e is the grade I nearly got in calculus.

@MagruderSpoots

4 жыл бұрын

e was the grade I got in calculus. I didn't have 3b1b.

@feynstein1004

4 жыл бұрын

Exceeds expectations? Nice, mate :)

@hexa3389

4 жыл бұрын

My ESL teacher used to give out E for Excellent.

@TerryJLaRue

4 жыл бұрын

@@hexa3389 That is probably what you thought it was for, anyway. :)

@hexa3389

4 жыл бұрын

@@TerryJLaRue bold of you to assume I ever got one. Or maybe that's a good thing. Idk.

3b1b's videos are like detective novels. First there is a mystery, and then the story slowly moves towards the solution by finding clues.

@FranFerioli

2 жыл бұрын

Grant makes exactly the same point in his TED talk.

@FaffyWaffles

Жыл бұрын

That’s why I’m studying math

@kmjohnny

10 ай бұрын

Math (especially calculus) is indeed a mystery.

@H3XED_OwO

8 ай бұрын

Socratic Teaching

Every child would fall in love with learning if the world had more people like you ♥️

@sparrowp2251

2 жыл бұрын

very true half of the teachers dont have a very sound understanding of what is what !but youtube did revolutinize the world in terms of education .accessing good education is easy and reliable,nowadays .none the less still one has to be cruious enough to use such good resources.

@parahumour4619

2 жыл бұрын

fax

@fazeblizz5898

2 жыл бұрын

You are just lucky to be born with high intelligence, effort plays a small part compared to your processing and memory

@cadmus2573

2 жыл бұрын

@@fazeblizz5898 Not at all, even the smartest person has to put in effort beyond a certain point. You learn maths invented by hundreds of top level geniuses. Even Einstein had to put in effort to learn everything that he then built upon. Just being smart can’t have you invent hundreds of years of mathematics.

@Living_for_Him_Alone

Жыл бұрын

If world had more people like you who appreciate others , then it would be far better place👏

Like I always say, mathematics is like a well Euler'd machine.

@TheManWithTheFryingPan

4 жыл бұрын

That's it. That's the joke to end all jokes

@anonymousash6678

4 жыл бұрын

rip to people who don't know how euler is actually pronounced and they say - eeewwlaar xD

@pianochannel100

3 жыл бұрын

That is so good ahaha

@T_tintin

2 жыл бұрын

@@anonymousash6678 sadly I was one of them...

@anonymousash6678

2 жыл бұрын

@@T_tintin All good, now you know its pronounced "Oiler" :)

Can we just appreciate the time it takes to animate this numbers, graphs and characters ...

@stargazer7644

3 жыл бұрын

He has a slave named Python that he makes do this grunt work.

@Hi-6969

3 жыл бұрын

@@stargazer7644 its called manim actually

@andyk2181

2 жыл бұрын

@@stargazer7644 At the risk of outraging the feminists, I'm making Julia my slave

@theroboman727

2 жыл бұрын

@@Hi-6969 no, python is grant's slave and manim is python's slave

@ElytrCSGO

2 жыл бұрын

@@andyk2181 bruh

After 6 years in electrical engineering, I finally understood the meaning of e, thanks to this video.

@cheesy4384

Жыл бұрын

6 years in eee to understand e

@parallellinesmeetatinfinity

10 ай бұрын

@@prun9095 W dp 🗿

@Periwinkleaccount

Ай бұрын

@@parallellinesmeetatinfinity What’s that supposed to mean?

The Internet is such a mixed blessing. There’s the vile poison that is 99% of Facebook and Twitter, but also the treasure that many truly gifted teachers bring us on You Tube. I don’t know which will ultimately triumph.

@somxr_738

2 жыл бұрын

The idiots will die out and we will win 😂

@elena6516

2 жыл бұрын

@@somxr_738 false. Have you seen Idiocracy?

@12-343

2 жыл бұрын

And then there's the blursed mess that is reddit, and 4chan, the retired hacker group.

