
Why does e to pi i equal -1? (2015) [video] - espeed
https://www.youtube.com/watch?v=F_0yfvm0UoU
======
cousin_it
1) e^x is a function whose derivative is equal to its value.

2) e^ix is a function whose derivative is equal to its value rotated by 90
degrees (ie^ix).

3) As x goes from 0 to pi, the trajectory of e^ix always has a velocity vector
perpendicular to its current position. For example, when x = 0, the current
position is 1 and the velocity vector is i.

4) So the trajectory a circle arc of length pi, which ends at -1.

~~~
amelius
I think you should explain step 2 in more detail.

------
unholiness
This explanation strikes me as a little too aggressive in throwing out the
notation with the bathwater, only reaching its result by redefining the terms
we already have intuition for into space-stretching operations that don't work
like arithmetic does in my head.

In that sense I think I have the same problem with this proof that I do with
the standard one, where you add the Maclaurin series of cos(θ) and i * sin(θ)
and match term-by-term with the series for e^(i * θ). The problem is, at the
point you can actually show equality, the things on one side aren't obviously
a rotation and the things on the other side aren't obviously an exponential.

I'm not just hear to yell at clouds. I was given a proof that I truly love by
a professor I adore, which I think really does give insight into what all
these operators are doing. The best video I can find with it is here:

[https://www.youtube.com/watch?v=-dhHrg-
KbJ0](https://www.youtube.com/watch?v=-dhHrg-KbJ0) (Skip to 7:30 if you're
already comfortable with the limit definition of e^x)

The basic summary is:

1) e^iθ is equal to (1 + iθ/n) ^ n for large n

2) That base, (1 + iθ/n), plotted as a complex number, has length approaching
1, angle approaching θ/n

3) The base squared, (1 + iθ/n)^2, by de moivre's theorem, forms another point
as if the transformation from (0, 1) were repeated twice — that is, the length
stays one, and another tiny angle is added for a total of 2θ/n

4) The full result is therefore n transformations, taking the path along the
unit circle, traveling θ and arriving at cos(θ) + i * sin(θ)

~~~
jacobolus
Your (1) is just one of many possible (technically equivalent) definitions for
the exponential function. And arguably not the most basic/natural/intuitive
one.

The primary motivation for the exponential function is to be the inverse of
the logarithm function. And the motivation of logarithms is to convert
multiplication problems to addition problems, so they could be solved with
table lookups (later performed on a slide rule) instead of difficult
arithmetic. That is, log _ab_ = log _a_ \+ log _b_. Which is to say, exp( _c_
\+ _d_ ) = (exp _c_ )(exp _d_ ). Just setting this constraint along with the
derivative exp’ 0 = 1 is enough to characterize the exponential function.

Or you can get to this function in many other ways, e.g. by solving the
differential equation d/dx exp _x_ = exp _x_ ; by the series 1 + _x_ \+ _x_
^2/2 + _x_ ^3/6 + ...; or by defining the logarithm as the definite integral
of 1/ _x_ starting at 1, and then taking the exponential function to be its
inverse. I like defining the (complex) exponential as a conformal mapping
between the cylinder and the plane minus a point.

~~~
unholiness
Of course! I think there's a good reason (1) is the first definition of e^x we
get in school, though. I don't think it's the most "natural" definition, but I
think it's the most grok-able. It's easy for high-schoolers to learn precisely
_because_ each part of it is so intimately familiar that it's clear without
hand-waving precisely what everyone's talking about.

When we then step into viewing real exponentials as "The thing that gets
bigger at a rate proportional to where we're at", it's un-abstract, which is
honestly important for intuition. Of course you're going to want to then view
it with the calculus definition, or as a matrix, or as an operator... But I
wouldn't start there.

So I feel the same about the complex version. There's another nice proof in
this thread that starts with the calculus definition and follows a similar
track. It's a much more beautiful proof! But I worry some will lose intuition
when we start the conversation with "a complex exponential's rate of change is
perpendicular to its position"...

That said, if I were to ever teach a math class on complex numbers, I think
fleshed-out versions of these two 4-step proofs are the ones that together
best build intuition for what "rotation in complex exponentials" really means.
That rotational aspect is the foundation of much of the handwaving done in
control theory, DSP, ASP, signals and systems, and really anything involving
fourier and laplace transforms. So, it's important to grok it well.

