two years ago, almost to the day actually

I've put up the first video on this channel about Euler's formula

e^(pi i) = -1

as an anniversary of sorts, I want to revisit that same idea

for one thing, I always kindda wanted to improve on the presentation

but I wouldn't rehash an old topic if there wasn't something new teach

you see, the idea underlying that video was to take certain concepts from a field of maths group theory

and show how to give Euler's formula a much richer interpretation than a mere association between numbers

and, two years ago, I thought it might be fun to use those ideas without referencing group theory itself

or any of the technical terms within it

but, I've come to see that you all actually quite like getting into the maths itself

even if it takes some time

so here, two years later

lets you and me go through an introduction to the basics of group theory

delving up into how Euler's formula comes to life under this light

if all you want is a quick explanation of Euler's formula

and you're comfortable with vector calculus

I'll go ahead and put up a particularly short explanation on the screen, that you can pause and ponder on

if it doesn't make sense, don't worry about it, it's not needed for where we're going

the reason I wanted to put out this group theory video though, is not because I think it's a better explaination

heck, it's not even a complete proof, it's just an intuition really

it's because it has the chance to change how you think about numbers, and how you think about algebra

you see group theory is all about studying the nature of symmetry

for example, a square is a very symmetric shape

but what do we actually mean by that?

One way to answer that is to ask about

"What are all the actions you can take on the square that leaves it looking indistinguishable from how it started?"

for example, you could rotate it 90° counter-clockwise, and it looks totally the same to how it started

you could also flip it around this vertical line, and again, it still looks identical

in fact, the thing about such perfect symmetry is that it's hard to keep track of what actions had actually been taken

so to help out, I'm going to go ahead and stick on an asymmetric image here

and we call each one of these actions a symmetry of the square

and all of the symmetries together make up a group of symmetries, or just "group" for short

this particular group consists of eight symmetries

there's the action of doing nothing, which is one that we count

plus three different rotations

and then there's four ways that you can flip it over

and in fact this group of right symmetries has a special name

it's called the dihedral group of order eight

and that an example of a finite group, consisting of only eight actions

bit a lot of other groups consists of infinitely many actions

think of all the possible rotations, for example, of any angle

maybe you think of this as a group that acts of a circle, capturing all of the symmetries of that circle, that don't involve flipping it

here, every action from this group of rotation lies somewhere on the infinite continuum between 0 and 2pi radians

one nice aspect of these actions is that we can associate each one of them with a single point on the circle itself, the thing being acted on

you start by choosing some arbitrary point, maybe the one on the right here

then every circle symmetry. every possible rotation, takes this marked point to some unique spot on the circle

and the action itself is completely determined by where it takes that spot

now this doesn't always happen with groups

but it's nice when it does happen, cause it gives us a way to label the actions themselves

which can otherwise be pretty trick to think about

the study of groups is not just about what a particular set of symmetry is

whether that's the eight symmetries of a square, the infinite continuum of symmetries of a circle, or anything else you dream of

the real heart and soul of this study is knowing how these symmetries play with each other

on the square, if I rotate 90°, and then flip around the vertical axis

the overall effect is the same as if I had just slipped over this diagonal line

so in some sense, that rotation plus the vertical flip equals that diagonal flip

On the circle, if I rotate 270°, and then follow it with and rotation of 120°,

the overall effect is the same as if I had just rotated 30° to start with

so, in the circle group, a 270° rotation plus a 120° rotation equals an 30° rotation

and in general, with any group, any collection of these sorts of symmetric actions

there's a kind of arithmetic. where you can always take two actions and add them together to get a third one, by applying one after the other

or, maybe you can think of it as multiplying actions, it doesn't really matter

the point is there's some way to combine the two actions to get out another one

that collection of underlying relations

all associations between pairs of actions, and the single action that's equivalent to applying one after the other

that's really what makes a group, "a group"

it's actually crazy how much of modern maths is rooted in, well, this

in understanding how a collection of actions is organized by this relation

this relation between pairs of actions, and the single action you get by composing them

