There are at least nine artists with this name:

1 - E's real name is… Read Full Bio ↴There are at least nine artists with this name:

1 - E's real name is Mark Oliver Everett and he is the driving force behind the band Eels. He released two solo albums under the moniker "E": Broken Toy Shop and A Man Called E.

2- E is a new project from American alt rocker Thalia Zedek (Come/Live Skull), a new power trio of experimental rock troubadours w/ Jason Sanford of Neptune and Gavin McCarthy of Karate.
https://www.facebook.com/ABandCalledE/
https://abandcallede.bandcamp.com/music
http://abandcallede.com/

3 - E is a Finnish/Portuguese Experimental Black metal project. ShadoW (Portugal) and MTYM (Finland) are the only members of this band.
Facebook page: http://www.facebook.com/pages/E/168957753128955?v=wall
Myspace page: http://www.myspace.com/edimension

4 - E is also the name of a Chinese alternative rock band.

5 - E was also a punk rock band from Caracas, Venezuela that existed around mid 2002-mid 2003 that featured Archi from ATTE and IL from Boom Boom Clan, Las Américas, La Fiera Del Vinil, Ulises Hadjis' live band and Elaine.

6 - E is also a Czech progressive rock-alternative band from Brno.
Visit http://www.facebook.com/pages/E/49547237061 for the Czech band E (CZ).

7 - E was also the name of the lead singer of early 1990s UK alternative rock band Mass.

8 - E was the name of an indie rock band from Melbourne, Australia in the early 1990s.

9 - An alias of Andy Bullock, a London-based experimental/noise musician.

"E" is also a common, often accidental, mistag. If the song you are scrobbling is not by one of these artists, you should change your tags to the correct artist.

Î
E Lyrics

Jump to: Overall Meaning ↴ Line by Line Meaning ↴

黑夜里想起你温柔臂弯　那只属于我的深情守候
在我心里你是ONLY ONE　现实却逼我离开你的世界
你陪我成长经历每个伤口　你的表情思绪只有我最懂
为什么在最相爱时别离　爱已深心已碎伤痕累累
太爱恋　伤悲无法渲泄　爱变得苦
还倦恋　依然情深不变　压抑不了我只是哽咽
挽着你　挽着你入睡
现在不是我　你拥着谁

一瞬间　心也变得灰灰
放纵自己　模糊了视线
无怨无悔不管时间空间　无所谓太投入或太浓烈
在我心里你永远ONLY ONE
BABY I LOVE YOU SO DON'T BRING THE PAIN TO ME
太爱恋　伤悲无法渲泄　爱变得苦
还倦恋　依然情深不变　压抑不了我只是哽咽
挽着你　挽着你入睡
现在不是我　你拥着谁
一瞬间　心也变得灰灰
放纵自己　模糊了视线
无怨无悔不管时间空间　无所谓太投入或太浓烈

在我心里你永远ONLY ONE
BABY I LOVE YOU SO DON'T BRING THE PAIN TO ME

Overall Meaning

The lyrics of e s t a.'s song "Only One" express deep emotions of love and heartbreak. The singer reminisces about their past relationship and the tender moments they once shared, but unfortunately, they had to leave the person they loved due to circumstances beyond their control. However, despite moving on, the singer cannot forget their former lover, and in their heart, the person will always be the "ONLY ONE" for them.

The lines "太爱恋伤悲无法渲泄爱变得苦" translate to "Love is too intense, the pain is unbearable, and love becomes bitter." These poignant words convey the singer's struggle to deal with the pain of their shattered heart. The lyrics also express the singer's resilience and determination to hold onto their love, even though they cannot be together. The singer cannot help crying, feeling choked up, and experiencing emotional turmoil. However, they know that the deep love they have for their former lover will always remain in their heart, and no one can replace them.

Overall, "Only One" is a touching and melancholic love song that portrays the complexities of love and breakups. The lyrics are emotionally charged and show the depth of the singer's feelings towards their ex-partner. Though the song is sad, it conveys a message of hope and the willingness to hold onto love against all odds.

Line by Line Meaning

黑夜里想起你温柔臂弯　那只属于我的深情守候
In the darkness, I remember the warmth of your embrace, which belongs only to me and is a deeply cherished memory.

在我心里你是ONLY ONE　现实却逼我离开你的世界
You are the only one in my heart, but reality forces me to leave your world.

你陪我成长经历每个伤口　你的表情思绪只有我最懂
You accompanied me as I grew, experienced each wound with me, and no one understands your emotions and thoughts better than me.

为什么在最相爱时别离　爱已深心已碎伤痕累累
Why do we have to separate when we love each other the most? The love is deep, but the heart is broken and the scars are numerous.

太爱恋　伤悲无法渲泄　爱变得苦
The strong love is overwhelming and the sadness can't be expressed. Love becomes bitter.

还倦恋　依然情深不变　压抑不了我只是哽咽
Despite being tired of being in love, the feelings remain strong and cannot be suppressed, leading to choking sobs.

挽着你　挽着你入睡
Holding you close, falling asleep with you.

现在不是我　你拥着谁
Now, who do you embrace if not me?

一瞬间　心也变得灰灰
In the blink of an eye, the heart also becomes gray.

放纵自己　模糊了视线
Indulging oneself leads to blurred vision.

无怨无悔不管时间空间　无所谓太投入或太浓烈
Without regrets or complaints, regardless of time and space, there is no such thing as being too invested or too intense.

在我心里你永远ONLY ONE
In my heart, you will always be the only one.

BABY I LOVE YOU SO DON'T BRING THE PAIN TO ME
Baby, I love you so much, please don't cause me any more pain.

Lyrics © O/B/O APRA AMCOS

Lyrics Licensed & Provided by LyricFind

To comment on or correct specific content, highlight it

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Most interesting comments from YouTube:

carmel pule'

3y↑2

I would say that there is simpler proof that also shows the real engineering operation behind that function.
We should know that the value of e is related to having $1 dollar in the bank and with 100% interest continuously compounded, then at the end of the year one gets $e = $ 2.7182

This compounding of interest in a numeral $ dollar operation, makes the money grow. If we had to represent the compounding of the value of $ money in the bank, we could use the length of a broomstick, where $1 dollar would be a unit length along with the broomstick, while the continuous compounding at the end of the year would be shown by a length of 2.7182 hence $2.718

But compounding on a broomstick can take place along the length, AND ALSO IN QUADRATURE WITH THE LENGTH OF THE BROOMSTICK, where this would result in a rotation or an orientation of the straight long broomstick. So if we invest in rotating the unit length of the broomstick acting as a rotating vector, rather than elongating it along its length, then e^iA would be the final orientation of the unit broomstick after it terminates it compounding "rotational banking interest" rather than the conventional linear compounding which just changed its length!

All we have to do is to draw the vector e^iA where "i" indicates the quadrature direction of the compounding with respect to the length of the unit broomstick length 1, and A is the final compounded angle in radians reached where the unit length of the broomstick will remain a unit length.

Hence taking the components of the final rotational compounding e^iA on the two basic axes which we used as a reference, we have e^ iA= cos A +isin A.
I would say that the students would appreciate all this OPERATION which describes the real engineering actions behind the " compounding of interest" when the interest is linear or when it is (i. interest angular displacement ) the rotational interest on a unit length of a broomstick, or it could even be a $1 dollar paper sheet, wrapped up like a thin cigarette to represent the unit vector, and the wrapped up thin $ sheet will act as the rotating vector accumulating the rotating angular interest which finally compounds to e^iA.
I wager there are a few smiles after readers read all this! One would agree that the philosophy of compounding the $1 dollar paper sheet in a linear manner of its growth in monetary value, or its broomstick length representation, and the rotational angular manner, is exactly the same! where all that engineering actions and reactions are described so accurately in the meaning of the symbols in that expressions. We all need to learn what they mean in engineering actions and reactions and not only the rules of differentiation of the product function used in the video as ...... (the first function)multiplied by (the differential of the second function) plus (the second function) multiplied by (the differential of the first function). I always wonder how many people see the engineering/physical activity in that operation!!!!!!!!!!!!

Roberto Carlos Rodal Salgado

Hello! 3blue1brown, I am a Mexican mathematician and I am completely in love with his videos, my studies make me understand a lot of mathematical theory, however, this practical and intuitive perspective is what we all need, because thanks to it we can make interesting Mathematics to more people.

I just discovered you watching a video related to the Fourier Transform and I fell completely in love. In truth this is the math disclosure that is needed.

I have the idea of creating a YouTube channel where I explain in a semi-formal way the theoretical mathematics, although I would love to complement these explanations by sharing your videos when they are related.

I really appreciate your work and have won a fan of this channel, I am not someone who comments a lot but I will be watching all your videos to complement my ideas.

You are amazing. Thank you for developing and sharing these videos.

You have become a reference for the typical question of And what are these math topics for?

Excellent day!

All comments from YouTube:

3Blue1Brown

5y↑1.9K

Complex exponents are very important for differential equations, so I wanted to be sure to have a quick reference for anyone uncomfortable with the idea. Plus, as an added benefit, this gives an exercise in what it feels like to reason about a differential equation using a phase space, even if none of those words are technically used.

As some of you may know, Euler's formula is already covered on this channel, but from a very different perspective whose main motive was to give an excuse to introduce group theory. Hope you enjoy both!

Gaurav Sharma

5y↑41

Please make a series on tensors and imaginary numbers because these topics are the least understood among students and misjudged due to their weird names. Also they are really very interesting, Intuitive and fundamental to the universe.

X15 CYBER RUSH

5y↑4

I am not feeling well😗😗😌😌

kelzang Tobgyel

5y↑3

Loved the Euler's formula video. This one too.
I'm about to start a course in computing this august. ( I took the advice from your numberphile podcast ) Just want to thank you for these videos the animations are honestly so beautiful to watch. I'm looking forward to the next videos!

Catherine Sanderson

5y↑4

Wolfgang Kleinschmit did you see his Essence of Calculus series?

Ahmed Shaikha

5y↑1

Last I checked, 3.14=/=4.08

40 More Replies...

Keelan G

5y↑2.5K

This is by far the most intuitive explanation of this identity I've ever seen!

John Lesslie

4y↑10

CozmicK G I would check out the explanation with the differential equation y’’ + y = 0

Matt Giles

3y↑3

@John Lesslie where is this at? i can’t find it

hallax

3y↑3

KRUH

carmel pule'

3y↑2

Ishwor Shrestha

3 More Replies...

Peckingbird

5y↑2.4K

"Multiplying by i has the effect of rotating numbers 90°" - Both lost and mind blown.

Paul Foss

5y↑603

Think of multiplying by negative one as turning around to face backwards. If you do that twice you will be facing forwards again. That is (-1)*(-1)=1. Now if we ask what number times itself equals negative one, it will be a rotation that when done twice has you facing backwards, either a quarter turn counterclockwise i or a quarter turn clockwise -i. (the complex plane doesn't actually distinguish between clockwise and counterclockwise because that would require it physically be facing in some direction relative to us, and it is an abstract mathematical object, but we aren't abstract and generally draw i above the real number line and -i below, and numbers on the real number line getting larger as you move right)

kindlin

5y↑26

pclouds

5y↑16

I think he has a video about group theory that explains more about this rotation.

SDoman

5y↑4

theres a really good series from another channel about imaginary numbers that goes into depth about it from the absolute basics, Id recommend giving it a watch

Michael Leue

5y↑16

@SDoman WelchLabs is probably the channel you're thinking of.

28 More Replies...

Eamon Fitzgerald

5y↑394

The idea that multiplying a number by i rotates it 90 degrees into an imaginary axis is probably the most important thing I've ever gone my whole life without knowing.

Mohammad Reza

4y↑9

Wow, how did anyone understand maths before 3b1b

Franziska von Karma

4y↑31

@Mohammad Reza They teach you polar coordinate multiplication literally in 11th Grade or so

Lenoquo

4y↑26

After four years of studying electrical engineering in college, no one ever explained it that way until a professor casually mentioned it in a elective I took in my last semester :/ It would have helped so much to learn that earlier

Joseph McGowan

3y↑55

"We're sorry, the number you have dialed is imaginary. Please rotate your phone 90 degrees and try again."

Achyuth Thouta

3y↑8

@Franziska von Karma They just say that cis(a)*cis(b) = cis(a+b). They don't go beyond that

4 More Replies...

Tax Collector

43w↑28

No wonder the Euler’s formula in Alan Becker’s “Animation Vs. Math” is given sentience, it’s all about position and movement. It can do so much.

hiqwertyhi

5y↑526

the e^iπ part comes right at 3.14 but also the e^i𝜏 part comes right at 3:14

nice

squibble

4y↑15

i wonder whats e^iφ

Ruchir Kadam

4y↑28

@squibble if you are wondering about anything else,
Generally e^ix=cos(x)+isin(x)
There's a really easy explanation for this, it has to do with trig

Irrelevant Noob

3y↑6

@hiqwertyhi you've forgotten to explain that since a minute is only 60 seconds... 3.14 minutes is in fact about 3:08... And then tell readers to look closely in the bottom right corner of the video at that time... :-B

K N

wow, actually amazing

Md. Sohrab Hossain

2y↑4

3:14 is not π minutes into the video.

2 More Replies...

Jeb Bush

5y↑3.2K

Oh my god that makes so much sense.
Why was this never explained to me like this?

umair

5y↑27

because it's wrong...

Primer

5y↑438

@umair Expand?

Anonymouspock

5y↑229

@umair go on

3Blue1Brown

5y↑986

The great thing about math is that you don’t have to take anyone’s word for whether something is right or wrong. You can think through if this explanation makes sense for yourself (spoiler alert, it checks out). To engage with it that way will force you to think critically about what it means to take a derivative of a complex-valued function, which in turn builds good intuitions for complex analysis.

umair

5y↑37

If e^πi=-1 then
π=ln(-1) / i
Then 2πr= 2r ln (-1) / i
= ln (-1^2r) / i
Which equals ln (1) / i
Which equals 0
Which means that if Euler's identity is true, the circumference of a circle is always 0.

64 More Replies...

xyzct

5y↑42

It's amazing how helpful Grant's use of color is. This was particularly pronounced on his superb Fourier series videos. The old analog chalkboards of my youth did not have this added dimension.

Finnyke

5y↑106

It is incredible how far you've grown as an educator. Your first take on e to pi*i as your first video on a channel was really confusing to a lot of viewers, me included. But now you are providing an easily conprehensive explanations together with enjoyable visuals on this and many other topics. Thank you for all the work you're putting in your channel

Liam Smith

4y↑55

I have known that e^i*pi = -1 for many years.

This video was the moment I understood that fact. Thank you.

lordoftheoreos

That's surprising, considering how I was exposed to it was through the MacLaurin series. That answers it immaculately too. As I've matured more in mathematics, still quite infantile, I realized that 3B1B's videos are very visual. Though this video is more approachable, I'm often leaning towards traditional textbook like material with the leap of intuition than his videos that are ambitious visualizations.

Calyo Delphi

5y↑19

Even with your previous videos about e^ipi I couldn't fully grasp WHY it behaves so. Why it has this undeniable link to rotations in the complex plane. As SOON as you brought derivatives into it, and e^x's nature of being its own derivative, it makes absolute perfect sense. Dude, you and blackpenredpen are the math teachers I needed so desperately when I was precalc in high school. <3

Jason Cheng

5y↑2

You just taught me vector fields better than an entire semester of my second year ODE course.

Louis

5y↑275

THAT WAS SUCH A GREAT EXPLANATION
I always struggled with that one, despite the fact you've made quite a few videos about it
this was so good though, it makes so much sense now
I gotta watch the other ones you've made to see if I can understand them better now

Hassan Omer

5y↑1

I did not understand why the yellow line (velocity ) was moving faster then the blow line (position ) even though he said the dervative is just equal to to the position at that time,.

Can u explain

dexter239

@Hassan Omer it's not yellow it's green

Usual Unusual Kid

4y↑2

@Hassan Omer It doesn't move faster. If the right side of green line moves at 2n, while left side moves at n, this means its length increases by n. The "speed boost" comes from blue line's end

andinosa

3y↑4

It would be awesome if you did a series on modern numerical methods. Finite elements (for for elliptic problems) add Runge-Kutta methods for hyoerbolic and parabolic problems, etc... With your unique talent you could make this fascinating topic accesible to millions of talented young people who might chose to pursue a career in applied math. Thanks for all the great work!

Andrea Di Biagio

5y↑9

Kudos! This is pedagogy brought to the level of art. I hope your work, together with that of other excellent YouTube educators, will pave the way for a new generation of teaching.

Daniel Beirão

2y↑5

The only 4min video that actually led me to understand the dinamic behind Euler's identity. Well explained! Congratulations!

RedAgent14

5y↑14

As an electrical engineer, I learned AC circuit solving through complex exponentials (phasors and impedances), so I'm really excited to see this video.

(Also, the pace of the video relative to the usual 15-20 minute vids reminds me of the first video on the channel.)

Cold Ham on Rye

5y↑16

This is the simplest and most intuitive expression for this I have ever seen. Thank you. I'm a grad student in robotics so ODE's and PDE's are my life blood. So, this has always bothered me that the intuition for this concept was so difficult. With such a simple explanation I wonder why I have never seen this before.

George Cowsert

40w↑2

I came back here after the Animation V Math video dropped, and I just realized.
All of the numbers in the video only obey Gravity when affected by either TSC or Euler's number.
This is because TSC is a stick figure that has gravity built-in to his existence, but e^(iPI) has a tangent line directing it downwards, giving it a form of gravity.

jrazz

5y↑11

That is by far the best, and most intuitive explanation of Euler's formula I have ever seen. Thanks!

mü

5y↑5

It's Magical !
The way you explain.
And I am not even a newcomer on this channel, yet my mind is always blown away by how you brilliantly decompose mathematical concepts to their core. So elegant, so beautiful.
This is the way to do it, the proper way to use technology in order to share science and wisdom.
So thankful for the Internet, that is allowing us to connect with such quality content.
So thankful for you, creators of such quality content.
Thank you for what you do !
Peace.

Pi Man

5y↑1

Interestingly, it's "creator" (singular). Just one guy does this

mü

5y↑1

@Pi Man Sure, but 3B1B is not the only great content creator on the Internet ;-)

Jon Dresser

5y↑3

I love this! Your ability to show the patterns in mathematics in an elegant, but understandable way is truly profound. Thank you for all you do!

Edward Boron

5y↑8

I really like the buildup in this video. You start at derivatives and everything just blends into each other until you're expanding it further. From the effect of positive factors in the exponent, introducing the negative factor and eventually the imaginary factor. Definitely hard to understand if you don't know about the underlying concepts, but once you do, this almost feels like the most perfect way to explain e to the i pi. Great video

piiinkDeluxe

YeS! Last time i watched this i didn't know about e, now we have been learning it in school and i understood the introduction way better. 🙂

LuismiSF

5y↑8

I thought at the end you'd show how, if you project the circular motion in the complex plane onto the real axis, you see simple harmonic motion. This could help high school students to relate better.
Btw I absolutely love this videos! Short but original and always make your concepts click no matter how much you think you know on the topic. Keep inspiring 3Blue1Brown!

Alexandre De Masure

4y↑2

Underrated comment

Emiliano Toledo

4y↑1

I can’t even stress how important your work is for me. You explain everything in such a beautiful and simple way; you were BORN to teach. Congratulations on all of your videos, sir.

Daniel Winicki

3y↑1

About the equation e^t, it's interesting to note that, in addition to the speed being equal to the position, the acceleration is equal to the speed (which is equal to the position), and the variation of the acceleration is equal to the acceleration (which is equal to the speed, which is equal to the position), and this recursion goes on and on

benzmansl65amg

5y↑4

You are an incredible instructor. Thanks for what you do.

Scott Richards

5y↑2

You deserve a Nobel Prize for this video, such an intuitive explanation.

Jessie Bauer

5y↑3

Amazing video. I've never had it explained this way to me. I've always just understood E^jX is equivalent to the unit circle and accepted it as so.

Team T-rex

5y↑46

I was comfortable with this identiy before, but I had a great “AHA!” moment. This is probably the most concise and surprising way to look at it I’ve ever seen. Thanks a bunch

Sergio Loza Villarroel

5y↑2

If you expand: e^(ix) using its Maclaurin series, you end up with the series for cos(x) and sin(x)*i
So:
e^(ix) = cos(x)+i*sin(x)
And, if x=pi
e^(i*pi) = -1

Arthur Brock

your videos are honestly better than my differential equations and partial differential equations classes. It makes me think about my physics classes, for example, RC and LC circuits and how the charging and discharging of the capacitors are related to this circular motion or of course the heat equation and wave equation like you have addressed in previous videos.

Keenan

5y↑1

Love the vid! I was wondering if you could do a graphical proof of the trig addition identities. (e.g., cos(v+u)=cosvcosu-sinvsinu) I recently came across these in my trig class and was wondering what the proof looked like. Thanks!

It's CKY

1y↑1

Its always so fascinating when you explains it from another perspective

Jesse Cook

This is so beautifully intuitive! I was first introduced to Euler's formula years ago in calculus 2, but algebraically derived using Taylor series. This geometric explanation is so wonderful (as are all of your geometric explanations in your videos)! :) I've been a fan for years and I love how you give these simple clear pictures for explaining interesting mathematical concepts and equations.

ElectricAvian

5y↑4

Can't wait for the full decomposition video! This was fantastic (many thanks from an electrical engineering student!)

Chlorine

I think this way of teaching "visually" should be introduced in every school in the world... it's amazing how you manage to find the right animation for whatever concept, bravo!

redbart

5y↑1

This is beautiful. This is the best explanation I've ever seen of this, the use of actual derivatives in the proof but in a way that does not need calculus is amazing! Please keep uploading because I learn so much from every single one of your videos!!!

Ekaansh Gaba

1y↑1

EULER WAS A GENIUS
he deserves to be the no.1 mathematician of all time uptil now.

Gpp

5y↑1

Thank you alot for your work! I really enjoy spending my time on your channel and discover "real" math and learn why the things we learn are as they are.
It would be cool if you do another/more videos on music theory. I think alot of musicians who aren't watching you would enjoy it too.
Have a nice day!

Dragonfire

Many trigonometry identities can be easily found just by using Euler's formula - also matrix operations like rotations - one doesnt have to learn it by heart since they can be easily derived from E.F. plus it gives u some kind of mental/visual representation of those formulas. It's really incredible how much information is integrated in it.

tnyeager

5y↑5

Brilliant. Absolutely brilliant. This is excellent teaching, as we've come to expect from your channel. Thank you!

Runstar Homer

5y↑1

I am constantly amazed at your ability to so elegantly show the beauty in mathematics. This was simply incredible.

Alex Thompson

I watched this video not too long ago and it did not quite make sense to me. However, after reading about what polar representation is within my textbook and going back to this video, this totally makes sense to me now. Thank you for your outstanding videos. I feel as if I'm always learning something new from you!

Doctor Monkey

5y↑1

Wow, elegant explanation! As always, thank you for great videos!

Harjeck

5y↑1

Well, I really appriciate the huge work You do for Maths and for all of the people interested in it and also help people to even be interested in it.. Such amazing explanations, animations and beautiful thoughts.
But I would really love to see You making similar Physics videos, from the very bases of Physics to some complex and huge physics thoughs or even unsolved mysteries..
Because I haven't seen any better educational channel with such a good explanations / animations, which helps to improve persons view on the world of maths/physics. It'd be nice to see this even in physics problems / theorems. :)

Safi Halim

5y↑5

Amazing! I never thought that there would be such a simple explanation to such a complicated equation. You are the best maths teacher!

Zoe Chen

5y↑49

Explained my confusion about Euler s formula so elegantly! Thank you!

Unknown Legend

Which one?

Joaquin Pirotto

5y↑1

This is the first time I actually understand why e^(i*theta) describes a circle, and it was as simple as to derivate the term!

Hilfigertout

5y↑145

To anyone who wants to think about something deeper with this concept:

Think about the function y=(-1)^x. Now this function is discontinuous, it’s not a line at all. It just seems to alternate between y=1 and y=-1 on the xy plane. But what happens if you include the complex numbers, say adding them in a z axis to make a three-dimensional graph. What you’ll find is that the function (-1)^x, or to write it another way e^(i*pi*x), traces out a line in 3d space. A line that is in the shape of a helix, a coil going around the x-axis, and intersecting the xy plane at 1 and -1 at appropriate points.

Nurali Medeu

5y↑5

Hilfigertout woah.

Li Hua

4y↑6

And not just any helix, sinusoidal at that!

NovaWarrior77

4y↑11

This is what I call high quality commenting.

NovaWarrior77

4y↑1

@Nurali Medeu "woah" - Neo.

Brorsan Beppe

4y↑1

Wouldnt it be two helixes mirrored since the function is ambiguous on complex numbers?

5 More Replies...

Jose Luis F

Clear an concise. ....simply a brilliant explanation!

DavidtheNorseman

5y↑27

I wish I had this for my students a quarter century ago!!! Beautiful :-)

Nikki Colton

4y↑1

Will you be making a 6th episode on differential equations? Are you working on a playlist for a new topic? I found your Essence of Calculus and Essence of Linear Algebra playlists to be very enlightening as well as this playlist.

Atakan Gürkan

5y↑3

Love this! So insightful.
Watching this video was like listening to a jazz piece for me. Everything was harmonious and made sense, but I would not have guessed the whole in advance.

Nosson Weissman

Hey 3b1b thanks for your content! You're videos never fail to leave me satisfied! I was looking for some explanations (intuitive explanations) on topics in Numerical Analysis, anyone know where I can find some of that kinda stuff?

B B

5y↑2

This explanation is amazing; so clear and concise. Keep it up!

GoingsOn

5y↑1

Now the unit circle intuition of e^(it) makes so much sense!

Manuel Jenkin Jerome

Key take away is to explore in terms of analytic extension through derivatives and not as repeated multiplications. Beautiful explanation, as always.

Saurabh Deshmukh

Can we expect more videos in this series? Would love to see move videos ib PDEs. Thank you so much for all the hardwork.

Mike

5y↑8

You have the best visuals for any channel on YouTube I've ever seen. Outstanding work!

TheAwfulUnicorn

I can't even put in words how brilliant that explanation was!

nollix

Genius, genius, genius brilliant explanation. I am moved by your ability to convey intuition so perfectly. This is the greatest

DraconicEpic

5y↑1

This without a doubt is the best explanation I've seen of this yet.

Nitin J

5y↑1

The feeling of getting closer to the ultimate truths of our universe. Simply enlightening. Thank you so much.

I watched the video

How I wish this channel existed way back when I had calculus. A truly amazing job in visualizing the concepts, in a way no professor I ever med can.

Paolo C

5y↑45

The last bit, when you pointed out that e^iπ does not mean multiplying e by itself iπ times (even though this would have been the case if the exponent had been an integer, like e^3 = e*e*e) made me realize that it is the same thing that happens when you find an analytic continuation for a series. For instance, ζ(-1) = -1/12, but this equality does not mean that 1+2+3+...=-1/12, since the series for the Riemann Zeta function is only defined for numbers greater than 1. I had never realized before that even with simple arithmetic, such as powers, you can already get a feeling for the rule "you cannot always interpret new things using your old knowledge", because sometimes a blind substitution can be misleading.

Rudraksh Jaiswal

5y↑1

That cleared up a whole lot more

squibble

4y↑3

ζ(e^iπ) = -1/12
never realized this before

IrokoSalei

That makes sense simply because the multiple of an imaginary is still an imaginary in the form ki. e isn't powered to anything in the real axis.

Ken Haley

5y↑1

This is beautiful! Has to be the best way to show e to the i pi = -1 that I've ever seen! Thank you!

syd sings

I’ve watched so many videos explaining this and I’ve never understood but this one is so clear and I understand it so well for my little background in math as it is.

NUKE

Awesome explanation and visuals as always!

Studious Boy

5y↑1

Hey Grant! I had a request to make from you.
I want you to make a video intuitively explaining what exactly are waves, different types of it and how to intuitively visualise all of them.
Waves are very basic to most of physics and getting a very clear idea about them may help millions of students get better at physics and in general science.

Marco Munari

Overall thank you for all your videos, I achieved to understand those things by writing my own program in '90, Is because your work is tailored to perfection that I'm writing the following comment: At minute 1:48 the animation about exp(-0.5 t) is for a super slow down animation (t changes constantly from 0.00 to 4.00 in 8 seconds AND the factor 0.5 slows the "exponential" of another further half). You would have the same vector animation for exp(-0.25 t) where t grow in seconds unit coherent to the video. However is more didactic to show exp(-t) which is more representative for the exponential function of a negative exponent, which yes slows down its approach to zero but also because it went toward it a lot (however there is nothing wrong in your video)

Mohammad Reza

4y↑8

I was researching e^ix= cos(x) + i*sin(x) after watching Lockdown math and wondered why there seemingly 'just happened to be' a neat connection between two quite different things: circles and e. From what I understood on Wikipedia and forums, there was an explanation that involved the Maclaurin series of e, sin, & cos, and whilst it is a proof, it didn't satisfy my question. But then this video made the connection obvious (with the help of another 3b1b vid about e and calculus) and I can now say that I understand e^iπ=1 :)

Connor Criss

I understood e^i*pi before this but this explanation is wayyy more clear and concise than the one I had in my head(which used Taylor series.) Thank you!

Roman SamaLamaDingDong

5y↑1

Now THIS makes total sense. Thank you so much!

Reuben Adams

4y↑1

Such an intuitive explanation of a formula that has always seemed so abstract to me. I will now always have this concrete image in my head when I think of it. Thank you :)

Yelectric

This is a bamboozling mind blowing horror story but wild realization to tell to a mathematician 500 years ago. Cheers to another 500! What manmade horrors lie beyond my comprehension that in all honesty, weren’t manmade to begin with but just a progressive unraveling of what’s around us.

Jaime Gonzales

A lot more easier to understand than the old version!

-Spiffy2

5y↑3

Best explanation yet!

RJSRdg

4y↑1

I'm glad you put that last bit in about e^i pi not equalling e multiplied by e i pi times, as that was something that was confusing me!

(If you take the log of both sides of Euler's identity, then multiply both sides by -1, you get some very strange results - now I know why!)

Fareed Abi Farraj

5y↑2

Awesome videos as always! Thank you for your amazing hard work👍

But I'd like to ask, why did you call your channel 3Blue1Brown what was the purpose?

And how do you do these perfect animations? What app do you use?

Thank you, @3Blue1Brown

Roberto Carlos Rodal Salgado

Isaac Kay

You have a beautiful way of explaining things and making them initiative - thank you!

Violet Broregarde

5y↑5

I've been arguing for a while that we should use physical models to teach math. You can represent addition and multiplication physically by superimposing 2 rulers on top of each other. Addition is a sliding action and multiplication is a scaling action, which you can represent by pulling them apart from each other while keeping the 0's overlapping and moving the 1 over whatever you're trying to scale up by. This makes complex numbers easier, since addition and multiplication work exactly the same way. This also makes it easier to go from only the nonnegative integers to the whole set of integers, since you can ask a question like "3+what=2" and it's not surprising that your incomplete number line has something that would answer that question, and seeing the structure laid out for you makes the negative numbers easier to use.

Honk Honk

2y↑1

thats why the dude made the Manim library. you can easily make these math animations as a teacher, and shit works easily.
it uses python too, so its hella easy.

Joni Paliares

5y↑1

So clear and elegant!
Thank you for this beautiful explanation

hedgeclipper418

watching your videos makes me feel similar to how I feel after meditating for a long while. It's so calming to know that it all makes sense somehow.

pitot

This is just incredible.
Simple, yet easy to understand.
Thx for your great video again

combatcello

I am looking very much forward to your next video! This is something I have thought about ever since I started Uni but never quite understood :)

Camilla Balliana

I love the different way you explain, very careful with the concepts and very detailed, which is good for those who are graduating in the bachelor's degree in the area, in addition to the illustrations are fantastic, congratulations, I follow you always - I am an undergraduate physics student in UFRN- Brazil

Cleiven

5y↑6

Kurzgesagt AND 3Blue1Brown uploaded today! This is fantastic!
Also, I can't neglect to say, very elegant proof and brilliant explanation!

Croissants for thought

They should do a collab

Ribal

This guy always has a way of blowing my mind. Incredible work!!!

Bluffmarster

5y↑1

This is the single greatest video and explanation I´ve seen on this!
Finally I understand why it produces a circle! :)

ayuminor

Watching this again after just having done some problems involving resonant circuits, where Tau is used as a time constant, I just had the revelation that this is actually not unrelated to this Tau = 2*pi idea! Freakin beautiful.

You help more people with these videos than you could ever know!

lukeluke246

5y↑1

This the most clear explanation that I’ve heard I just completed high school geometry but I’ve always had an interest in higher level math and this just never made sense but I knew it worked and some properties so I just accepted it but this explanation now gives it meaning

Manu Isaacs

I've always seen Euler's formula explained using taylor series, but this is sooo much cleaner.

Great video 😃

Weeb Kien

43w↑8

I actually just come here after watching "Animation vs Math" 😅

DigginJoe

5y↑2

That was an awesome way to think about this. And in such a short form makes it easy to process. Thanks heaps, and can't wait for the next video.

OctopusParty

After a year or so of using it, I can now officially say I fully understand Euler's Formula. Bless up. 🙌

René S

This was just a beautiful explanation of a beautiful concept. thanks and keep on making your great vids!

Fred Rodriguez

Amazingly good explanation! I wish these kinds of resources had been available when I was in engineering school!

ESTEFEN MENES

The level of explaining complex things like this will take a lot of to achieve. But in just 4 minutes you made me understand something that I could use on my Advance Engineering for mechanical engineers course.

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