@aidenheffernan7556

2 жыл бұрын

how you felt after saying that 🐶👺

@parallaxladder256

2 жыл бұрын

@@12-343 very blursed

It's amazing the amount of intuition and conceptual learning that is left out of higher education math classes. These videos are pure gold for that.

@manthansharma4835

Жыл бұрын

Yes , they don't teach like this and then students fear vuz they can't solve numerical which later creates fear for science as whole....

Your videos never fail to inspire. Calm yet profound, penetrating insights stick out like jewels studded into a ring. Thank you, Grant, for all the effort that you have poured into this series; As a Patreon supporter, I know first-hand the thought that went to these videos on Calculus. I and each member of the audience of 3blue1brown can tell you that your work is appreciated. For me, this means that you were and are at the heart of the reason that I love mathematics: you showed me the beauty. And, as for the rest of the audience, one needs to look no further than the thankfulness of my fellow commenters. 3blue1brown, you are awe-inspiring. Thank you.

@3blue1brown

7 жыл бұрын

Wow! What an absurdly heart-warming comment to read. It really means a lot to me, and words like that are incredibly inspiring to work hard in expanding the channel's offering. Thank you.

@Joe-bb4yi

3 жыл бұрын

Joe

@CubeZero

3 жыл бұрын

@@Joe-bb4yi Mama

@Pladinium

3 жыл бұрын

@@CubeZero nice

@aarohansharma4551

3 жыл бұрын

@@CubeZero nice

Instead of saying stuff like "Pi population", just say "Pipulation".

@CasioAns

5 жыл бұрын

This actually made me laugh out loud thank you

@big_cuh

5 жыл бұрын

@@CasioAns Instead of saying stuff like "This actually made me laugh out loud, thank you!", just say "This made me LOL, thank you!"

@DoctressCalibrator

5 жыл бұрын

Why would anyone want to say that?

@squdardt.9719

4 жыл бұрын

Catface nah man, I think no one should say that.

@Ryan6.022

4 жыл бұрын

I went to like this but I already had and have no memory of when I watched this.

Thanks. It's been so long since I studied all this that it is a revelation to me again. All the effort poured into making these videos is highly appreciated by me and all your viewers, I'm sure!

@jblangcua2726

5 ай бұрын

There goes tomorrows lunch

@Slick6464

5 ай бұрын

@@jblangcua2726😂😂

@display9138

Ай бұрын

😂😂​@@jblangcua2726

I have a Ph.D. in Aeronautical Engineering with a focus in spacecraft and missile dynamics and control, and applied mathematics and your video never fails to teach me more about mathematical properties and relationships that I never really gave a second thought about.

@spacejunky4380

Жыл бұрын

That sounds incredible! What do you work on now?

@jmadratz

Жыл бұрын

@@spacejunky4380 I retired from Lockheed Martin (LM) in March 2022 after working for 32 years. I started in their Astrospace division and designed guidance, navigation, and control algorithms for commercial, civil, and defense satellites (e.g. Telstar, Tiros NOAA and EOS-AM aka Terra, and DSCS communication satellites, respectively). Then I switched divisions and for the remainder of my time with LM I worked on Aegis ballistic missile defense (BMD) “hit to kill” systems. I developed the guidance and navigation algorithms for Aegis BMD, Patriot PAC-3, and Thaad defense missile systems. I have 23 US patents and 5 LM trade secrets in all areas of BMD and satellite control systems.

@alecmiller5296

Жыл бұрын

@@jmadratzThaad systems are fucking sick

@jmadratz

Жыл бұрын

@@alecmiller5296 Sick in a good or bad way?

@alecmiller5296

Жыл бұрын

@@jmadratz 100% good way they’re so cool imo

Oh My God ! I could not help myself from pausing the video somewhere less than 10 minutes into the video, and type this comment. My heart was racing because of all that wonderful and beautiful insights I got from this video. Definitely not for the faint hearts !! Your videos are AMAZING !!! I wish there was some award for the best educational video of the year. If then, you would be the winner for sure !!!!