groups are extremely general

a lot of different ideas came be framed in terms of symmetries and composing symmetries

and maybe the most familiar example is numbers, just ordinary numbers

and there are actually two separate ways to think about numbers as a group

one, where composing actions is going to look like addition,

and another, where composing actions will look like multiplication

it's a little weird, because we don't usually think of numbers as actions, we usually think of them as counting things

but let me show you what I mean

think of all of the ways you could slide a number line left or right along itself

this collection of all sliding actions is a group, where you might think of as the group of symmetries on an infinite line

and in the same way that actions from the circle group can be associated with individual points on that circle

this is another one of those special groups where we can associate each action with a unique point on the thing that it's actually acting on

you just follow where the point that starts at zero ends up

for example, the number three is associated with the action of sliding right by three units

the number negative two is associated with the action of sliding two units to the left

since that's the unique action that drags the point at zero over to the point at negative two

the number zero itself? well, that's associated with the action of doing nothing

this group of sliding actions, each one of which is associated with a unique real number, has a special name

the additive group of real numbers

the reason the word additive is in there, is because of what the group operation of applying one action followed by another looks like

if I slide right by three units, and then I slide right by two units

the overall effect is the same as if I had slide right by three plus two, or five units

simple enough, we're just adding the distance of each slide

but the point here is that this gives an alternative view for what numbers even are

they are one example in a much larger category of groups, groups of symmetry acting on some object

and the arithmetic of adding numbers is just one example of the arithmetic that any group of symmetries has within it

we could also extend this idea, instead asking about the sliding actions on the complex plane

the newly introduced numbers: i, 2i, 3i, and so on, on this vertical line

would all be associated with vertical sliding motions

since those are the actions that drag the point at zero up to the relevant point on that vertical line

the point over here, at 3 + 2i

would be associated wth the action of sliding the plane in such a way that drags zero up into the right to that point

and it should make sense why we call this 3 plus 2i

that diagonal sliding action is the same as first sliding by three to the right

and then following it with a slide that corresponds to 2i, which is two units vertically

similarly, lets get a feel for how composing any two of these actions generally breaks down

consider this slide by 3 + 2i action, as well as this slide by 1 - 3i action

and imagine applying one of them right after the other

the overall effect, the composition of these sliding actions

is the same as if we had slid 3 + 1 to the right, and 2 - 3 vertically

notice how that involves adding together each component

so composing sliding actions is another way thinking about what adding complex numbers actually means

this collection of all sliding actions on the 2D complex plane goes by the name

the additive group of complex numbers

again, the upshot here is that numbers, even complex numbers, are just one example of a group

and the idea of addition can be thought of in terms of successively applying actions

but numbers, schizophrenic as they are, also lead a completely differently, as a completely different kind of group

consider a new group of actions on the number line

always that you can stretch and squish it

keeping everything evenly spaced, and keeping that number zero fixed in place

yet again, this group of actions has that nice property where we can associate each action in the group with a specific point on the thing that it's acting on

in this case, follow where the point that starts at the number one goes

there is one and only one stretching action that brings that point at one to the point at three, for instance

namely stretching by a factor of three

likewise, there is one and only one action that brings that point at one to the point at one half

namely squishing by a factor of one half

I like to imagine using one hand to fix the number zero in place, and using the other to drag the number one wherever I like

whilst the rest of number line just does whatever it takes to stay evenly spaced

in this way, every single positive number is associated with a unique stretching or squishing action

now, notice what composing actions looks like in this group

if I apply this stretch by three action and then follow it with the stretch by 2 action,

the overall effect is the same as if I had just applied the stretch by six action, the product of the two original numbers

and in general, applying one of these actions, followed by another, corresponds with multiplying the numbers that they're associated with

in fact, the name for this group is the multiplicative group of positive real numbers

so, multiplication, ordinary familiar multiplication, is one more example of this very general and very far reaching idea of groups, and the arithmetic within groups