~~~
jacobolus
Anything which repeats on some periodic interval (e.g. a signal which repeats
in time) can be represented geometrically as being defined with respect to
uniform motion around a circle (sometimes there are physical circles involved,
and sometimes it’s just an abstract circle).

Complex numbers (a two-part complex of a scalar part and a bivector part) and
the complex logarithm/exponential are the natural formalism to use for
describing uniform circular motion, and are therefore the natural formalism
for any kind of periodic signal.

The way to teach this is to start with vectors in the plane, and then teach
about geometric products/quotients of vectors (this is a subject called
“geometric algebra” or “Clifford algebra”, and the basics are plenty
accessible to high school students). All of the mystery is removed from
complex numbers when they are taught this way.

[http://www.shapeoperator.com/2016/12/12/sunset-
geometry/](http://www.shapeoperator.com/2016/12/12/sunset-geometry/)

------
phkahler
It's a nice video but it goes so fast I don't think people who don't already
understand this stuff will get it.

~~~
ekianjo
> goes so fast

Especially goes too fast right when it becomes less obvious.

~~~
phkahler
Like when he presents equations that he said we wouldn't need...

~~~
espeed
See MIT's "Big Picture of Calculus" video:

[https://www.youtube.com/watch?v=UcWsDwg1XwM&list=PLBE9407EA6...](https://www.youtube.com/watch?v=UcWsDwg1XwM&list=PLBE9407EA64E2C318&index=2)

------
obastani
The most elementary explanation I ever got for why e^(i pi) = -1 is from the
definition

e = lim_{n -> infinity} (1 + 1/n)^n

First, we can generalize this to

e^z = lim_{n -> infinity} (1 + 1/n)^(z n) = lim_{m -> infinity} (1 + z/m)^m

using the substitution m = zn. Therefore,

e^(i pi) = lim_{m -> infinity} (1 + i pi/m)^m

Now, converting the complex number (1 + i pi/m) in terms of polar coordinates
(r, theta) yields

r = (1 + pi^2/m^2) ~ 1

theta = sin^{-1}(pi/m) ~ pi/m

Since the product of two complex numbers is

(r, theta) (r', theta') = (r r', theta + theta'),

we have

(1 + i pi/m)^m ~ (1^m, m * pi/m) = (1, pi) = -1.

------
timlod
I love the Khan Academy video on this:

[https://www.youtube.com/watch?v=mgNtPOgFje0](https://www.youtube.com/watch?v=mgNtPOgFje0)

Sal's enthusiasm at the end is contagious!

~~~
mstade
I've learned so much from Khan Academy – not just whatever is explained by the
video I'm watching, but different ways of educating people as well. In fact, I
helped out with tutoring basic math a while back and I struggled at first
because I didn't realize how much I _wasn 't_ teaching simply because it was
second nature to me. After watching a bunch of Khan Academy videos on the same
topics, I found different ways of explaining things, and was much more
successful. Khan Academy has really challenged my own way of thinking about
many things, math in particular.

It's a great resource, and while I don't use Khan Academy much these days I
still donate $100 every year.

------
datainplace
Is it pure chance that this was posted only two or three days after I watched
it along with a few other e to pi = -1 videos?