@3blue1brown

7 жыл бұрын

+Mohan7 Wow, thank you!

@mujtabanasir2970

6 жыл бұрын

^^^^^^^^^^^^^^^^

@ThangTran-qn7vq

5 жыл бұрын

Same to me. It removed the mental block about "e" which has been in my brain for years. So beautiful video.

@wiseacredave

5 жыл бұрын

I agree. I love the idea that the value of e is essentially defined by the fact that the ratio of the derivative of an exponential of base e to the exponential of base e equals 1, and that this is analogous to the definition of pi as the ratio of any circle's circumference to its diameter. That really puts e in its rightful place as a transcendental number on par with pi and is a satisfying way to understand the number e.

@wodeyaeric5351

5 жыл бұрын

This man doesn't give lip service to calculus like many do without the intuition he arouses here. When I get some money I should give u some. No calculus text book could have made it easier than this. So what should have gone into buying books shall be passed to u. Just be patient I receive some payment.

Easy way to remember digits of *e* After 2.7 repeats a string of 1828 two times and after it follows the angle of a right isosceles triangle i.e45 90 45 e=2.7 1828 1828 45 90 45 ..........

@AJ-fo3hp

5 жыл бұрын

Good thing you mentioned

@hlfan

5 жыл бұрын

followed by the first three primes, 2π in degrees, the days in a standard February, the famous Boeing, the first three odd numbers, then 26, then that reversed, after that 7², a double 7, 7² in Duodecimal (57), the nonstop service synonym 24 7, two-digit 3², 6², a triple 9, throw in a 5 for good measure and you got 50 decimal places of e: 2.7 1828 1828 459045 235 360 28 747 135 26 62 49 77 57 247 09 36 999 5 …

@shyamdas6231

5 жыл бұрын

@@hlfan loved it.

@taiman9423

5 жыл бұрын

bruh... chill

@user-kf1xn1dq9t

5 жыл бұрын

@@hlfan also, 1828 is a year of birth of Leo Tolstoy. And he was born 28 august, on the old style. So if he was born one day earlier, exponent was also called Tolstoys number.

All these videos make me feel sad about modern education. We were so cheated by just being told the solution. Seeing how to derive it on your own is so much more impactful and intuitive.

@samyakjain727

2 жыл бұрын

That’s exactly what I was thinking. I’m in medicine now and haven’t done maths for years. But if I had a chance to learn it like this, I would have enjoyed it so much more, and perhaps even done better too

@pietroalessandrini

Жыл бұрын

And then if they actually tell you how to arrive at the formula they ask you to repear he process step by step word by word making it just stressful

@Thetarget1

Жыл бұрын

For sure, the problem is that it takes a lot more time, and students also have a very large curriculum they need to learn in a rather limited time. Maths education is always going to be a compromise between efficiency and letting the students explore the problems for themselves.

@donkkut5003

Жыл бұрын

​@Ben Smyth knowing the derivation makes the applications much more intuitive so i disagree also every math class should ideally begin with a problem or a set of problems being shown so that the theory is a natural follow up as the generalization of the solution to those interconnected problems

@gabbyk.7358

4 ай бұрын

@@Thetarget1I agree; I realize that time constraints are one of the main problems. Of course, there will always be the students (tending to be the majority) who don’t and will never care about the reason behind mathematical principles. But for students like me who actually would want to know underlying principles, I am limited by the deadlines. What’s interesting is, because I have been reading our calculus textbook more thoroughly than others, i understand the underlying principles that help me to better perform calculations, whereas most if not all of my classmates often miss certain problems because they are just blindly following the process they have memorized. Sadly, sometimes I’m not able to dive deeper into certain topics because I run out of time to do so.

The amount of work put behind this is simply impeccable, I can feel how careful each development in the story is planned out to tailors to everyone's intuition and understanding. This is a prime example the word 'universal', and should be included in textbooks as visual aids.

