---
title: 'Intuition for i to the power i | Ep. 9 Lockdown live math'
source: 'https://youtube.com/watch?v=pq9LcwC7CoY'
video_id: 'pq9LcwC7CoY'
date: 2026-08-10
duration_sec: 4418
---

# Intuition for i to the power i | Ep. 9 Lockdown live math

> Source: [Intuition for i to the power i | Ep. 9 Lockdown live math](https://youtube.com/watch?v=pq9LcwC7CoY)

## Summary

In this episode of Lockdown Math, Grant Sanderson explores the expression i to the power i, revealing its surprising real-valued result and the deep ambiguities in complex exponentiation. He uses dynamic visualizations and interactive quizzes to build intuition for Euler's formula and the multi-valued nature of complex exponentials.

### Key Points

- **Introduction to i^i** [00:00] — The video introduces the expression i^i as a culminating topic for the series, highlighting its counterintuitive nature and the need to rethink exponentiation beyond repeated multiplication.
- **Rewriting the base in terms of e** [01:09] — To make sense of i^i, Grant suggests rewriting the base i as e^x, leveraging the definition of the natural logarithm and the properties of exponential functions.
- **Exponential as an infinite polynomial** [02:06] — The exponential function is defined as an infinite polynomial: 1 + x + x^2/2 + x^3/6 + ... This definition allows plugging in complex numbers, making sense of e^(iθ) as a sum of vectors.
- **Euler's formula and the unit circle** [05:06] — Euler's formula e^(iθ) = cos(θ) + i sin(θ) is presented as walking a distance θ around the unit circle. For i, this corresponds to θ = π/2, giving e^(iπ/2) = i.
- **Computing i^i** [06:31] — By substituting i = e^(iπ/2) into i^i, we get e^(iπ/2 * i) = e^(-π/2), which is a real number approximately 0.2079.
- **The puzzle of two 90-degree rotations** [08:07] — The result e^(-π/2) is puzzling: how do two 90-degree rotations combine to yield a specific real number? This motivates a deeper look at the dynamics behind Euler's formula.
- **Dynamics of e^(it)** [09:23] — e^(it) can be interpreted as a position vector whose velocity is a 90-degree rotation of itself, leading to circular motion. This dynamic view makes Euler's formula intuitive.
- **Effect of raising to the power i** [14:29] — Raising e^(it) to the power i changes the dynamics: the velocity becomes negative one times the position, resulting in exponential decay toward zero. After π/2 units of time, the position is e^(-π/2).
- **Multiple solutions for e^x = i** [19:33] — There are infinitely many values of x such that e^x = i, e.g., 5π/2 i, -3π/2 i, etc. Each leads to a different value for i^i, such as e^(-5π/2) ≈ 0.000388 or e^(3π/2) ≈ 111.31.
- **Ambiguity in multi-valued functions** [28:34] — Similar to square roots, expressions like i^i can have multiple values. The choice of which value to use depends on conventions, like choosing a branch of the logarithm.
- **2^(1/2) also has multiple values** [32:15] — Even simple expressions like 2^(1/2) can be ambiguous: by rewriting 2 as e^(ln 2 + 2πi), we get another valid square root, -√2. This shows the ambiguity is not unique to complex numbers.
- **Exponentials as exp(r*x)** [40:38] — All exponential functions can be written as exp(r*x) for some constant r. This representation is more fundamental and avoids ambiguity when dealing with complex bases.
- **Uniqueness theorem for exponential functions** [52:05] — If a function is differentiable, non-zero, and satisfies f(a+b) = f(a)f(b), then it must be of the form exp(r*x) for a unique complex number r. This theorem justifies the exp(r*x) representation.
- **Visualizing exp(r*x)** [57:04] — An animation shows how varying r transforms the complex plane. For real r, the real axis is stretched; for imaginary r, the real axis wraps around a circle, illustrating the behavior of complex exponentials.
- **Conclusion and further questions** [01:06:01] — The video concludes by emphasizing that i^i is multi-valued, and the choice of branch affects the result. It also touches on power towers and the ambiguity in iterating i^x.

### Conclusion

The expression i^i is not a single number but a multi-valued expression, depending on the chosen branch of the logarithm. The video encourages a shift to thinking of exponentials as exp(r*x) to avoid ambiguity and build a stronger intuition for complex exponentiation.

## Transcript

Welcome back to what is now the second to last in the Lockdown Math series. So we've been talking a lot about exponential functions and complex numbers and the interplay of the two of them, and I thought what might be a very nice culminating
topic to help bring it all together and show really the full breadth of what exponential functions are and what they can be would be to talk about raising i to the power i. Now this is a, this is a funny little expression because usually if we
you know, we know how to think about it in terms of repeated multiplication. We're just multiplying something by itself four different times. And in general you'd like to think of i to the x as some variant of that,
but as soon as x becomes a complex number, we should realize that it's really nothing So it's really nothing like this.
be very related to the way that we think about raising e to various different numbers, And in fact if we go, oh no, that's not sure where things are going too much here,
if we go ahead and rewrite that base i in terms of e, it can help us make sense out of this expression. So if you head on over to 3b1b.co.live, link is in the description,
it'll forward you on over to item pool, which now asks us write a value of x so that e to That way we'll be able to replace it in the base of our expression.
And then once we do that, maybe we can use our knowledge of e and exponentials of e in order to make some sense out of this expression. with answers for that, that when we write e to the x and complex numbers are in the mix,
And instead I think the healthiest way to think about it is that It's the, it's sort of the main exponential function.
way that I'll describe later on in this episode actually. It's 1 plus x plus x squared over 2 plus x cubed over 6.