Randomness aside. 3blue1brown makes some wonderful math videos that I find
really explain the intuitiveness of some of the ideas. I was unfortunately
cursed with a math teacher who for whatever reason required us to memorize
until we passed the test. Imaginary numbers were taught as "something that
will help you in college"

~~~
vinchuco
Richard Feynman used to go up to people all the time and he'd say "You won't
believe what happened to me today... you won't believe what happened to me"
and people would say "What?" and he'd say "Absolutely nothing".

I agree wholeheartedly with the quality assessment of 3blue1brown videos. All
mathematics should be clear and intuitive, by definition :)

~~~
dvt
> All mathematics should be clear and intuitive, by definition :)

I disagree with this completely, and so does all of higher mathematics. It's
neither clear nor intuitive. In fact, as soon as you start learning about
infinities (Calc I), intuition becomes hit and miss.

~~~
CN7R
Intuition always comes after the feeling of awe or confusion, in my opinion.
Really depends on how you learn.

------
skybrian
If you prefer slides over a video, here is a slideshow I wrote:
[https://docs.google.com/presentation/d/1oMNjkDp-
LieSGnZEwNpc...](https://docs.google.com/presentation/d/1oMNjkDp-
LieSGnZEwNpceG8KTIvbnus9olu3KqnM5bg/edit)

Or you might prefer Better Explained's article:
[https://betterexplained.com/articles/intuitive-
understanding...](https://betterexplained.com/articles/intuitive-
understanding-of-eulers-formula/)

~~~
mih
Upvoted for the Better Explained link. I always liked their explanation of _e_
[https://betterexplained.com/articles/an-intuitive-guide-
to-e...](https://betterexplained.com/articles/an-intuitive-guide-to-
exponential-functions-e/)

------
nilkn
In Lie theory, given a Lie group, there's a general notion of an "exponential
function", which maps elements in the tangent space to the identity to "full"
elements of the group.

In this case, our group is the unit circle in the complex plane. This is not
circular logic, by the way. If a, b are complex numbers on this circle, then
|a| = |b| = 1 and so |ab| = |a||b| = 1.

The identity of the group is the complex number 1 (with plane coordinates
(1,0)). So the tangent space to the identity is the vertical line 1 + it, for
t a real-valued parameter. In two-dimensional coordinates, that expression
looks like (1,0) + t * (0,1).

(To be completely precise, the tangent space is a vector space, not a line
displaced from the origin. In particular, the tangent space must contain a
zero vector.)

If v is an element of this tangent space, then in Lie theory the exponential
of v, exp(v), is defined to be g(1), where g is the unique geodesic on the
circle (passing through the identity element 1) whose velocity at time 0 is v.

Visually, this is sort of like taking the tangent vector v = ti, placing its
base at 1, then wrapping it around the circle and marking where its endpoint
ends up at. If the vector is v = pi*i, then it has length pi, so it will end
up demarcating an arc length of pi on the circle. Since we're working with the
unit circle, this takes us straight to (-1,0).

I'm still leaving a lot out, of course -- most importantly why this notion of
exponential has anything to do with the ordinary one.

~~~
chombier
I was about to write a similar story, but then I realized you need a
Riemannian metric (and exponential) for everything arclength-related.

IIRC since the unit circle is a compact Lie group, there is a bi-invariant
Riemannian metric whose exponential is the Lie group exponential and we land
back on our feet, but the Lie group structure alone is not sufficient.

------
mitchtbaum
e^τi = 1.

In words, this equation makes the following fundamental observation:

 _The complex exponential of the circle constant is unity._

Geometrically, multiplying by e^iθ corresponds to rotating a complex number by
an angle θ in the complex plane, which suggests a second interpretation of
Euler’s identity:

 _A rotation by one turn is 1._

The Tau Manifesto [http://tauday.com/tau-manifesto#sec-
euler_s_identity](http://tauday.com/tau-manifesto#sec-euler_s_identity)

~~~
wodenokoto
Thanks for posting this. I recently saw another video about this equation than
the one posted here and thought "maybe this is one of those cases where tau
doesn't work out nicely", but didn't have the intuition to see if it was or
not.

------
hprotagonist
This seemed like voodoo until i took signals and systems. Then it's just a
normal observation about vector sums on the complex unit circle.

~~~
gydfi
It seems like voodoo when you have high-school mathematics, because you've
been taught about "to the power of" in terms of multiplying something by
itself a certain number of times, and multiplying something by itself an
imaginary number of times is nonsensical.

Once you've done enough maths you think of it in a different way and e^ipi
seems obvious.

It's just a different conception of what the symbols mean, I suppose.

~~~
mstade
There are so many assumption that were hammered into me in high school (and
below) that I find hold back my thinking, rather than help me. I kind of feel
like the education I received was full of hacks to get me past whatever test
was coming up, rather than truly teaching me. I've spent years playing what
feels like catch up to actually truly learn the things I thought I knew – a
lot of high school mathematics included.

~~~
fl0wenol
What was not obvious to me, and took a lot of higher math to get to, was that
in the grouping of N=>I=>Q=>R=>C, at each stage you're pulling in a more
complete mathematical representation (counting, rings, fields) until you're
algebraically closed, and while you could go higher, you start losing
properties you like again, like commutability over multiplication.

Complex numbers are that nice saddle point, which is an interesting thing to
ponder.

I wish there was more focus on these "why" aspects in at least the optional
advanced math or physics you could get in HS. It helps put some things in
perspective.

------
lowpro
I've been watching 3Blue1Brown a lot recently, and he's been linked to many
times here too. His work is absolutely amazing, and he even wrote a python
library to make the animations!

~~~
espeed
Yes, the Python library is called Manim, and it's on GitHub:

[https://github.com/3b1b/manim](https://github.com/3b1b/manim)

------
1001101
Gauss said that if the reason weren't immediately obvious to someone, they
would never be a first rate mathematician. For the rest of us, there's
YouTube.

------
0xbadf00d
Very Nice Video - One of my favourite sub 5 minute explanations is Oliver
Humpage's @ Ignite Bristol circa 2011:

[https://www.youtube.com/watch?v=vHaPyuEkK4A](https://www.youtube.com/watch?v=vHaPyuEkK4A)

Hope you enjoy as much as I did.

------
somewhat_slow
At 2:37, right after finishing explaining adders and multipliers on the line
it started going way too quick like where did e^x and some these infinite sums
come from???

~~~
gizmo686
He introduces e^x as a way of converting adders into multipliers.

The infinite sum is not important. I believe he is showing it purely as a
visual tool to indicate that e^x is just "some function".

More specifically, based on the video, the important feature for a adder ->
multiplier converter to have is that f(x+y) = f(x)f(y). However, many
functions satisfy this requirement. He states (without justification [0]) that
e^x is the most "natural" choice for such a function, and provides the
infinite sum as a visual aid to show that it is just some function.

[0] Rather, justification is provided by reference to another video.

------
dvt
I've seen this a while ago, and while it's pretty instructive, it's actually
also pretty confusing. The magic happens in a seemingly-innocuous throwaway
sentence at around 4:20 (after being introduced to the 2D plane):

> ... This can now include rotating along with some stretching and shrinking
> ...

It's entirely non-obvious WHY we should be okay with _rotating_ all of a
sudden. The real answer is not super complicated, but it deals with a couple
of amazing relationships between exponential functions and trigonometric
identities[1]. So really, we don't have to accept "rotating" as some weird new
action we can do when moving into the complex plane, we just have to accept
that trigonometry is weirdly related to analysis due to some very cool
properties.

[1]
[https://www.phy.duke.edu/~rgb/Class/phy51/phy51/node15.html](https://www.phy.duke.edu/~rgb/Class/phy51/phy51/node15.html)

~~~
moresilenter
as a masters in mathematics, this is exactly what i came here to write.
thanks!

To have a notion that multiplication by imaginaries causes rotation, you'd
need Euler's formula. I honestly think the best way to get a visual sense for
why multiplication by r _exp(i_ theta) is to look at the first few terms of
the taylor series added together and see that the adders combine into a spiral
that converges on r _cos(theta) + i_ sin(theta). that is still plenty visual
for me.

~~~
gizmo686
>To have a notion that multiplication by imaginaries causes rotation, you'd
need Euler's formula.

No you don't. You might notice this by simply working with imaginary numbers.
You might invent imaginary numbers for rotation [0].

Alternatively, consider the multiplication (x + yi)(a + bi), as the value (a +
bi) performing a transformation on (x + yi). We want (x + yi)(a + bi) = xy -
by + ayi + bxi. If we consider (x + yi) to be a 2 dimensional matrix (with
basis 1 and i), we can write the above equation as a matrix multiplication:

    
    
        [ x   y ] [  a   b ] = [ ax-by ; ay + bx]
                      -b  a
    

Notice that

    
    
        [  a   b  ] 
          -b  a
    

is just a rotation matrix multiplied by the scalar (a^2 + b^2).

[0] That is, define a group (in the group-theory sense) of functions of
rotations, denoted as xi for real numbers x, and a group of group of functions
for sliding, denoted x for real numbers x. You might then notice that you can
combine these groups in a field structure, that happens to have 1i * 1i = -1,
and that the subfield of elements with no i component happens to be isomorphic
the the reals.

~~~
mtourne
Yay skew symmetric matrixes.

It's also fun to introduce e as a matrix operator for 3d rotations.

It's useful for kinematics and having compact representations for axis angle
notations. It's a little far in my head but at some point it felt like a ha-ha
moment with the Euler identity, in the "2d version".

~~~
jacobolus
In the three-dimensional case, for any practical purpose, use quaternions.
[https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotati...](https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation)

(At least if you’re ever needing to compose rotations; to apply a quaternion
to a big array of 3-vectors, go ahead and convert it to a 3x3 matrix first,
which should end up slightly more efficient.)

3x3 rotation matrices are really hard to keep normalized properly, whereas
quaternions are trivial to normalize (just divide by the norm).

If you need to compress a unit quaternion down to 3 numbers, instead of taking
the logarithm (which is computationally expensive and a pain to deal with) use
the stereographic projection.

~~~
mtourne
Quaternions are a useful tool for manipulating rotations in a lot of common
applications. But that wasn't my point here. Also quaternions are hard to
grasp by humans. I find axis-angle much more palatable in general.

Imaginary numbers can be represented with a 2x2 skew symmetric matrix with no
stretch of the imagination at all. And 3x3 skew symmetric matrixes represent
rotations most compactly with only 3 actual variables. Instead of 4 for
quaternions, 9 for "classic rotation matrixes", or the need to tell which is
the order of the angles if you're given 3 euler angles.

There are interesting applications of Lie Algebra on SO(3) [1], notably in
computer vision where a global energy is minimized across two successive rgb-d
"shots" in order to recover the infinitesimal rotation [2]. It's going to be
easier to minimize energy on something that is most compactly defined, and
always amounts to a valid rotation.

[1]
[https://en.wikipedia.org/wiki/Rotation_group_SO(3)#Lie_algeb...](https://en.wikipedia.org/wiki/Rotation_group_SO\(3\)#Lie_algebra)
[2]
[https://vision.in.tum.de/_media/spezial/bib/kerl13icra.pdf](https://vision.in.tum.de/_media/spezial/bib/kerl13icra.pdf)

~~~
jacobolus
From what I understand that would be (sorta; I’m not an expert in Lie theory)
the logarithm of a rotation. I find the stereographic projection to be a more
useful way to compress an arbitrary rotation down to 3 dimensions, for most
purposes.

~~~
mtourne
Yes, precisely. exp() is the mapping from so(3) to SO(3), so you can use log
to go the other way around.

Here is the relationship with the other representations [1]

Where it becomes interesting for our rigid-body transform application (or
recovery of it, in the case of computer vision), is with Twist coordinates (6
element vector) which will map to a 4x4 Transform, again using the matrix
exp() operator [2].

[1]
[https://en.wikipedia.org/wiki/Axis%E2%80%93angle_representat...](https://en.wikipedia.org/wiki/Axis%E2%80%93angle_representation#Relationship_to_other_representations)
[2]
[https://en.wikipedia.org/wiki/Screw_theory#Twists_as_element...](https://en.wikipedia.org/wiki/Screw_theory#Twists_as_elements_of_a_Lie_algebra)

------
neximo64
For anyone that finds it beautiful isn't there a bit of humanisation and
definition involved (for example the Sine function used in deriviation uses
'pi' instead of 90 degrees), not to mention Sine is a human created function.
You could have e to pi (90 * -1) too.. or a different method to define angles
instead of having 360 degrees (base 60)

~~~
panic
Yeah, you can have the same thing with degrees if you use a different exponent
base than _e_ :

    
    
        e^((pi/180)ix) = (e^(pi/180))^ix = E^ix
    

and therefore

    
    
        E^180i = -1
    

where _E_ is about 1.0176065.

The particular base _e_ has some nice properties, though, like that its rate
of change _d /dx e^ix_ at a given point _x_ is just _i e^ix_. The rage of
change for _E^ix_ , on the other hand, is _d /dx E^ix = (pi/180) i E^ix_,
which is a little less "natural". This strange fact -- that using radians
makes the expression _e^ix = cos x + i sin x_ have a nice derivative -- is one
of the reasons why mathematicians like to define these functions in terms of
radians instead of degrees.

------
cmollis
I had never understood how imaginary numbers had any bearing on physics
(probably because I'd never been taught). But lately I've thinking about how
all mathematics must have some natural physical equivalent or relation. E.g.
how does multiplication happen in nature.. what does it mean really to
multiply something?...how does this happen in nature? For the most part, we
take these things as facts and learn the mechanics, but in physics, it seems
to me, you really need to understand how these algebraic interpretations
translate into physical realities (also, perhaps sort of obvious for everyone
reads HN). For someone who enjoys discovering these perhaps obvious things
later in life, it's clear how mathematicians and physicists could clearly come
to the same mathematical conclusions from completely opposite vectors.. I
guess this probably happens all the time.

~~~
tim333
It's an interesting and unresolved question whether imaginary numbers are
fundamental to the nature of physics or just a handy tool to calculate stuff.
They are certainly handy for calculating - they are an easy way to represent
oscillations.

~~~
digler999
Maybe it's a mistake to allow imaginary to ever be separate from real numbers
? Another post here brought up how easily it is to work with imaginaries if
you just treat them as the vector [re im]. Maybe (philosophically speaking),
there are no purely real numbers. Could it be that all quantities in nature
must contain a (sometimes zero) Im component ? That might be a more satisfying
interpretation than just allowing them to creep in when they are absolutely
required, such as polynomial equations.

------
jjaredsimpson
e^x = an infinite series

if you substitute x = iz

you can split the even and odd terms of the infinite series into cos and sin

e^iz = cos z + i sin z

evaluating at z = pi yields

cos pi + i sin pi

-1 + 0i = -1

the exponential function maps the imaginary axis to the unit circle. pi gets
mapped to -1.

This shows its true, but the "why" and real understanding requires calculus
and the first week of complex analysis. Otherwise you are just parroting a set
of facts.

~~~
gizmo686
Math major here.

This presentation does skip some nessasary legwork for complete rigor, but
presents a valid non-standard construction of e^ipi.

Specifically, he defines two sets of objects: adders and multipliers, and a
function (written e^x for historical reasons) that maps adders into
multipliers.

He then generalizes this construction to work in 2 dimensions instead of 1
dimension.

I would add to this construction that e^x is the particular converter that
maps pi -> -1. However, I think this requirement is implicit in him stating
that e^x is the most "natural" of the converters, and that pi -> -1 is the
most natural of mappings.

The only place where you might need calculus is to provide an explicit
construction of e^x (and possibly to justify the notational choice of writing
it like an exponential).

EDIT:

To make the point clearer. Under the construction presented by the video, e^x
is not defined as an infinite some, but rather as a function satisfying
certain properties. That this function is equal to a particular infinite sum
is a statement that requires proof; and not a statement that is needed for
many applications.

------
gizmo686
As always with this author, I followed a link to a math explanation video
expecting to point out how it is only a superficial treatment that breaks down
under rigorous analysis, only to be dissapointed in my inability to find any
such flaws. Seriously, this guy is awesome. If you need to learn math, see if
he has a video about it.

I do have 1 gripe with this video though, and that is his handwaving around
"natural".

More specifically, as far as this particular case goes, there is nothing
natural about the choice of e^x, or the significance of pi.

As he identifies, we are interested in some function that maps adders into
multipliers with the property f(x+y)=f(x)f(x).

To uniquely identify such a function, we need to add an additional constraint.
He chose to add f(pi) = -1. He justifies this by arguing that pi is the length
you would travel along the unit circle to arrive at -1. This is true (and the
underlying reason why pi and e end up being natural), however using this
argument seems to break the abstraction for me.

Under this construction, in the equation f(i * pi)=-1, "i * pi" is an object
in the set of multipliers, and "-1" is an object in the set of adders.
Specifically, "i * pi" is a _function_ which takes a plane (or perhaps a
point) and rotates it, while "-1" is a function which takes a plane (or point)
and slides it.

He then invokes an unstated mapping, g, to convert the multiplier [0] "i * pi"
into the real number "pi". He than insists that g(x) gives the distance a
point would travel along the unit circle when the multiplier x is applied to
it.

At this point, because arriving at the point "-1" [1] from the point "1"
through rotation requires traveling pi distance, it makes sense that f(i * pi)
= f(g^-1(pi)) = -1

I am still not convinced that (within this construction alone), this is a more
natural choice that saying that f(i) = 1, but would agree that it is one of
the two natural choices. To get to f(x) = e^x being _the_ natural choice
requires showing that it comes up in all sorts of unrelated parts of math, so
it is probably more natural.

[0] It might be better to speak of "rotational multipliers" here, as I am not
sure how to natural define g(x) for multipliers that stretch the plane,
instead of or in addition to simply rotating it.

[1] This "1" is again distinct from the adder "1" and multiplier "1", but
plays a central role in defining them, so I do not object to its usage.

------
jejones3141
Look at the Taylor series for e^x, and see what you get for e^(i * theta). The
terms with i raised to an even power are real, those with i raised to an odd
power are imaginary. Group the odds together and the evens together, and
compare with what the Taylor series for sin x and cos x give you for sin theta
and cos theta. (I'm being a bad person here, ignoring convergence issues, but
I want to say that all three are absolutely convergent.)

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etatoby
I think Mathologer's take is much more intuitive:

[https://youtu.be/-dhHrg-KbJ0](https://youtu.be/-dhHrg-KbJ0)

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ageofwant
Why not just write it e^iτ=1 which conceptually and pedagogically makes just
so much more sense.

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ttoinou
BTW for thoses who like Complex Analysis here's how to visualize them as a 2D
deformation :
[https://www.youtube.com/watch?v=CMMrEDIFPZY](https://www.youtube.com/watch?v=CMMrEDIFPZY)

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hgdsraj
Honestly the calculus explanation is the most clear to me (Taylor series
expansion of e^x)

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agumonkey
I deeply thought the iterative nature was advanced ..

I see how intuitive the scale / rotate mindset is; but I'm a bit sad that
iteration is a mishap in their view.

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hawkice
> (2015)

I'd prefer if admins could change the link to a description of this
mathematical fact that's more up to date, and fits the needs of me and my
family in 2017.

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espeed
It originally said "Why does e to pi i equal -1? (visualized)".

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faragon
Pure beauty.