Everyone, everyone, I just got calculus right just now, it finally clicked for me.. if the rate of change over a function is constant across the function the derivative is a constant but if it changes based on where you are in a function that's okay no big deal we just have to represent it with another function based on where you are. The derivative finally clicked for me I was so happy I was shouting out the window telling the world I understand calculus.

@muhannadak8087

4 жыл бұрын

Bruh that's exactly how I felt!

@manideepp2229

3 жыл бұрын

Hopefully you didn't become hulk😉. His voice itself creates interests to read.

@NovaWarrior77

3 жыл бұрын

I'm proud!

@StrummerDave

3 жыл бұрын

Good for you. Too many people do Calc 1 and never have anything explained in a comprehensible way so they just give up. Keep at it. Math is the beautiful language of the universe and studying change is central to it.

@aarohansharma4551

3 жыл бұрын

Good job bro ! 👌

I really like those π creatures...

@shugaku2461

7 жыл бұрын

TheMurderArt Yup, me too. Who wouldn't?

@Tsskyx

7 жыл бұрын

I've always wondered, π has 2 legs, but τ has only one, but its value is double of π... like, what gives? Shall we redefine mathematics? :P

@larry5911

7 жыл бұрын

I want to eat them. I love Pi.

@shugaku2461

7 жыл бұрын

TheMurderArt Id like to snuggle and live with one. Heck, I'd buy a plusher of those things!

@TheMurderArt

7 жыл бұрын

one of them would be the perfect pet

It will be 10 years since I first learned differentiation.. But it is today that I realized the greatness of 'e'. I am late, but happy to feel the beauty of mathematics thanks to you. Furthermore, If my juniors can access your channel in the new semester, I would like nothing more. appreciate and have a great weekend!

After years of doing calculus, this video finally made "e" click for me. So stupidly simple, for something so anxiety provoking. Thank you!

The wonderful thing about exponentials is just how many different perspectives you can take to define or introduce them. This video is of course just one look at a possible exploration that could lead you to stumble onto e^x, and if you'd like another, I'd highly recommend Mathologer's coverage of the topic in these two videos: - e to the pi i for dummies: kzread.info/dash/bejne/X5icqtSgXa7IepM.html - The number e explained in depth: kzread.info/dash/bejne/dqN1xKNvorvYp6Q.html And if you want yet another perspective, quite different in flavor from the calculus view, you can see my video on "Euler's formula with introductory group theory": kzread.info/dash/bejne/n6qh16WJprXVh7Q.html (By the way, not that it really matters, but the investing example should probably have been written as dM/dt = rM(t), with solution e^{rt}. Although, given the rate at which those dollar signs were coming up, maybe the implication that the interest rate is over 100% was apt!)

@subhasish-m

7 жыл бұрын

I don't mean to be intrusive, but where did you learn all this amazing math? You can see snippets of what you teach through quick searches, but your beautiful visual way of explaining things is found almost nowhere else. I am simply interested, because I aspire to have the knowledge you do.

@loopingdope

7 жыл бұрын

Reading and studying with passion. And I'm sure that it requires a bit of an unusual mind combined with a clear idea what you want to do and how to do it. Maybe some thinking process makes you prepared for it, e.g thinking automatically in a way like you're explaining the concept to someone, or to an audience.

@RaidChampion

7 жыл бұрын

On the definition of exponentials: only a few months into university I noticed I was never even given a definition of an exponential function with real powers, we just went with it in high school. You can give a nice explanation of what the number 2^4 represents and even 2^(-2/3) has a clear definition, but what even IS something like 2^pi? It turns out that the function exp(x) is not just special because it satisfies dy/dx = y(x); its existence is actually essential to rigorously define things like 2^pi and establish properties of exponential functions. This blew my mind.

@rickyleung5890

7 жыл бұрын

3Blue1Brown how about derivatives of matrices?

@Antonego64

7 жыл бұрын

3Blue1Brown Why "e" is si important?

GRANT! YOU ARE PHENOMENAL! Honestly, you have been the best teacher so far that satisfied me with enormous amounts of "heavenly" beneficial information. Your videos are golden,reflected by your amazing mind and passion for the subject!

