allowed in the context of someone who
55:46
only knows about Taylor series and
55:47
nothing else to take this idea and look
55:50
at the Taylor expansion for F and kind
55:52
of justify the idea that there's a
55:55
unique complex number such that our
55:57
function f can necessarily be written
55:59
like this
56:00
and then the connection to normal
56:03
Exponential's is whenever you have such
56:06
a value R we do essentially what we do
56:09
in the complex context of real numbers
56:11
is if you look at expo that function f
56:13
that value R and write that as a base it
56:18
feels like you should be able to write
56:19
that as B to the X
56:21
but the whole the whole point here of
56:23
course is that when we play this game
56:24
and you're trying to interpret something
56:26
like I to the X that's an ambiguous
56:27
function because there's lots of
56:30
different values of R we can interpret
56:32
that to mean not just X both PI halves I
56:35
times X but we could also interpret it
56:38
to mean expo of five PI halves I times X
56:44
and these are separate functions and
56:46
there's an infinite family of separate
56:48
functions that feel like we should write
56:50
them as I'd to the X so the expression I
56:52
to the I unless you've adopted a
56:53
standard for what that's necessarily
56:55
going to mean when you say it has
56:57
infinitely many outputs another way to
56:58
think of that is that the function I to
57:01
the X with the notation we have is a
57:03
little bit ambiguous now with all of
57:05
that let's let's just start visualizing
57:06
some of this because I think that's fun
57:08
and you know you you tell me if this is
57:11
if this is a helpful visual or a more
57:14
confusing visual but what we're going to
57:16
do is look at this function X bar times
57:19
X which is basically this is another way
57:23
to write e to the power of X in fact I
57:25
think I I think I rendered a different
57:27
animation at some point that specified
57:29
that because I was planning on planning
57:32
on doing that
57:32
so let me oh yeah there you are get back
57:35
in my file system get back to where
57:36
you're supposed to be get on in there is
57:42
a complaining because there's multiple
57:43
different it's gonna be like there's a
57:46
oh replace it shows up on the other
57:49
screen wait why is it ok place place
57:54
whatever you see there and now we go
57:58
back to all of that all of that just so
58:00
that I could have nicely written out if
58:02
you're uncomfortable we're thinking of
58:03
it as X of R times X this infinite
58:06
polynomial just in the back of your head
58:07
e to the R times X and we're gonna vary
58:09
around R so I'm going to follow the
58:12
points of the imaginary axis and I'm
58:14
gonna follow the points of the real axis
58:16
and let's see what this does
58:19
well that's all kind of fast so let me
58:21
think through it a little bit more
58:22
slowly all of the negative numbers
58:24
anything that's a negative real number
58:25
is going to get squished into the range
58:27
between 0 and 1 which would make sense e
58:30
to the negative B to a negative real
58:32
number is
58:33
thing between 0 & 1 and we're
58:35
specifically tracking F of negative 1
58:37
which is going to show up around
58:38
whatever 1 over E is around 30 0.37 F of
58:43
1 lands on e as expected that's what X
58:47
both 1 is f of I is going to land a 1
58:51
Radian around the unit circle and it's
58:53
kind of fun to follow along the whole
58:54
imaginary axis here how the imaginary
58:57
axis gets wrapped around a circle and
58:59
what happens as we tweak this value of R
59:03
that's determining not just that we're
59:05
talking about an exponential function
59:06
but which exponential function there's a
59:08
nice one to one correspondence between
59:09
all the exponential functions we might
59:11
want and values of R here it stretches
59:15
things differently so when we put it up
59:16
to two you know it stretches out the
59:18
real axis a lot more so that f of one
59:20
ends up around where Y squared is a
59:22
little above seven f of negative one is
59:24
much closer to 0 F of I is a two Radian
59:28
rotation around the circle F of negative
59:30
I is a negative two Radian rotation and
59:33
of course we can get to our favorite
59:34
formula that if that were pi that we had
59:38
as our scaling constant then the real
59:40
axis gets stretched out quite a lot you
59:42
know F of 1 is sitting off at e to the
59:45
PI which is very close to 20 plus pi
59:47
which is always fun and F of negative 1
59:49
extremely close to zero so it's really
59:50
stretched out that real axis and it's
59:53
also stretched out things in the unit
59:57
circle direction so that getting to F of
59:59
I or F of negative I walks halfway
60:01
around the circle so that's all well and
60:03
good now how would we think about a
60:05
function like 2 to the X which is what
60:09
we would also write as X of X above the
60:13
natural log of 2 times X so if we kind
60:16
of move our yellow dot representing the
60:18
value of R to around 0.69 still no
60:22
imaginary part just a real number zero
60:23
point 6 9 or so that's the natural log
60:26
of 2 well you can see that F of 1 lands
60:28
on 2 which is why we want to call this
60:30
function 2 to the X F of 1/2 actually
60:33
sorry F of negative 1 lands right on 1/2
60:36
F of I it's some walk around the unit
60:39
circle very specifically it's going to
60:41
be 0.6 9 radians around the unit circle
60:43
and now we could have a little bit more
60:46
fun and say
60:47
what would happen if we were to change
60:50
this to instead of being 0.69 instead of
60:52
being the natural log of to make it I
60:54
times the natural log of two so that
60:56
we're really thinking of something that
60:58
might have an exponential base to it
61:00
this would be what we might think of as
61:02
two times I raised to a power well if we
61:07
move that yellow dot which is
61:08
representing our off of the real axis
61:11
and on to the imaginary axis it swaps
61:13
the roles of the teal dots and all of
61:18
the maroon dots in this context which
61:20
remember came from being the positive
61:21
and imaginary axes and it should make
61:23
sense that it swaps their roles because
61:24
what does it mean if we take the input
61:25
space and we multiply it by I
61:27
it means we're rotating that input space
61:29
so everything that was the real number
61:31
axis turns into the imaginary axis and
61:33
everything that was the imaginary axis
61:35
is getting turned into the real axis so
61:38
for us what that means is our new
61:40
exponential function where our value of
61:42
R is now purely imaginary takes all of
61:44
the real numbers and it just wraps them
61:46
around a circle and it takes all the
61:47
imaginary numbers and it's putting them
61:49
onto the real number line so in
61:52
particular let's say we scale this thing
61:53
up so that we're sitting at around PI
61:56
halves times I well what does that
61:58
actually mean that means it takes the
62:00
real number 1 to the value I which is
62:03
the sense in which we want to write this
62:05
function as I to the power X right it
62:07
just tempts us to write it not as this
62:09
abstract looking thing X bar times X
62:11
where R is equal to PI halves I know no
62:14
we just want to write it as I to the X
62:16
even if that's a little ambiguous what
62:18
that really means it's just that the
62:19
function we're dealing with outputs I at
62:21
1 and if it's an exponential function
62:24
that and we're asking what does it do to
62:28
the value I what is I to the power I in
62:31
this case it shoves it to around 0.2
62:34
around 1/5 but there's many different
62:37
exponential functions that would have
62:38
this property of putting F of 1 onto the
62:40
number I so if we were to scale it up
62:42
even further I don't think I haven't
62:44
animated here but if we were to take
62:45
that yellow dot and raise it up until it
62:47
got to 5 halves times pi I what you
62:50
would see is that the unit circle is
62:53
rotated around on itself so that F of
62:55
negative 1 would rotate around another 2
62:58
pi radians and land
62:59
where it is but it would stretch out the
63:01
real axis a lot more which was the sense
63:04
in which another output of eye to the
63:05
eye is a much much smaller number
63:07
it was around what was it 0.0003 or so
63:11
but we can also see what I think is
63:13
quite fun what happens if we consider
63:15
alternate expressions that we want to
63:18
interpret as two to the power X right so
63:21
when R is a purely real number the
63:23
natural log of two kind of makes sense
63:24
that when you plug it in here the
63:27
expression we get is what we want to
63:29
write as 2 to the power X but what if we
63:31
start moving it in the imaginary
63:32
direction ok and what I'll first do is
63:35
I'll move it up by PI I units now what's
63:39
going on here we have X bar times X and
63:42
R is equal to this value which is the
63:44
natural log of 2 plus pi times I what
63:47
that means is that when we plug in 1 f
63:49
of 1 is that negative 2 so we want to
63:52
write this function as negative 2 to the
63:54
power X right and that's actually
63:56
something that you know it's it's a
63:59
little deceptively simple when we write
64:01
a negative number to a power negative 2
64:06
to the power X it doesn't at first look
64:09
like this necessarily it brings us into
64:11
the complex numbers in any way but of
64:13
course when we plug in even a value like
64:14
1/2 where we're kind of asking for a
64:17
square root of negative 2 we realize
64:20
that we want to write this as something
64:21
like I times the square root of 2 but if
64:26
you were to look at this function
64:27
negative 2 to the power X in the full
64:29
complex domain that is dealing with what
64:31
you're looking at is a function that
64:32
takes the value of 1 to negative 2 and
64:34
if it does that what it does to the rest
64:36
of the real number line is it kind of
64:38
spirals it outward so we see that F of
64:40
negative 1 sits at negative 1/2 about
64:43
where you would expect if you were to
64:45
follow to F of 1/2 it would sit exactly
64:49
on the imaginary line and F of 1/2 would
64:51
be square root of 2 all my Mouse's not
64:54
where I want it to be it would be around
64:55
a square root of 2 times I and as you
65:00
continue further on this is showing you
65:02
all of the real value powers of negative
65:05
2 to the X it necessarily spirals around
65:08
but we could also move our value of R
65:11
even higher and get it up to around
65:13
cow times I around six point two eight
65:15
times I and in that context this is
65:19
another function that we would want to
65:20
write as something like two to the X
65:22
because for any hole number two hole
65:26
number that you plug in for X it will
65:27
look like repeated multiplication and it
65:29
even has kind of reasonable values for
65:31
things like 1/2 where it spits out the
65:34
negative square root instead of positive
65:35
square root but what it's actually doing
65:37
is a transformation to the plane where
65:39
it puts everything is the real number
65:41
line ends up being a very tightly wound
65:43
spiral that goes around and it just
65:45
spirals in such a way that f of one
65:46
lands right on the number two so it is
65:50
in that sense that we could say two to
65:53
the X is is plausibly interpreted as a
65:56
separate exponential function from the
65:58
one that we are traditionally used to so
66:01
I think with all of that I will I will
66:04
leave things for today and I'll just
66:06
leave you with a couple lingering
66:07
questions to think about ok so if you
66:11
want to think of eye to the eye as being
66:13
a multivalued expression right you could
66:15
you could say we adopt a convention
66:17
fancifully you'd say you choose a branch
66:19
of the natural logarithm function and
66:21
maybe that locks you into this being e
66:23
to the negative PI halves but if you say
66:25
this kind of wants to be infinitely many
66:27
different values like the various ones
66:29
that we saw how many values does 2 to
66:32
the 1/3 want to be in the same sense
66:35
where we are replacing 2 with various
66:37
different various different options for
66:40
e to the X such that e to the x equals 2
66:44
how many different values does that want
66:46
to be or how many values does 2 to the
66:48
3/10
66:49
want to be phrased differently of all of
66:53
the let me say of all of the exponential
66:55
functions f of X which satisfy Oh have I
67:00
written it down somewhere f of X that
67:03
satisfies all of these properties that
67:06
I've written so if it satisfies all of
67:08
these and if f of 1 is equal to 2 all
67:14
right how many different outputs are we
67:16
going to get when we plug in x equals
67:17
3/10 for the various options for what
67:19
function that is and how many outputs
67:21
are we going to get for 2 to the PI for
67:25
the
67:25
functions the two to the X could
67:27
represent if we're thinking of two to
67:28
the X as some kind of exponential
67:30
function exponential in the sense of
67:32
these sort of abstract properties and if
67:37
we yeah if we if we have a class of
67:41
different such functions and we want to
67:43
plug in PI it makes me laugh just
67:44
because it's such a and a kind of a
67:46
funny answer that pops out as you're
67:48
trying to think about it so those are
67:50
the questions that I'll leave you with
67:51
and I think this is you know my my
67:56
central question in approaching today's
67:57
lecture was whether I wanted it to be
67:59
kind of describing like these abstract
68:02
properties of exponential functions and
68:03
it's just cool to me that starting from
68:05
those abstract properties you get locked
68:07
into the idea of e to the rx or more jet
68:10
you know I think more honestly but an X
68:12
of R times X for different values of R
68:14
that it locks you in that far but it
68:17
doesn't lock you in as far as having an
68:18
unambiguous notion of what 2 to the
68:21
power X should be much less something
68:23
like I to the power X the risk in that
68:25
of course is that sometimes people don't
68:27
love abstraction and sometimes it
68:29
doesn't come off as approachable but if
68:32
that's the case you know you just let me
68:34
know I think I think there's a whole
68:36
interesting circle of thoughts that
68:37
surrounds all of this stuff to include
68:39
power towers because if you want to
68:41
actually talk about power towers like we
68:44
were last time in the context of complex
68:45
numbers or even with negative bases you
68:47
have to be thinking through things like
68:49
this so it was a question that we had up
68:51
on screen yeah what happens if we do
68:54
this for I to the power I tetration you
68:56
know let's just try this let's just go
68:58
ahead and try a power tower where we're
69:00
raising I to a given power and see what
69:03
what pops out of it so I wasn't planning
69:05
on doing this but we can we can always
69:07
pull up Python and essentially do what
69:09
we were doing last time so the way that
69:12
this would work is we were starting off
69:14
with some base value and then for some
69:17
kind of range what were we doing we were
69:20
taking a and we're going to reassign it
69:22
to be whatever the base which in this
69:26
case is I raised to the power of a
69:29
should be okay cool so we're going to do
69:31
that and then we're going to print off
69:33
the value of a and let's just do this
69:34
for yeah it's a much bigger number
69:38
like 200 so it seems like what happens
69:44
is there's there's potential for chaos
69:48
but these things like sometimes it's not
69:50
that you've landed on a stable a stable
69:53
value or it's not even that you've
69:54
diverged it could be that you're
69:56
bouncing between a cycle of values or
69:58
that you're like literally bouncing in a
70:01
way that's it's not periodic or anything
70:03
and it's actually chaotic I that doesn't
70:06
happen for I but it's a thing to
70:08
potentially look out for it looks like
70:10
it does kind of stabilize maybe there's
70:13
some little subjection to numerical
70:16
error but we stay pretty consistently
70:18
around something with a real part of 0.4
70:20
3 and 0.36 now what I would want to
70:25
emphasize though is this expression so
70:26
let's set a back to be equal to 1 this
70:29
expression of taking I to the power of a
70:30
remember that's a little bit ambiguous
70:32
it depends on what choice of the
70:33
function I we actually have so let me
70:36
let me import numpy so I have the
70:40
exponential function let me go for our
70:43
big range like we had before rather than
70:46
writing it as you know something that's
70:48
like I to the power of X I'm gonna write
70:50
it as the exponential function of a
70:52
different constant write a different
70:55
constant that I'm gonna make I want it
70:58
to be 5 PI halves so I'll do five PI
71:02
halves times I so it's a complex number
71:06
and it's got five PI halves as the
71:09
imaginary part so this is five PI halves
71:11
times I and what am i doing I'm
71:13
exponentiating that so I want to
71:15
multiply a onto the inside there okay
71:18
this is basically another way that you
71:20
could interpret the expression I to the
71:22
X ffensive LY you would say you've
71:24
chosen a different branch of the natural
71:25
log function but it is another function
71:28
which we could iterate on itself and see
71:30
what happens and we might get a
71:32
different result
71:32
I'm very interesting okay so we actually
71:34
have a different result it looks like
71:36
what ends up happening is it bounces
71:37
between two values oh wow is it Period
71:41
three that's interesting period three
71:43
implies chaos so we've got seven point
71:45
three five and zero than point nine nine
71:47
so it looks like it gets into the cycle
71:50
of bouncing between
71:51
three separate values even though in in
71:55
theory in both of those cases we were
71:57
doing something that was I to the ex
71:59
iterated on itself it has everything to
72:01
do with what actual function you think I
72:03
to the ex is referring to so in that
72:06
sense the power tower question is
72:08
ambiguous usually you just choose it to
72:10
be the pi-halves i variant but it's more
72:12
fun to see that it can be multiple
72:14
things all right we have addition
72:15
multiplication exponentiation tetration
72:17
can we think of something in between
72:19
halfway between multiplication and
72:21
exponentiation oh that's a I mean I love
72:25
the spirit of it because each one of
72:26
them it comes down to like a discrete
72:28
step of you're repeating the previous
72:30
operation but often times in math when
72:32
you have something initially defined in
72:34
terms of repetition like exponentiation
72:36
you can extend it beyond that I can't
72:38
think of anything off the top of my head
72:39
but that's an interesting question I
72:41
don't know if that's been that's an
72:43
extended notion I mean you have things
72:45
like fractional derivatives you have
72:46
things like fractional and the complex
72:48
exponents so it doesn't seem outlandish
72:50
but I'm not familiar with one myself and
72:52
then lastly do complex numbers for the
72:55
power of complex number values arise in
72:57
physics and if yes how does one decide
72:59
which values are valid so I can't think
73:03
of if they necessary necessary seen
73:06
switching I can't think of if they come
73:09
up in physics in like a direct sense and
73:11
this is probably just because I'm not a
73:12
physicist so I would say like ask ask
73:15
your neighborhood physicist and see what
73:17
they have to say
73:17
I mean clearly the point of teaching the
73:20
lesson here is that the thoughts that
73:21
you have to go through to make sense out
73:22
of it do build a stronger relationship
73:24
with Exponential's things like it's not
73:26
just more natural to represent them as e
73:28
to the RX once you get to complex
73:30
numbers you kind of have to represent
73:31
them that way
73:32
whereas previously in like physics or
73:34
other real-world contexts it's something
73:35
that's just nice