
Intuitive Understanding of Euler’s Formula - sdeepak
https://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/
======
quietbritishjim
The article is quite right that multiplying by _i_ gives a rotation. But it
doesn't quite explain the reason for this: it's because that's the whole point
of defining imaginary numbers in the first place!

Remember you start off wanting to find a solution for the equation:

    
    
        i^2 = -1
    

This is actually easier to think about if you multiply it by a general real
number r:

    
    
        r i^2 = -r
    

In other words you want _i_ such that if you multiply _r_ by _i_ twice, it's
the same as multiplying _r_ by -1 once.

This is tough if you try to solve it by analogy with real positive numbers. If
you picture the real number line (all the possible _r_ ) and multiply it by a
positive number, let's say 4, then the whole thing stretches out quite a bit.
It's pretty obvious that the way to break this operation into two equal parts
is to stretch it a bit less (in this case, by a factor of 2).

The analogy of a stretch for -1 is a reflection: Imagine the whole number line
collapsing in towards zero and bouncing back out again. But if you stop this
half way then everything has just settled on zero, and doing that twice is
obviously not going to get to the whole reflection. No other intermediate
point seems any good either. (These are all the multiplications by _x_ where
-1 < _x_ < 1.)

The key idea of imaginary numbers is to consider multiplication -1 to be a
rotation by half a turn rather than a reflection. That is a lot easier to do
half of! As soon as you have multiplication by -1 as a rotation by half a
turn, it is obvious to identify _i_ as rotation by a quarter turn.

~~~
empath75
> it's because that's the whole point of defining imaginary numbers in the
> first place!

People used imaginary numbers for a long time before Cartesian coordinates
even existed.

~~~
crazygringo
I'll be honest, I've never understood how humanity didn't invent Cartesian
coordinates until 1637, with all the other engineering we had.

Once we had linear equations, for example with the ancient Greeks, not one
person ever thought to plot a line with it? Or to use it to calculate the
necessary building materials for something like a pediment or cathedral?

~~~
jerf
I think one of the keys to understanding math, and to a lesser but still
significant extent, physics history is to remember what you believed as a
child, and how you struggled with the concepts taught to you. And that's even
with a math curriculum designed to lead you to modern math. (One can debate
how _effective_ it is at that, but that's a separate topic.) Those
misconceptions we had as children are pretty fundamental to the human wetware.
Even today, with centuries of refinement and educational advancement, really
only a small fraction of people come away from school with the ability to
think truly mathematically.

For instance, even "just" negative numbers is a fairly counterintuitive
concept. Despite how they may seem universal today, they actually didn't pop
up in all that many cultures historically before their current line. And that
story gets repeated over and over for all sorts of developments. It takes time
for fields of study to process and abstract these things, because they weren't
just handed it on a silver platter in school.

(To the extent that that doesn't seem to be the case today, I'd say that as we
have become more and more mathematically sophisticated and the area of
mathematical inquiry exponentially increases, the dominant factor 'holding
back' math today is our inability to cover territory. Today nobody can
completely cover a major discipline before the next generation is already
coming in with fresh brains. That's a relatively recent development.)

~~~
eternalban
> It takes time for fields of study to process and abstract these things,
> because they weren't just handed it on a silver platter in school.

A sense of the phrase "knowledge is power" aligns with this.

I read Alan Kay's "User Interface - A Personal View" [1] recently, wherein he
discusses Seymour Papert's [2] ideas on learning, specifically the 3 stages of
learning. I found Papert's conception (with only mild exaggeration) to be
illuminating. For example, I now have a explanatory model as to why certain
inventions that did not require the du jour technology of the industrial age
were developed so late in the game.

[1]:
[http://www.vpri.org/pdf/hc_user_interface.pdf](http://www.vpri.org/pdf/hc_user_interface.pdf)
[2]:
[https://en.wikipedia.org/wiki/Seymour_Papert](https://en.wikipedia.org/wiki/Seymour_Papert)

------
sytelus
TLDR; Author writes:

 _Argh, this attitude makes my blood boil! Formulas are not magical spells to
be memorized: we must, must, must find an insight. Here 's mine:

Euler's formula describes two equivalent ways to move in a circle._

Euler's identity is a massive elephant and there have been many ways to look
at it from different angles. It wouldn't be fair to say that this single
interpretation suffices for views from other angles.

Here are couple of articles that goes in more details:

The remarkable Euler's Formula (3 part series):
[http://www.integralworld.net/collins30.html](http://www.integralworld.net/collins30.html)

An Appreciation of Euler's Formula: [https://scholar.rose-
hulman.edu/cgi/viewcontent.cgi?article=...](https://scholar.rose-
hulman.edu/cgi/viewcontent.cgi?article=1357&context=rhumj)

------
Michael_Groom
3blue1brown also has some nice visual explanations:

[https://www.youtube.com/watch?v=F_0yfvm0UoU](https://www.youtube.com/watch?v=F_0yfvm0UoU)

~~~
JshWright
He made a "sequel" video two years later that improves on the explanation.

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

------
man-and-laptop
I once had an interesting thought about the function e^x. I think this is a
key idea in the theory of Lie groups.

If the x in e^x = (1+x/N)^N is understood as some transformation, then e^x is
essentially repeating an infinitesimal transformation lots of times. So it's
like a for-loop where the body of the loop is some infinitesimal
transformation.

I tried to define the integration operator in terms of e^x. The 1 + x/N needed
to be one "infinitesimal" iteration of integration, that adds an extra
infinitesimal rectangle to the area. But it didn't seem to work out.

Ultimately that helps to explain Euler's formula. For large N, 1+ix/N is an
"infinitesimal" rotation by angle x/N. Repeating it N times produces a
rotation of angle x. That's essentially what TFA says. And it's a special case
of the Lie theoretic view of e^x.

~~~
532nm
>> I tried to define the integration operator in terms of e^x. The 1 + x/N
needed to be one "infinitesimal" iteration of integration, that adds an extra
infinitesimal rectangle to the area. But it didn't seem to work out.

You're close! This can indeed be done properly and is then called the Euler-
Maclaurin formula. For this, you define the "shift to the left by n operator"
e^(nD) where D is the differentiation operator d/dx.

You then always take the current value of f(x), multiply it by the small shift
n to get the first rectangle. Then you shift to the left by n, i.e. to
e^(nD)*f(x) = f(x+n), multiply that by the small shift n to get the next
rectangle etc.

The book "street-fighting mathematics" [1][pdf] has a very hands-on and
playful derivation of this in chapter 6.3.

[1] [https://mitpress.mit.edu/books/street-fighting-
mathematics](https://mitpress.mit.edu/books/street-fighting-mathematics)

[pdf][https://www.dropbox.com/s/722rlvrwy9l9w73/7728.pdf?dl=1](https://www.dropbox.com/s/722rlvrwy9l9w73/7728.pdf?dl=1)

~~~
man-and-laptop
Amazing, thanks.

------
chasereed
The proof of the formula is beautiful. It's common to defined the complex
exponential as the extension of the Taylor expansion of exp(x) to the complex
plane. Thus,

    
    
      exp(iy) = 1 + iy + (iy)^2/2! + ...
    

Now just group the even-numbered terms together and group the odd-numbered
terms together, the i's multiply to become 1 in the even numbered terms, and
what you get is

    
    
      (the Taylor expansion of cos) + i(the Taylor expansion of sin)

~~~
escherplex
Interesting. Looking at the Euler equation again as Argand plane rotation,
would e^-ix be a form of clockwise rotation? Using the methodology of:

[https://www.mathsisfun.com/algebra/eulers-
formula.html](https://www.mathsisfun.com/algebra/eulers-formula.html)

as a reference template, e^-ix would seem to involve:

(taylor cosine series) - i * (taylor sine series)

or e^-ix = cos x - i sin x = -1

which suggests another twist to the familiar identity:

e^ix * e^-ix = (cos x + i sin x) * (cos x - i sin x) = (cos x)^2 + (sin x)^2 =
(-1) ^ 2 = 1 = e^0

------
sevensor
Euler's identity was my secret weapon in graduate-level EE classes. Out of
laziness, I only ever memorized a couple of trig identities. (Trig functions
are a huge part of EE.) Whenever I needed a trig identity on a test, I whipped
out Euler's formula. From there, you have one step to definitions of sin and
cos that you can manipulate any way you want.

~~~
sannee
Wasn't some sort of engineering-grade complex analysis part of your EE
(undergraduate?) curriculum? I am sort of surprised Euler's formula would be a
"secret weapon" in EE classes.

~~~
nabla9
Analog electronics and signal processing fundamentally tied into complex
numbers but it's completely possible to do the math to pass exams etc. and
become EE engineer without deep understanding complex math. You just use
formulas and do arithmetic with them.

Most practically oriented engineering students bitch about the math heavy
parts because they don't have any use for them. They are happy with just the
formulas and arithmetic. They can calculate electronic circuits, do Laplace
transforms and Fourier transforms mechanically. As long as they understand
what goes in and what comes out, it works just fine.

------
vnorilo
my favourite intuition about Euler, which is not really an explanation,
somewhat tautological, and may or may not be wildly incorrect, but I like it
nonetheless:

e^x is a function whose value is its rate of change. (De^x=e^x). Now imagine
the unit circle by taking a point an unit away from O, and set "rate of
change" perpendicular to that vector. You will end up with Df(x) = i f(x),
which really only works when f(x) = e^ix, supposing i means perpendicularity.

~~~
pjbk
That is exactly what it is!

In the real plane you get exponential growth: the rate of change is equal to
the current value.

In the Argand plane you get a curling action due to the quadrature effect of
the imaginary unit. The rate of change at each point is the tangent, and the
result is therefore a circle.

Lie infinitesimal displacements capture this nicely, and also render the
generic case which is a similarity transformation, e.g. rotation through two
half reflections, e^(-w/2) * x * e^(w/2) like those found in quaternions and
Clifford algebras.

~~~
vnorilo
Thank you for this! The wikipedia rabbit hole beckons :D

------
ttoinou
Nice. Made some similar explanations with GeoGebra drawings a while back. See
those pages in french (just look at the pictures !) :

[https://fr.wikiversity.org/wiki/Calcul_avec_les_nombres_comp...](https://fr.wikiversity.org/wiki/Calcul_avec_les_nombres_complexes/Annexe/D%C3%A9monstration_de_la_formule_d%27Euler)

[https://fr.wikiversity.org/wiki/Calcul_avec_les_nombres_comp...](https://fr.wikiversity.org/wiki/Calcul_avec_les_nombres_complexes/%C3%89criture_exponentielle_et_trigonom%C3%A9trique)

------
sweetheart
To anyone who finds this sort of explanation interesting or helpful, I
recommend you check out "A Most Elegant Equation" by David Stipp, who covers
Euler's Formula from step 0 for those with zero formal math knowledge. I'm
definitely in that camp of people, and I was able to get a lot out of it. It's
actually the book that helped several mathematical concepts "click" for me.
Plus David Stipp just writes very romantically about math, which I thought was
warming, and gave a lot of life to a field that I know nothing about.

~~~
dstipp
Thanks for your kind words about my book. I never quite knew when writing it
whether it would find its way to the people I mainly wrote it for -- those
interested in math who don't know a whole lot about it. They aren't thick on
the ground. So it's a real, sort of rare blast for me to hear about it
arriving where I'd hoped.

------
rofo1
People that like this article might be interested in 'Visual Complex Analysis'
by Tristan Needham. It's (IMHO) a rare book of collections of "concrete"
analogies for complex analysis.

~~~
User23
Yes, this page appears to be lifted directly from that book.

------
jordigh
The way I thought about it is that the trigonometric, hyperbolic, and
exponential functions are all nontrivial solutions of the differential
equation y'''' = y. Sharing differential properties is a very powerful
kinship, which is also why you can substitute any of them for any of the
other: the solution space of a linear differential equation is a vector space,
and these three families of functions are just a change of basis in this
space.

Well, I don't know, this is what makes sense to me.

------
amai
This is a very confusing way to explain a simple thing. In fact "Eulers
formula simply shows how one can parametrize a helix using the exponential
function", see

[https://math.stackexchange.com/questions/3510/how-to-
prove-e...](https://math.stackexchange.com/questions/3510/how-to-prove-eulers-
formula-ei-varphi-cos-varphi-i-sin-varphi/625151#625151)

And that's it.

~~~
romwell
There are many answers in that thread; I think different things work for
different people.

My favorite angle on this is the following graphic/animation, also present in
the thread:

[https://upload.wikimedia.org/wikipedia/commons/0/0e/ExpIPi.g...](https://upload.wikimedia.org/wikipedia/commons/0/0e/ExpIPi.gif)

This shows how (1+i * Pi/N)^k, k=1..N traces out a semi-circle for large
values of N.

Geometrically, all it says is:

* Draw a right triangle ABC with AB=1, BC=Pi/N, and ABC the right angle

* Make a copy of ABC, call it A'B'C', and scale it so that A'B'(the long leg) = AC (the hypotenuse)

* Put A'B'C' over ABC so that A'B' and AC coincide

* Let ABC=A'B'C'

* Repeat the process N times

* Look where you end up when N is large enough

The answer is: when N is large, Pi/N is small, and the right triangle ABC is
almost isosceles, AB ~= AC. So you end up with N slices of a pie that make up
a fraction of a circle.

Which fraction? Well, the perimeter is N/Pi * N = Pi - so half a circle. So if
A=(0,0) and B=(1,0), you end up at (-1,0).

Now (1+x/n)^n approaches e^x, so it makes sense to _define_ e^(i * Pi) to be
the same limit - which we found out to be -1 + i * 0.

------
MAXPOOL
Yet another way to visually understand complex functions is to think them as
2D plane to 2D plane transformations. You draw a picture, grid or curves into
complex plane, then run it trough complex function you are interested in and
see how it looks like.

Here is w = e^z:

[https://i.imgur.com/pAALOh2.png](https://i.imgur.com/pAALOh2.png)

Just look at the above picture and every detail until it starts to makes
sense.

------
syntaxing
I love reading posts like these. I was talking to my friend about how doing
math proofs during HS is pretty similar to a math "lab" even though it's
tedious and seemingly useless to some. I wish we were taught higher levels
proofs like this. Even though we couldn't appreciate it at that time, it would
of definitely helped us in the future.

------
foxes
i^i is actually multi valued - it depends on which branch of log(z) you
choose. The rotation analogy is still correct.

~~~
OscarCunningham
And all of the values of i^i are real!

------
Myrth
It looks as a circle only from one direction.

I imagine it as a spiral.

Imaginary number adds 2nd dimension, and exponential growth adds 3rd
dimension.

------
aap_
When was this posted? I think i read exactly that explanation like 7-8 years
ago and it really made it click for me. It was a wonderful insight!

~~~
okket
It was posted a few times here before, first time was 2010. Todays posting is
the first that gained traction and comments.

See
[https://hn.algolia.com/?query=Intuitive%20Understanding%20of...](https://hn.algolia.com/?query=Intuitive%20Understanding%20of%20Euler’s%20Formula&sort=byDate&dateRange=all&type=story&storyText=false&prefix&page=0)

------
saagarjha
(i^i)^i is pretty easy to do with Euler's formula: it's just
i^(i*i)=i^(-1)=1/i=-i.

~~~
sischoel
I think you have to be careful here when using rules that hold for real number
for complex numbers. In fact, Wolfram|Alpha says, that -i is just one of
multiple results:
[http://www.wolframalpha.com/input/?i=(i%5Ei)%5Ei](http://www.wolframalpha.com/input/?i=\(i%5Ei\)%5Ei)

~~~
OscarCunningham
Indeed you can easily produce completely incorrect results if you aren't
careful, like 1 = sqrt(-1)/sqrt(-1) = (-i)/(i) = -1.

------
jshowa3
I don't understand why this is so profound. I knew Euler's formula was a trig
formula simply by looking at it. I don't know why there needs to be this
complex explanation of where it came from where its obvious that it comes from
trig and the concept of a unit circle.

The problem is that reliance on intuition doesn't prove anything and its very
deceptive. You end up having to store several cases of explanations instead of
deriving one through a proof.

For example, one would intuitively think dropping a heavy object and a light
object would conclude with the heavy object hitting the ground first which is
not the case. This is just one example.

I'd much rather derive things the traditional way and not be duped by
"intuitive" explanations. Because ones intuition is often different from
natures.

~~~
jonsen
Intuition is not an absolute thing. Intuition changes as we learn. Sometimes
we learn things without building an accompanying intuition. This post is about
building a sound intuition for something learned.

~~~
jshowa3
That's nice. I didn't say there was no value in intuition. I just said it was
unreliable and I demonstrated an example of why its unreliable.

Its better to focus on solutions that don't rely on intuition because
intuition is based solely on experience and often can't be applied to
everything. This is why there's a strive to find general solutions to things.
Intuition can help you, but you should not base your understanding off it.
Intuition can lead you astray easily as in my example. People just assumed it
was correct for many years until Galileo actually proved it through
experimentation.

In fact, I found this explanation way more complicated than just stating that
you are performing trigonometry in two different domains. Your sine is scaled
by an imaginary value and your cosine is scaled by a real value. The radius
equates to an exponential. When you move it pi units you get sin = 0, cosine =
-1 which you use algebra to get Euler's identity.

You don't need to introduce complex intuition in order to get it. Intuition is
often a crutch people use to try and think they understand something and just
blame others when they don't get it much like this post. If intuition was so
important, why not submit it as a legitimate mathematical way of proving
things?