This is one of my very favorite @3blue1brown videos. I’m 41, a college prof in music, and last fall I took calculus I as a student. Math has long fascinated me in my adulthood, despite having had only poor to mediocre experiences with it before. Learning about e has been amazing. Such a beautiful constant. I’m partial to it over pi.

The only magic close to the beauty of mathematics is Grant's incredible ability to describe it to laymans such as me

This is more beautiful than any piece of art I've ever consumed. Math done properly has a deep and beautiful sense of warmth in it.

Yes! So excited :-) love the series so far, you're an inspiration.

Grant I am really in love with your this series. how could someone made something so helpful. Grant I was saying You earned a subscriber now I am saying you earned a fan. Thank you so much grant ,we love you

I cannot believe how simple you made this concept. Kudos!

Your videos are what calculus teachers dream they could teach. This is the type of content that really gets an individual to appreciate the beauty of mathematics. Love your work man!

What is e? Baby don't derive me, don't derive me, no more

@rimelgana1656

5 жыл бұрын

Oh wooow what a great underrated comment

@muhammadqasim7056

5 жыл бұрын

Shouldnt it be baby dont differentiate me don't differentiate me,no more?

@muhammadqasim7056

5 жыл бұрын

Derivation is different

@MarcoBarbato

5 жыл бұрын

😂😂😂

@Zaraki35711

5 жыл бұрын

Muhammad Qasim Dilawari u

Thanks to the very explanatory videos of this channel, they have helped me understand the whole concept of derivatives (and not only that) at a deeper level and I am grateful for that! Keep up the great work!

This is art. This series is the product of pure creativity and visualization, just like having a 3rd eye inside of your head that allows you to feel the flow. Over and Above the excellent grasp on the English language, another language which is so powerful, when one knows how to express it using another set of skill, is the language of programming. You have a beautiful life I know for sure...

I am absolutely in love with this series! Thank you for these beautiful videos.

Loved the video. I always wondered about exponential derivatives and integrals. I always just resorted to memorization due to the necessity of passing the tests, but this makes the entire thing a whole lot more intuitive! Thanks!

I like how you said in the video that you felt sorry that there is no diagram or picture that can help visualize this relationship, but expressing it numerically has actually made me so much more interested; I can't believe these can result in so clean numbers.

You sir, along with Eddie Woo, Khan Academy and other similar amazing channels present science like its supposed to be presented: endlessly intriguing, simple and consequential, yet complex and mysterious, but all in all, you never fail to show with your beautifully made videos, with superb, original, dynamic animations, that the seemingly random formulas and symbols of math can all be boiled down to a simple, very logical and intuitive concept, and you are doing such a damn well job explaining the why's and how's, so we don't have to rely on memorisation of a bunch of random characters anymore. Thank you, from the bottom of my heart. You really changed my whole attitude towards mathematics.

This gets me excited the way TLC, History Channel, Discovery Channel used to when I was a kid. Thank you for these videos and I hope you keep them coming.

That is a depressingly good interest rate.

@mikehaskel8455

7 жыл бұрын

Either that or it's a time-lapse video over many decades!

@nameguy101

7 жыл бұрын

0 years: Big Bang 10^-130 years: Weak force separates from electromagnetic force 1 second: Neutrinos stop interacting with other particles 10^9 years: Milky Way is formed 10^9.96 years: Earth is formed 10^10.01 years: Life on Earth forms 10^10.14 years: Jonathan puts $200 in his savings account 10^15 years: Sun becomes a black dwarf 10^100 years: Last black hole evaporates 10^211 years: Jonathan doubles his money

@b1odome

7 жыл бұрын

An interest rate is usually expressed as a percentage (i.e. 5%), so you have to write the exponent for the function as 1+r, as otherwise you would be losing money.

@b1odome

7 жыл бұрын

Ahh, I did indeed mix up something there in that example. You're right, all sources that I can find don't write the equation as 3B1B did.

@illuminati.official

6 жыл бұрын

This is the greatest comment I've ever seen on youtube

This channel is blowing me away. I’ve never seen calculus described so concisely, logically and yes beautifully. Almost lyrical. Never really understood the geometric logic behind the various ‘rules’ of calculus….until now!! And the animation is as perfect as the discussion! Thanks for this amazing resource!

I can not thank this man enough. What this man is doing is truly, truly incredible. This comment won't probably reach you. But thanks man. I have finished the whole series. I don't think anyone in the earth could make this as clear as you did. I am gonna reccomend this to all my classmates. And if I ever have kids and they need to learn calculus, this is the series they will be watching...

These videos have helped me more through calculus than two year's worth of my teachers explaining it to me. Thank you @3Blue1Brown

I love your channel. I'm very familiar with the topics you present but you explain them wonderfully and I always learn something new.

I love your approach to explaining difficult problems, it’s refreshing to get a tangential view of something that seems difficult at first but becomes easily apparent once viewed in this way. Thanks for making all these great explainers, it makes learning or refreshing our knowledge base so much easier. Thank you for your time.

This is an amazingly clear explanation of not only the 'what' and 'how', but also the ever elusive 'why'. Thank you!

They found the value of ϕ, π, i, the square root of 2 and e. Now if mathematicians could *please* finally agree on the value of x, I'd be so happy. ;)

@blankiecat9302

5 жыл бұрын

But x is the wildcard that we dont know of thats is whole purpus it can be anything

@amor76

5 жыл бұрын

@@blankiecat9302 That's the joke

@cbw_

5 жыл бұрын

r/wooosh

@Kualinar

5 жыл бұрын

The first are constants. X, Y, Z,... are variables. It can be a variable that you change, or that you find.

@boysurfs

5 жыл бұрын

Björn P. If we found x, I bet we could finally find Y. Like goshdamn!

Just figured an interesting property geometrical of e^x: If the slope equals the height, the tangent of point (x,e^x) will touch the x axis on point (x-1,0). In other words, the base of the slope triangle always has length distance of 1.

@ihsanauliarahman1057

3 жыл бұрын

Ow cool. I'll remember that

@martinkunev

3 жыл бұрын

He says that at 8:42

@jonasrla

3 жыл бұрын

@@martinkunev It's been a while I wrote that, so I won't remember exactly my train of thought, but I think actually I constructed something over 8:42. I'm not talking just about e^1, it's about all the slope triangles constructed using: the tangent, x axis and the perpendicular line that goes through the x axis and the tangent point. The claim was that this triangle always have base one.

@jonasrla

3 жыл бұрын

Yeah, that goes exactly because the value and the slope have the same value, its just a corollary based on that fact

@mahdiyousef4516

3 жыл бұрын

Yeah exactly what I noticed! Also to say, this side of the slope triangle always equal to the natural logarithm of the constant base of the function🤓

Absolutely amazing concept explained beautifully! I'm amazed by how algebra manipulation can allow us to see a constant in something that doesn't seem to have any constants (an exponent function)

I appreciate how you highlight that the additive property of exponents allows you to relate additive ideas to multiplicative ideas. Something I hadn't seen expressed so clearly!

I've never seen such an amazing channel like this one! Now that exponential curves are common subject on all the news, I came here to better understand it and got really impressed seeing how good and well done this video is. Thank you a lot for this high level content.

@kopek702

2 жыл бұрын

no need say all this. What if i went around saying to randoes, "hey guys you know eveery day shit comes out of my rectum and i have to get rid of it!" now that would be dumb. no? some things are so obvious that mentioning them is just like going around telling random fuckers, Hey bro. You know this one time i held my farts in all day and afterwards when i went home non of them would come out. and then it started to hurt like hell and i had to go to the docrtors and a male nurse put his pinkie finger up my ass and pulled it out real fast and diahhrea and farts came poring out all over his angular sharp featured face.

My favorite part of these videos is showing the numbers move as operations are applied. This is exactly how I visualize math when I work out a problem.

@vanillaannihilation5871

4 жыл бұрын

same, just as in his videos when I simplify the numbers/variables disappear.

I just want to say that your videos have been a massive catalyst for my growing interest in mathematics. Thank you so much for everything.

Thank you Grant and thank you to 3blue1brown. The insights you provide on your video pertaining to mathematical concepts are truly appreciated.

After 3 days of struggling with this concept. And after 4 different viewings of this video, I finally, finally get it. Thank you and every other math teacher online like NancyPi and various other websites. Man that's useful

Even though I knew what e was as a definition, not until 8:40 that I was able to actually picture it for the first time. Might not mean much to some but I've been trying to make that connection for years, Thank you!!!!

He is taken the visual understanding of mathematics to absolutely different level. Even I understood most of the materials and have to repeat it several times to get it. Thank you teacher who is able to put himself at a student level.

Those last minutes explaining how the euller's numer its actually something that changes proportionally to it self blew my mind! And the way you led the video to get to this point was fascinating, thank you

This is my favorite math less I have ever had. I find this so fascinating.

Explains a lot about physics and chemistry in which you often encounter formulas that have some kind of exponential in them (CR circuit, cinetic of chemical reaction, temperature throughout time...) It's really nice to get a sense of where those fomulas come from after blindedly learning and applying them. Thank you!

I would WISH I can find this channel when I was studying calculus back in 2016. I am a Master's student rn and I am sitting here, watching Calc lectures on KZread, AND learning like a newbie. I learned a loooooot from your videos, thank you! You give me a new perspective on Calculus and Linear Algebra, thank you so much!

Grant, your videos on Euler's number 'e' and eigenvectors/eigenvalues, along with your entire collection of content, have been incredibly helpful for me. As a current high school student, I know there's still a lot for me to learn, but your teaching style has made complex concepts much clearer. Recently, while exploring simple harmonic motion, I had trouble understanding why the solution to second-order ODEs involves e^rt. Your explanations finally made sense of it for me. Your dedication to making math more accessible shines through in all your videos, and I'm truly grateful for the impact they've had on my learning journey. Thank you for all your hard work!

This video really helps me to understand why we need e, and why it is important to use e in calculus when dealing with exponential functions.

You are the best! I cannot imagine Calculus without your help! Thank you.

no because why did it take me 3 years of knowing about your channel and having this confusion over e to finally see if you had a video on this 😭😭 this feels life changing. Thank you for helping me understand so clearly.

If you aren't amazed or you don't get goose bumps or even smile in astonishment then your're not really getting what he's saying. Absolutely powerful!!!

Your lucidity to teach is admirable! I'm learning so much calculus with this serie, very grateful.

Thank you for making math so intuitive and beautiful!

I friggin love how the visuals have helped me start to see math/calculus as a machine visualization. How the mechanical parts in a machine's relations to one another can actually be looked at as composite functions. It's so cool to see how sin(x^2) moves around in a strange rhythmic pattern.

I love this channel and these series. They are really helping to me to visualise the maths I am using in my A level studies. It is excellent content that isn't afraid to use actual maths but also never fails to make it interesting and intuitive.

@silverhusky7993

5 жыл бұрын

so, how did your exam go

One fun exercise is to define a function as f(x + y) = f(x)f(y) and work out what its derivative can be. Might give you a different perspective on exponentials.

@dlevi67

7 жыл бұрын

The really interesting thing about this (for me at least) is that it manages to tie back to the "naive" definition of exponentiation as repeated multiplication, just like the Gamma function does for factorials.

@hybmnzz2658

3 жыл бұрын

In fact all we require is differentiability at x=0 and you can prove f(x) = Ce^x for real values C.

Idk what about 3b1b’s voice, but it’s incredibly soothing to me. I mean this in the best way, you help me fall asleep and calm down and slow my heart down. Thank you so much.

Thank you. This episode has given me enlightenment into the all the damping constants & time constants in my daily work. