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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]