Each term looks like a power of x divided by the factorial of that power. make sense to plug in things like complex values. And one of the things that we looked at earlier in the series was how if you're
plugging in imaginary values, you know, let's say I go over here and I'm going to plug in i, where i is the square root of negative 1 times some value theta, we can very literally read off what it means to exponentiate it in that context.
For example, this very first term 1 we might think of as a vector pointing from 0 to 1. it's a 90 degree rotation of that first vector scaled a little bit differently. Then the next vector is a 90 degree rotation of that scaled differently.
the way that these things scale is going to look very different. but it's at least a sensible thing to think about, right?
and we're going to start thinking about i in various complex numbers, And then there's going to be a separate illustration that I'll pull up
But first let's just see how people are doing on our live quiz, finding a value of x so that e to the x is equal to i. How have people done here?
Okay, so the top answer is in a sense absolutely correct, you could say. By definition, the natural log is the inverse function of exponentiation. So whatever the answer is here, we should be able to write it as natural log of i.
It's not going to move us forward, because if we if we tried to use that, you know, I'm going to write e to the natural log of i, and then I'm going to be raising
It begs the question, and the only thing I can really do from here is simplify that down to be i, which sort of gets us nowhere. some concrete point on the plane that I can plug in,
for that we'd want to look at, I accidentally skipped ahead of my questions here. Let me go back, pull up people's answers. Got a little excited, a little trigger happy with some of my clicking around here.
actually help us move forward here, which was one half of i times pi. third most common one, one half pi times i, just swapping the two numbers there.
and I think, yeah, I think the one following that I'm guessing people either forgot to mention the i but let's just, let's just write that out and kind of see where this moves Now, if you're confused about where that came from,
I think this is a good time to remind ourselves of Euler's formula and It's one half pi times i, where again, computationally, this very literally means if we plug in pi over two for theta here,
Let me see if I can, I don't think I'll be able to get it right on the head, but something in the 1.57 range, if you were to play out this whole sum, is that when you play all of this out and you add all of the terms together,
But usually this is kind of a complicated way to think about it. I think it's helpful just in case you find yourself doubting the reality
of what we're talking about or the fact that it's something sensible. Almost always, anytime that people are thinking about complex exponentials, instead they find themselves thinking something like e to the i times theta,
but you come to immediately know of Euler's formula, that it's telling you you walk a distance theta around the unit circle, which is to say an angle of theta radians, and whatever point you're sitting
So people who knew about this would say, hmm, i as a number is also sitting on the unit circle, so I just have to walk an angle of 90 degrees pi halves radians.
And from here we can actually move forward and get kind of a funny looking answer, because that i will hop to the inside and we'll be looking at e to the pi halves times i i squared is negative one by definition.
That is a purely real number. You could just plug that into your calculator and it's actually around 0.2 or so. It's about a fifth, which is weird, uh, that you take an imaginary
number and raise it to an imaginary power and that gets you a real number. not that much if we're thinking in terms of Euler's formula and all of that,
I think it's one thing to plug in some numbers and see what pops out. about i to the i from some of my favorite channels. Stand Up Math with Matt Parker, he talked about i to the i, did a beautiful job.
One thing that I try to, you know, I try to do is add something And in this case, I think one of the most interesting questions you can ask before
we jump in, I mean, there's lots of interesting things about this expression i to the i, but if we're seeking intuition, thinking about how i is a 90 degree rotation, right, that's the way that it acts when we're multiplying things on the complex plane.
multiplication turns stuff 90 degrees, or pi halves radians. So when you look at this expression that i to the power i equals e to the negative pi halves, it begs the question, in what sense do two 90
this very specific real number of e to the negative pi halves around 0.2079, on and on? And it's like those games that sometimes you play with arithmetic where someone says,
oh, I'll give you the numbers 1, 3, 4, and 6, see if you can combine them in some way with addition, multiplication, division, to get some other specific number, In this case, it's kind of like that style of puzzle on steroids.
How do we take the idea of a 90 degree rotation and combine it with itself twice in some mathematical sense such that we get out this number e to the negative pi halves? And the first step, I would say, is to remind ourselves of the intuition
for why it is that Euler's formula works, why it is that I can take an expression like e to the pi halves i and say that that's the same thing as i. You know, I just showed what it means in a very, um, computational sense,
why it would be the case that this is how things behave. Now I've at this point made, you know, quite a number of videos on variants of this, So if we take an expression like e to the i times t and we're thinking of that
as describing a dynamic, you're going to have a position that changes over time, then what I'll do is I'm going to start drawing that with a blue vector, and we know as an initial condition that when we raise e to the zero, that's at one.
but let's say that's all we know, right? be thinking about it with circles or anything like that. Well, one thing we might know is that e to the x is its own derivative.
So if we're thinking of dynamics here, we might ask about the velocity of our position taking its derivative, and because e to the x is its own derivative, this is quite nice. and then the inner expression as we're applying the chain rule is just i times t,
And this gives us a literal way to read the equation in terms of dynamics, right? if you rotate that 90 degrees, you apply this action of i,
So wherever you're standing, draw a vector from zero, the origin, up to where you are, and that's enough to tell you how to move, assuming you know where you're starting.
the fact that we've rotated 90 degrees already brings us there. And I could draw out a bunch of different potential position vectors, you know, suggestively maybe putting them on a circle and saying really imagine yourself
sitting at any one of those and following the rule for this dynamic. And what it would tell you is, okay, I've got to sit there, rotate my vector 90 degrees, And it doesn't necessarily have to be any one of the points on that circle,
you say rotate that vector 90 degrees, which if we're drawing a vector field, And here you could phrase this question without even talking about exponentials or complex numbers or anything like that, and in fact I'm going to go back to the quiz
just to really emphasize that Euler's formula and what it's claiming and then how it's really intuitive as soon as we're putting some dynamics into it, and we can ask questions removed from the idea of exponentials that actually have
So here I want you to imagine starting a walk from the point 1,0 on the coordinate plane, in such a way that at all moments your velocity vector is a 90 degree counterclockwise
rotation of the vector drawn from 0,0, the origin, up to where you are. After pi halves units of time, where will you be on the plane? dynamic and where you'll end up after a little bit of time.