and we can also extend this idea to the complex plane

again, I'd like to think of fixing zero in place with one hand, and dragging around the point at one, keeping everything else evenly spaced while I do so

but this time, as we drag the number one to places that are off the real number line

we see that our group includes not only stretching and squishing actions, but actions that have some rotational component as well

the quintessential example of this is the action associated with that point at i, one unit above zero

what it takes to drag the point at one to that point at i, is a 90°rotation

so, the multiplicative action associated with i is a 90° rotation

and notice if I apply that action twice in a row, the overall effect is to [rotate] the plane 180°

and that is the unique action that brings the point at one over to negative one

so in this sense, i * i = -1

meaning the action associated with i, followed by that same action associated with i, has the same overall effect as the action associated with negative one

as another example, here's the action associated with 2 + i, dragging one up to that point

if you want, you can think of this as broken down as a rotation by 30°, followed by a stretch by a factor of √5

and in general, every one of these multiplicative actions is some combination of a stretch or a squish

an action associated with some point on the positive real number line

followed by a pure rotation, where pure rotations are associated with points on this circle, the one with radius one

this is very similar to how the sliding actions in the additive group can be broken down as some pure horizontal slide

represented by with points on the real number line

plus some purely vertical slide, represented by points on that vertical line

that comparison of how actions in each group breaks down is going to be important, so remember it

in each one you can break down any action as some purely real number action, followed by something that's specifically complex numbers

whether that's vertical slides for the additive group, or pure rotations for the multiplicative group

so that's our quick introduction to the groups

a group is a collection of symmetric actions on some mathematical object

whether that's square, a circle, the real number line, or anything else you dream up

and every group has a certain arithmetic, where you can combine two actions by applying one after the other

and asking what other action from the group gives the same overall effect

numbers, both real and complex numbers, can be thought of in two different ways as a group

they can act by sliding, in which case the group arithmetic just looks like ordinary addition

or they can act by these stretching, squishing, rotating actions

in which case the group arithmetic looks just like multiplication

and with that, let's talk about exponentiation

out first introduction to exponents is to think of them in terms of repeated multiplication, right?

I mean, the meaning of something like 2^3 is to take 2 * 2 * 2

and the meaning of something like 2^5, is 2 * 2 * 2 * 2 * 2

and a consequence of this, something you might call the exponential property

is that if I add two numbers in the exponent, say 2^(3+5)

this can be broken down as the product of 2^3 times 2^5

and when you expand things, this seems reasonable enough, right?

but expressions like 2^½, or 2^-1, and much less 2^i

don't really make sense when you think of exponents as repeated multiplication

I mean, what does it mean to multiply two by itself half of a time, or negative one of a time

so we do something very common throughout maths

and extend beyond the original definition which only makes sense for counting numbers to something that applies to all sorts of numbers

but we don't just do this randomly

if you think back to how fractional and negative exponents are defined

it's always motivated by trying to make sure that this property 2^(x+y) = 2^x * 2^y still holds

to see what this might mean for complex exponents, think about what this property is saying from a group theory light

it's saying that adding inputs corresponds with multiplying the outputs

and it makes it very tempting to think of the inputs not merely as numbers, but as members of the additive group of sliding actions

and to think of the outputs not merely as numbers, but as members of this multiplicative group of stretching and squishing actions

now it is weird and strange about functions that take in one kind of action and spit out another kind of action

but this is something that actually comes up all the time throughout group theory

and this exponential property is very important for this association between groups

it guarantees that if I compose two sliding actions, maybe a slide by negative one, and then slide by positive two

it corresponds to composing the two output actions, in this case squishing by 2^-1, and then stretching by 2^2

mathematicians would describe a property like this by saying that a function preserves the group structure

in the sense that the arithmetic within a group is what gives it its structure

and a function like this exponential plays nicely with that arithmetic

functions between groups that preserves the arithmetic like this are really important throughout group theory

enough so they've earn themselves a nice fancy name

"Homomorphism"

now, think about what all of this means for associating the additive group in the complex plane with the multiplicative group in the complex plane

we already know that when you plug in a real number to 2^x you get out a real number

a positive real number, in fact

so this exponential function takes any purely horizontal slide and turns it into some pure stretching or squishing action

so, wouldn't you agree that it would be reasonable for this new dimension of additive actions, slides up and down

to map directly into this new dimension of multiplicative actions, pure rotations?

those vertical sliding actions correspond to points on this vertical axis

and those rotating multiplicative actions correspond to points on the circle with radius one

so what would it mean for an exponential function like 2^x to map purely vertical slides into pure rotatios

would be that complex numbers on this vertical line multiples of i, get mapped to complex numbers on this unit circle

in fact, for the function 2^x, the input i, a vertical slide of one unit, happens to map to a rotation of about 0.693 radians

that is a walk around the unit circle that covers 0.693 units of distance

with a different exponential function, say 5^x, that input i, a vertical slide of one unit, would map to a rotation of about 1.609 radians

a walk around the unit circle, covering exactly 1.609 units of distance

what makes the number e special is that when the exponential e^x map vertical slides to rotations

a vertical slide of one unit, corresponding to i, maps to a rotation of exactly one radian, a walk around the unit circle covering a distance of exactly one

and so a vertical slide of two units would map to a rotation of two radians

a three unit slide up corresponds to a rotation of three radians

and a vertical slide of exactly pi units up, corresponding to the input pi * i

maps to a rotation of exactly pi radians, half way around the circle

and that's the multiplicative action associated with the number negative one

now you might ask "Why e? Why not some other base?"

well, the full answer resides in calculus

I mean, that's the birthplace of e, and where it's even defined

again, I'll leave up another explanation on the screen if you're hungry for a fuller description, and if you're comfortable with the calculus

but at a higher level, I'll say that it has to do with the fact that all exponential functions are proportional to their own derivative

but e^x alone is the one that's actually equal to its own derivative

the important point that I want to make here though, is that if you view things from the lens of group theory

thinking of the inputs to an exponential function as sliding actions, and thinking of the outputs as stretching and rotating actions

it gives a very vivid way to read what formula like this is even saying

when you read it, you can think that exponentials in general map purely vertical slides, the additive actions that are perpendicular to the real number line

into pure rotations, which are in some sense perpendicular to the real number stretching actions

and more over, e^x does this in a very special way, that ensures that a vertical slide of pi units corresponds to rotation of exactly pi radians

the 180° rotation associated with a number -1

to finish things off here, I want to show a way you can think about this function e^x as a transformation of the complex plane

but before that, just two quick messages

I've mentioned before how thankful I am to you, the community for making these videos possible through patreon

but, in much the same way that numbers become more meaningful when you think of them as actions

gratitude is also best expressed as an action

so, I've decided to turn off ads on new videos for their first month

in the hopes of giving you all a better viewing experience

this video was sponsored by Emerald Cloud Lab

and actually I was the one to reach out to them, since it's a company I find particularly inspiring

Emerald is a very unusual startup, half software half bio-tech

the Cloud Lab that they're building is essentially enables biologists and chemists to conduct research through a software platform instead of working in a lab

scientists can program experiments which are then executed remotely and robotically in Emerald's industrialised research lab

cuuI know some of the people in the company and the software challenges they're working on are really interesting

currently, they're looking to hiring software engineers and web developers for their engineering team

as well as applied mathematicians and computer scientists for their scientific computing team

if you're interested in applying, whether that's now or a few months from now

there are a couple special link in the description of this video, and if you apply through those it lets Emerald know you heard about them through this channel

alright, so e^x transforming the plane

I'd like to imagine first rolling that plane into a cylinder, wrapping all those vertical lines into circles

and then taking that cylinder and kindda smooshing it onto the plane around zero

where each of those concentric circles spaced out exponentially correspond with what started off as vertical lines

[cc first draft by Geoffrey Yeung]