that have popped in from the audience. No questions at the moment, at least that I see, so if ever you do want to ask, go to Twitter, use the hashtag lockdownmath, those will be forwarded to me and then
and that always makes for a fun time. which is how about other solutions to the expression x equals i, such as 5 pi halves?
we will certainly be talking about that. And if you just hold on for like one little moment, that's exactly what we're going to get to, so I'll keep that on file for us to
But right now while we're just trying to get intuition around the first, The first answer we saw was e to the negative pi halves. So it seems like most of you correctly said that
It's essentially walking you a quarter of the way around. I'm sort of picturing in my head as a big circular lake that you're going for
a walk around, because when you have this rule for dynamics where your position vector rotated by 90 degrees gives you your velocity vector, that is circular motion, because the tangent line to a circle is always perpendicular to its radius.
So this rule of motion corresponds with walking around a circle, That you keep increasing that value of t, and this dynamical rule, expect of a function like e to the t, necessarily walks you around a circle.
So in particular, when we're wondering how long does it take to get to i, you would basically say just wait for an amount of time equal to, Okay, so in this case we're thinking pi halves.
We're thinking of pi halves as kind of an amount of time for these dynamics to get you on the number i. And that gives us the first half of our intuition if we want to go back to our formula,
to make e to the negative pi halves? Once we're thinking of our dynamics here as each one of your velocity vectors
the way that we're kind of thinking about it reading it off as something with it's where you get after traveling for pi halves units of time according to the
dynamics of this expression, according to the idea that your velocity is always Okay, so that's awfully nice. which, or I guess there's three different i's at play.
We have this base, which is describing the 90 degrees that we walk around the circle. We have the i that's sitting here, which is describing But now we're going to introduce another i, which
essentially has this effect of changing what your dynamics are. Because as we go from e to the i times t, and instead we, you know, raise things to another power of i, which I'll just write with a little caret i,
if we take e to the i t and we alter what that expression is by raising it to the i, what we get is e to the negative t. And if we try to interpret that with the same sort of dynamics that we
what that's telling us is that the derivative of our new dynamic e to the negative t is equal to, well now the constant sitting in front of t is negative one,
so our chain rule will have it be negative one times itself, times e to the negative t. So whatever your position is, now your velocity is negative one times itself. So the effect of raising to another power i, it's kind of like we took the dynamics,
we looked at every velocity vector and we said rotate another 90 degrees, so that in this context it would actually be in the beginning a So if you're starting off at the number one, your
And as you walk even lower, if you were sitting at one half, vector would be negative one times where you are, which is negative one half.
you might kind of imagine it, I haven't animated this nicely yet or anything, but if you look at this whole vector field and you ask each one of those vectors, they would all be pointed in towards the origin.
So if you were to have a little dot move in such a way that its velocity what you would end up with is something where with each time step it kind of
takes a step towards zero and it just with each of your time steps is walking towards zero and each step has a size that gets smaller and smaller as you And of course in practice, this would be a bunch of
infinitesimal steps rather than very concrete sizing. You might be very exact about it if you wanted and say what we're looking at is scaling down by some number just less than one and we're doing this n
different times and then we're gonna multiply that by however much time we're waiting. That's just what we talked about in the compound interest lecture if you're curious
and it's kind of the standard way to talk about e to powers as this sort of limit. You could also have that t living on the inside here instead if you wanted. But now if we think about our original point i, our base, what that meant,
it was saying look at our dynamics and it's the point you get to when you wait pi halves So the effect of raising to the i changes our dynamics in such a way that instead of walking around a circle we're doing this kind of exponential decay.
We're moving towards zero at a slowing and slowing rate and the place that you end up after pi halves units of time will be e to the negative pi halves around 0.2079.
two 90 degree rotations combine to get you this very specific value. one came from the idea of walking pi halves radians.
otherwise when you look at the expression i to the i equals e to the negative pi halves, This doesn't correspond to any, you know, thing that might see in the real world. through things like what are the dynamics of circular motion?
Things that come up in the real world all the time. which was raised and let's see if it's still up there. about other solutions to e to the x equals i?
let me let me ask other people to contribute some solutions here. Let me go to question number three, which is going to be very similar to
a question that was already seen, which is to write a value of x other than the one we just saw x equals pi halves times i so that e to the x equals i. get an answer that other people haven't written, right?
few different options here and then in a moment we'll talk But I'll give you a moment to think of which of the many possible values of
x you could choose is the one that you want to put your fingerprint on that you want to be contributing to the live stats page that we're seeing right now. I'll give you a little moment.
So, and as you're answering that we've got a question coming in that says wouldn't it be more accurate to say that i to the power After all it's multiplying by i that's the rotation.
So if we stretch the meaning of words too thin, aren't we multiplying by i i times? I mean, I think the most honest answer here is just no not at all. means as soon as we're extending to the idea of complex numbers.
It's as if you're multiplying by itself i different times. But I cannot think of a way that that actually like satisfactorily makes sense in my mind.
What I do think makes some sense is to try to think of the function i to the x. And in the context of counting numbers, you know,
if n and k are just things like 3 and 5, we know that it should satisfy this idea of multiplying the outputs correspond to adding the inputs. I've said this in many different forms in many different places,
we want it to satisfy the property that when you add the inputs you multiply the outputs. trying to stretch the idea of repeated multiplication.
So, it like, I mean, correct me if I'm wrong, if you can definitely, if you can find some way where if you read this off in a stretching language context saying we're taking a 90 degree rotation and applying that i different times,
that's not nonsense and that somehow like intuitively gets you to the answer e to the But I think the healthier relationship to have is to say we have this central property
for exponentiation, which definitely holds in the context of repeated multiplication. because it just it just doesn't make sense when we're talking about something that's
not a counting number like i, or other crazy things that we exponentiate in math like Focus on the property more so than what some people think of as the origins of that property or the original intuition behind it.
is there just one such function that feels reasonable to write for this? Because, you know, if we're going to write it as i to the x not only should it satisfy this, it should also satisfy, you know, when we plug in the number one we get i,
presumably i to the power one, however we're thinking of this function should be i. That's what the quiz question is starting to get at and we've
So let's see some of the variety that people have thrown in here. Great, that absolutely is another value that we could plug in for x here.
if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves, which is 1.57, what if instead we took another full turn and
we go another pi halves to get us to pi, which you know, we might kind of record, We walk another pi halves, we walk another pi halves, which at this point we would have gone a full circle getting us back to one,
and then we walk for five pi halves, which numerically is about 7.85. Yeah, that absolutely is another number that gets us on top of i. And if we were to go through the whole rigmarole of re-expressing i
to the power i by first writing e to the five pi halves i to the power i, those i's multiply to become negative and we'd be looking at e to the
negative five pi halves, which is a very different number, right? I'm not sure off the top of my head, but let's take a look at Desmos maybe.
What is e to the negative five pi halves? Okay, 000388.
0.000388. Which begs the question of okay i to the i, what are you? Are you about a fifth like we saw before around 0.2?
And in terms of our intuition, it's basically a question of how we're interpreting that base i, right? Are we thinking of it as you wait about 1.57 units of time and then you
spinning and see where you decay to after that amount of time, which is 0.2. Or do you wait even longer for about 7.85 units of time, which is five pi halves?
which gets you to a much smaller number. We have other people coming in here with negative three halves times i pi.
Which, you know, in terms of a unit circle, we could think of as saying, hey if I want to get to i, rather than walking 90 degrees, pi halves radians that way, what if I walk 270 degrees the other way?
because the convention is usually that counterclockwise is positive. And that would get us a different answer. If we had e to the negative three pi halves i,
Now the i squared cancels with the negative that's already there, and we have a positive three pi halves. And numerically this gets us an even different looking answer from what we had before,
which if we go over and we say hey, what is e to the three pi, Very different kind of number than what we saw before.
111.31. 111.31 or so. suppose we have this rotating dynamic, but we move backwards in time.
such that if I played things forward from there, I would land on the number one, And you have to go back in time three pi halves units. which is what raising to the i is doing in this context,
you say if I'm starting at the number one, but I want to move backwards in time and say where should I have started if I want to decay down such that I end up at the number one after three pi halves units of time?
and eleven for that kind of exponential decay And you can see where this is going, where there's actually infinitely many different values that we could plug in for x if we're thinking of e to the x as being i.
And people have entered a lot more here. Excuse me, throwing my pen onto the ground as one does. Nine pi halves, great choice.
Lots and lots of different options, infinitely many different values, Because we look at an expression that seems like, I just plug that into my calculator and see what pops out.
So what's going on here, right? And I think this really cuts to the idea of how we think about exponentials in general, But before that I do want to emphasize that this isn't the only time in math
where we come across a kind of ambiguity for how to interpret something, right? Because if I say something like, what is the square root of 25? You know, I think a lot of us say, well, it's five.
It should be some number x such that when you square it you get 25. Who's to say that our conventions should be that the square root function is positive,
So we have one expression that seems like it wants to have multiple different values, And this could actually happen in another context where instead of square roots,
what if I was asking for the fourth root of something like 16? Two is a number such that when you multiply by itself four times you get 16. But if we're thinking of this as answering the question,
There is another answer to this. That's a number that when you multiply by itself four times you get 16.
That seems valid, but there's another answer. All four of these numbers satisfy that property. So who's to say that the fourth root of 16 should be two?
When there's multiple options like this, when you have a multi-valued function, we often just choose one of those values to be what we mean when we want to treat it as a function, as something with a single input and a single output.
In fancier lingo, this comes up all the time when we're dealing with complex numbers, the idea of something as an operation kind of wanting to have multiple values. the square root function, which is to say you choose a certain convention.
because you say just choose the positive one. positive complex numbers when we want to take square roots. Just to give one example, let's say we wanted to take the square root of i.
Because there's multiple different answers. You know, we think of i again as this 90 degree rotation. it feels like the square root should be, you know, something sitting at a 45 degree angle.
Maybe that's the square root of i, which we could write out very explicitly as root 2 over 2, root 2 over 2 i. But if we were thinking of i instead as being a negative 270 degree rotation,
should actually get us on the other side. And that's actually just the negative of what we saw before.
Negative root 2 over 2 minus root 2 over 2 times i. just choose the square root to be whatever the positive answer is. You know, maybe it feels like we should consider this
But however you try to define positive in a nice way here that's going to be consistent, number, you're not really going to be able to do it the way that you can for real numbers.
And in fact, this phenomenon here where we're taking roots is we were talking about multiple values for i raised to the power of i. Because forget i raised to the power of i, let me ask what might
look like a much simpler question of taking 2 to the power one half. Yeah, I think you say, well, we know what this is, But what if I said let's approach this the same way
I want to first express things as e to the something, right, and then I'm going to raise that to the one half by multiplying the one half And I say, well, okay, I can, I guess I can do that.
Well, that's the natural log of 2. It's a constant which is around 0.69 or so. So we could be thinking of this as e to the natural log of 2 times one half.
And if you wanted to, if you were thinking of e to the x, you know, this might be kind of overkill in the context of real numbers, but if you were thinking of e to the x as shorthand for this x function,
you could plug in the value 0.69 times one half, which I guess would be around 0.345ish, You plug in that very concrete value into your polynomial, see what it outputs, and it will output around 1.415.
But if we do the same thing we were just doing with i and when we want to write something as e to a power, we could also write this.
This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi i. Right?
You could break it down as it's e to the natural log of 2 multiplied by e to the 2 pi i. So it's just going to equal 1, so we're looking at 2 times 1.
And yet when we play the same game of taking this and raising it to a power and treating that by multiplying the power into the exponent, look at what happens. We have e to the natural log of 2 times one half plus Well, what's 2 pi i times one half?
Now this first part, e to the natural log of 2 times one half, that will end up being the familiar square root of 2. But we're going to be multiplying that by e to the pi i.
And quite famously e to the pi i is negative 1. So in this case, it seems to be suggesting that if we are solving this expression 2 to the one half by playing around with the different answers we could plug in for
something like e to the x equaling one half, what we end up with is another answer. What we might traditionally write as this negative square root of 2. to look at 2 to the one half and say that's not equaling one thing,
but based on choices we make it could equal multiple different things. If there's going to be anything that 2 to the one half is, the negative variant of that.
And in fact, we could we could play this game even further, where let me ask you for even more creative answers to this expression. Because maybe we can find other funny powers of something like 2 to the power x as
we start plugging in various different values of x based on what substitution we make. If we're abiding by the same rules that we were using in evaluating i to the power i. the equation e to the x equals 2 is the real number natural log of 2.
It's not boring, but it's boring in comparison to what else we could do. Can you write some other answer to the question e to the x equals 2?
So I will give you another little moment for that.
All right, I will go ahead and lock in some answers here if that's all right with you. the math entry depending on what device you're looking at. But don't be too stressed if it's before you got the chance to enter
the question that you want into the answer that you wanted to answer. So it looks like 131 of you have entered the variant And I guess I in writing this question mistakenly like marked one of the
answers as being correct when in fact there's quite a few different correct ones. So that's on me for the fact that I don't know if it looks to any of you like, 42 i pi which is of course a great choice.
But you could also have something like 4 pi i plus the natural log of 2 or 6 pi i. Or really any integer multiple of 2 pi i if you add that it doesn't affect e to the x.
to the 2 pi i which is the effect of multiplying by 1. seems to output kind of reasonable results when we do it.
entered expression there was that we might replace 2. Okay. There was a suggestion that we replace 2 with e to the natural log of 2 plus 4 pi i.
And we raise all of that to the one fourth, right? Well if you were to play the same game you would get e to the natural
log of 2 times one fourth and we'd be multiplying by e to the pi i. Now the first part of that is going to be the usual positive fourth root of 2. The thing we mean when you plug in an expression like fourth root of 2 into a calculator,
But then this second part is negative 1. So it seems to be saying, you know, if we were to interpret 2 in this different way, that we get but it's a reasonable answer.
It's another number that when you raise it to the fourth power you get 2. And if we had done this with even different values, if instead we had been using 2 pi i, you know, it's kind of fun to think about how that would have changed things.
well then over here we would have been looking at pi halves times i. And instead of multiplying by negative 1 we would have instead been multiplying by i.
It seems like a reasonable output for something like 2 to the one fourth. So when you're looking at the fact that i to the power i seems to have multiple different values for it, right, we have this funny phenomenon where we can plug in e to the 5 pi
halves i, negative 3 pi halves i, and we get what seem like wildly different answers. Something super small, something super big, all very different from the one fifth, approximately one fifth answer that we found before up here.
what's 2 to the one fourth and acknowledging that there's actually multiple different solutions to the expression x to the fourth equals 2. And what you're looking at is the fact that there's multiple different
solutions to the expression e to the x equals some kind of base, whether that base is i, whether that base is 2, whatever it might be. you're dealing with real numbers things are just lovely.
It's great. let me just cover some of this stuff up.
We have this nice back and forth where you can choose to express any exponential as a base to x like 2 to the x or you could express that same exponential as x of r times x which you know, that is the polynomial that
we refer to whenever implicitly refer to whenever we write something like e to the x. logarithm of b and it gives you one answer assuming that b is a positive number.
So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible exponentials, right, we could write them as x of r times x and change what r is.
times x if that's something you're more comfortable with. We could think about changing what that is. But on the other hand if you were to think about all possible exponentials as some base,
let me do base to the power of x and we're going to change what that base is. manipulate but it's just another way of expressing the same family. Right, and a way that you might think about this for how do we think about what base
does it correspond to if we're thinking a little bit more abstractly as exp of r times x. apply this to complex numbers where it's going to look weirder. If instead of looking at that base, one thing I could do is say what is the
value of exp of r, right, which is basically this function of when we plug in one. So exp of r times one if you prefer to think of it that way. nd what you could see is okay if I get R that Factor in front of X in my exponential
function exp of R X to be zero point six nine Which I know is around the natural log of What this means is that exp of one is about two. we would usually write as two to the power x.
Okay, and basically as I change around my r, you know, I could try to change it to something so that it looks like three. which we would usually write as three to the power x.
think about varying this value r rather than varying the base. And the main reason is that as soon as we get to complex contexts and we're thinking of exponentiation, you have this overloading that goes on where
if we change around what sits in front of the x, that's all well and good. I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by 2 pi i.
That would still correspond to two. That doesn't change the base that it corresponds to. Because in all of those cases when we plug in x equals one, we get the same thing.
However, all of these for different values of x are distinct functions. Because i to the x is an ambiguous function in that context.
It would be unambiguous if we decided which value of r such that what we're representing is exp of r times x, which value of r do we choose? But at that point it just feels like maybe what we want is to stop
thinking about things in terms of some base raised to the power x. we should just write them all as exp of some constant times x. numbers if we want to do a computation, or just to do math on top of it.
We've got this nice infinite polynomial that we plug them into. And I'll make another case for you that this is maybe the the correct way to think about exponentials, as soon as we're extending into other domains, things like complex numbers.
Oh doorbell, something's arrived. Go back to the original way that we extend the idea of exponentiation and just think of like what is two to the x?
you know something like two to the three, repeated multiplication. the x for fractional amounts or for negative amounts and things like that?
Well, you're usually taught that two to the one half should be something where you know if I multiply it by itself and this follows the usual rules that exponentials do with counting numbers where we're able to add things in that exponent,
So it should be some number that when I multiply it by itself, I get two. Maybe it's negative. you're going to be able to get a nice continuous function out of this.
Same deal if we ask about negative numbers, what should two to the negative one be? it gets me two to the zero. that negative exponents look like one half.
But what's really going on here is we're saying whatever this is, it should be some kind of function that satisfies this property f of a plus b equals And moreover the fact that the base is two is basically
It's a function where when we plug in one we get two. see if you're following along with some of the implications here.
I want to ask you what is, I won't call it like a softball, but this is, this isn't meant to be like an incredibly deep question necessarily. abstractly starting with properties of a function and then kind of
deducing ways that we might want to write it down based on those properties. If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs, and it also satisfies f of one equals two, which of the following is true?
true no matter which such function you're starting with. It's whichever one we were talking about how to I asked a question of this style where I neglected a single condition,
x is non-zero everywhere and then that caused some amount of confutlement. Which is cool, get confutlement on screen that happens to all of us. But the the intent of it was to basically show that this abstract property of
something that turns addition into multiplication is uh is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power.
Um Now we've got a couple questions actually about power towers that seem to have popped up here, which is great connected to last time.
Um, let's let's hold off on the power tower question for just a moment, so that we first get like a deeper feel of like what exponentiation should mean here. is we can answer it in like multiple different ways.
Uh, and then just as a number line can be represented in a logarithmic scale, Maybe mapping the complex plane onto an infinite cylinder in the logarithmic sense.
Yeah, uh, in fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that. Because what we'll do is play around with different exponential functions x of r times x,
represented by a little yellow dot. It's not going to map the whole plane, but just a couple But the idea is that as we move around what that constant is,
it does to the plane. logarithmic scale and then wrapping the imaginary axis along a circle. And then as soon as that value of r becomes imaginary, it swaps the role of those.
numbers get put on a logarithmic scaled positive axis. All three of which I guess are sort of jumping the gun ahead for where I want to go, So on this one, let's go ahead and just grade it.
The idea is that this property of f of a plus b ends up letting you express a lot of different things purely in terms of what f of one is. And just to spell that out very explicitly, something like f of five is the
same thing as f of one plus one plus one plus one plus one, which is the same thing as f of one multiplied by itself five times because Which if f of one is two is the same as two to the power five.
And then something like f of negative five, it should be the case that when we multiply it by f of five, we get whatever f of zero is. And it's not immediately clear what f of zero is,
but we could say that f of one plus zero is equal to whatever f of one is But f of one is equal to two, and so this is also equal to two.
Well that something has to be a one. So in this context this guarantees that f of negative five is two to the negative five. So we could explicitly write this as two to the negative five.
Which is all to say, these two properties together make us really want to write the function as two to the x, because any counting number that we put in, it's going to satisfy, it's going to look like two multiplied by itself that number
Any fractional number we put in, it's going to satisfy these properties that we wanted. And in the context of real valued functions, it actually would be. there would be multiple such functions f that we could write for this.
One of which is what we were looking at before, where we could have a function defined to be exp of the natural log of two plus
two pi i all of that times x. I just get excited writing about this. by what happens if you plug in x equals one half.
what you get is the negative square root of two. but i times the fourth root of two. So it is a different function, but it still satisfies these properties,
And it makes it suggest that maybe two to the x is an ambiguous bit of notation, and we should just write everything in terms of exp of r times something. creative enough with all of the functions that satisfy this property.
and there's different values of r that could come into play. and then maybe give like a sketch of what the proof would look like if you want, and it satisfies the following properties.
It's differentiable, which just keeps it from being some, uh, you know, totally messy discontinuous thing that's like taking on some random values depending on, fractional amounts you might want to think of in crazy ways.
It's not equal to zero everywhere. So the condition that sort of slipped my mind and I forget which lecture, And then it has this central property that it turns addition into multiplication.
If you have such a function, I claim that there's a unique, maybe I should really specify, there exists a unique complex number r so that you could write f of x as basically being this exponential function of r
Which is, you know, basically saying that if you have exp as a function, this infinite polynomial with nice derivative properties and all of that, if you have this you have every exponential that you want in a very like abstract
generic sense of the word exponential just based on a property that we could want from it. If you want to first look at what is the derivative of this value,
And you explicitly write out what the limit of that is. want to like pause and think through the details, feel free to. The central property that we have lets us expand out the f of x plus h term.
the output over the change to the input that caused it. And because we can factor that out, we can factor f of x out of the
expression entirely and the whole limit is expressed only in terms of h. Which if you think about what it means in the context of derivatives and the fact that f of zero necessarily equals one, this whole limiting expression is just some
constant, but more specifically it's whatever the derivative of our function at zero is. at zero that determines what its derivative is everywhere. And in the context of exponential functions this is hopefully quite familiar because all
that we're really saying is the derivative of an exponential function is proportional to itself and that proportionality constant is equal to whatever the derivative at zero is. This is all very abstractly phrased and such, but the purpose of it is to
emphasize that it's not necessarily just functions that we already think of as a to the power x, but it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication.
But if you have that, it actually guarantees that you also have a second derivative. the derivative function is just proportional to itself. So in order to take the nth derivative you just look at
that proportionality constant and raise it to the power n. And then from here you could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series and that
idea especially if you want to intermix the idea of any differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic.
but fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for f and kind
of justify the idea that there's a unique complex number such that our function f can And then the connection to normal exponentials is whenever you have such
a value r we do essentially what we do in the complex context of real numbers is if you look at x of that function of that value r and write that as a base it feels like you should be able to write that as b to the x.
But the whole the whole point here of course is that when we play this game and you're trying to interpret something like i to the x that's an ambiguous function because there's lots of different values of r we could interpret that to mean
not just exp of pi halves i times x but we could also interpret it to mean exp of five pi halves i times x and these are separate functions and there's an infinite family of separate functions that feel like we should write them as i to the x.
So the expression i to the i unless you've adopted a standard for what that's necessarily going to mean when you say it has infinitely many outputs another way to think of that is that the function i to the x with the notation we have is a little bit ambiguous.
some of this because I think that's fun. And you know, you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of
r times x which is basically this is another way to write e to the power of x. In fact, I think I I think I rendered a different animation at some point that specified that because I was planning on planning on doing that.
file system get back to where you're supposed to be. Get on in there is it complaining because there's multiple different?
It's going to be like there's a oh replace it shows up on the other screen. Yeah, okay replace place whatever you see there. And now we go back to oh there we go all of that all of that just so that I could have
nicely written out uh, if you're uncomfortable with thinking of it as exp of r times x this infinite polynomial Just in the back of your head e to the r times x and we're going to vary around r So i'm going to follow the points of the imaginary axis and
i'm going to follow the points of the real axis And uh, let's see what this does Well, that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers Anything that's a negative real number is going to get squished
into the range between zero and one which should make sense e to the negative e to a negative real number is something between zero and one and we're Specifically tracking f of negative one which is going to show up around whatever one over e is around zero
point three seven f of one lands on e As expected that's what x of one is f of i Is going to land one radian around the unit circle and it's kind of fun to follow along the whole imaginary axis here How the imaginary axis gets uh wrapped around a circle?
And what happens as we tweak this value of r that's determining not just that we're talking about an exponential function But which exponential function there's a nice one-to-one correspondence between all the exponential functions we might want and
values of r here It stretches things differently So when we put it up to two You know it stretches out the real axis a lot more so that f of one ends up around where e squared is a little Above seven f of negative one is much closer to zero f of i Is a
two radian rotation around the circle f of negative i is a negative two radian rotation And of course we can get to our favorite formula that If that were pi that we had as our scaling constant then the real axis gets stretched out quite a lot you know f of
one is sitting off at e to the pi which is very close to 20 plus pi which is always fun and f of negative one extremely close to zero so It's really stretched out that real axis and it's also stretched out things in the Unit circle direction so that
getting to f of i or f of negative i walks halfway around the circle So that's all How would we think about a function like? We would also write as exp of Exp of the natural log of two times x So we
kind of move our yellow dot representing the value of r to around zero point six nine still no imaginary part Just a real number zero point six nine or so. That's the natural log of two Well, you can see that f of one lands on two,
which is why we want to call this function two to the x f of one half actually, sorry f of negative one lands right on one half f of i It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle And now
we could have a little bit more fun and say what would happen if we were to Change this to instead of being 0.69 instead of being the natural log of two make it i times the natural log of two So that we're really thinking of something that might have an
exponential base to it This would be what we might think of as two times i Raised to a power Well, if we move that yellow dot which is representing r off of the real axis and onto the imaginary axis it swaps the roles of the Teal dots and all of the maroon dots
in this context which remember came from being the positive and imaginary axes And it should make sense that it swaps their roles because what does it mean if we take the By i it means we're rotating that input space So everything that was the real number
axis turns into the imaginary axis and everything that was the imaginary axis Is getting turned into the real axis So for us what that means is our new exponential function where our value of r is now purely imaginary Takes all of the real numbers
and it just wraps them around a circle and it takes all the imaginary numbers And it's let's say we scale this thing up so that we're sitting at around pi halves times i Well,
That means it takes the real number one to the value i Which is the sense in which we want to write this function as i to the power x Right. It just tempts us to write it not as this abstract looking thing X of r
we just want to write it as i to the x even if that's a little ambiguous What that really means is just that the function we're dealing with outputs i at one and if it's an exponential function that Uh,
What is i to the power i? In this case, it shoves it to around uh, 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of
one onto the number i So if we were to scale it up even further, I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to five halves times pi i What you would see is
Uh is rotated around on itself so that f of negative f of one would rotate around another two pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of i to the i is a much much smaller number.
0.0003 or so But we can also see what I think is quite fun. that we want to interpret as two to the power x right?
that when you plug it in here The expression we get is what we want to write as two to the power x But what if we start moving it in the imaginary direction?
Okay And what i'll first do is i'll move it up by pi i units Now what's going on here? which is the natural log of two plus pi times i What that means is that
f of one is at negative two, so we want to write this function as negative two to the it's it's a little deceptively simple when we write a negative number to a power Negative
two To the power x it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like One half Where we're kind of asking for a square root of negative two We we realize that we want
to write this as something like i times the square root of two But if you were to look at this function negative two to the power x in the full complex domain that it's dealing with What you're looking at is a function that takes the value of one to negative
two And if it does that what it does to the rest of the real number line is it kind of So we see that f of negative one sits at negative one half About where you would expect
if you were to follow to f of one half It would sit exactly on the imaginary line and f of one half would be square root of two Well, my mouse is not where I want it to be. It would be around a square root of two times i and As you continue further on this is
showing you all of the real value powers of negative two to the x it necessarily spirals around Um, but we could also move our value of r even higher and get it up to around tau times i around 6.28 times i And in that context,
this is another function that we would want to write as something like two to the x because For any whole number to whole number that you plug in for x it will look like repeated multiplication And it even has kind of reasonable values for things like one
half where it spits out the negative square root instead of the positive square But what it's actually doing is a transformation to the plane where it puts everything Uh is the real number line ends up being a very tightly wound spiral That goes around and
it just spirals in such a way that f of one lands right on the number two so it is in that sense that we could say, um two to the x is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to so I think with
all of that I will um I will leave things for today and i'll just leave you with a Okay, so If you want to think of i to the i as being a multi-valued expression, right?
You could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function and maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different
values like the various ones that we saw How many values does two to the one-third want to be in the same sense where we are replacing two with various different uh various different options for e to the x Such that e to the x equals two How many
different values does that want to be or how many values does two to the three-tenths Phrased differently of all of the uh, let me say of all of the exponential functions
So f of x which satisfy oh have I written it down somewhere f of x that satisfies All of these properties that i've written so if it satisfies all of these um and
if f of one is equal to two Right, how many different outputs are we going to get when we plug in x equals three-tenths for the various options for what function? For two to the pi for the various functions that two to the x could represent If we're
thinking of two to the x as some kind of exponential function exponential in the sense if we if we have a Class of different such functions and we want to plug in pi it makes
me laugh just because it's such a I don't know Kind of a funny answer that pops out As So those are the questions that i'll leave you with and I think this is you know,
my my My central question in approaching today's lecture was whether I wanted to be um kind of describing like these abstract properties of exponential functions and it's just cool to me that Starting from those abstract properties you get locked into the
idea of e to the rx or more You know, I think more honestly written exp of r times x for different values of r That it locks you in that far but it doesn't lock you in as far as having an unambiguous Notion of what two to the power x should be much less
something like i to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes it doesn't come off as approachable But if that's the case, you know, you just let me know I think I think there's a whole interesting
circle of thoughts that surrounds all of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things
like this so, um, it was a question that we had up on screen Uh, yeah, what happens if we do this for i to the power i titration, you know, let's just try this Let's just go ahead and try a power tower where we're raising i
to a given power and see what uh, what pops out of it so I wasn't planning on doing this but we can We can always pull up python and essentially do what we were doing last time so the way that this would work Is we were starting off with some base value
and then for some kind of range What were we doing? We were taking a and we're going to reassign it to be whatever The base which in this case is i raised to the power of a should be Okay, cool.
So we're going to do that and then we're going to print off the value of a and let's just do this for Uh, yeah, it's a much bigger number like 200 uh So it seems like what happens is There's there's potential for chaos with these things like sometimes it's not that
you've landed on a stable A stable value or it's not even that you've diverged it could be that you're bouncing between a cycle of values or That you're like literally bouncing in a way that's um, it's not periodic or anything and it's actually chaotic I I suspect
that doesn't happen for i but it's a thing to potentially look out for It looks like it does kind of stabilize um, maybe there's Some little subjection to numerical error, but we stay pretty consistently around something with a real part of 0.43 and 0.36 Now
what I would want to emphasize though is this expression So let's set a back to b equal to 1 this expression of taking i to the power of a remember That's a little bit ambiguous. It depends on what choice of the function I we actually have so let me Let me
import NumPy so I have the exponential function Let me go For our big range like something that's like i to the power of x I'm going to write it as the exponential
function of a different constant right a different constant That i'm going to make So i'll do five pi halves times i so it's a complex number And it's got five pi
halves as the imaginary part So this is five pi halves times i and what am I doing? I'm exponentiating that So I want to multiply a onto the inside there. Okay, this is basically another way that you could interpret the expression i to the
x Thankfully you would say you've chosen a different branch of the natural log function But it is another Function which we could iterate on itself and see what happens and so we actually have a different result It looks like what ends up happening is it
Oh, wow, is it period three? So we've got seven point three five then zero then point nine nine So it looks like it
gets into the cycle of bouncing between three separate values even though um In theory in both of those cases we were doing something that was i to the x iterated on itself It has everything to do with what actual function you think i to the x is referring to
so in that sense The power tower question is ambiguous Usually you just choose it to be the pi halves i variant but it's more fun to see that it can be multiple things All right. Can we think of something in between halfway between multiplication and exponentiation?
oh, that's an uh, I mean Spirit of it because each one of them it comes down to like a discrete step of you're repeating the previous operation But oftentimes in math when you have something initially defined in
I can't think of anything off the top of my head, but that's an interesting question I don't know if that's been That's an extended notion. I mean you have things like fractional derivatives you have things like fractional
but i'm not familiar with one myself And then lastly do complex numbers to the power of complex number values arise in physics and if yes How does one decide so I can't think of if they necessary.
I can't think of if they Come up in physics in like a direct sense and this is probably just because i'm not a physicist. icist so I Would say like ask ask your neighborhood physicist and see see what they
that you have to go through to make sense out of it do build a stronger relationship with exponentials things like it's not just more natural to represent them as e to the rx Once you get to complex numbers you kind of have to represent them that way whereas
previously in like physics or other real-world contexts It's something that's just nice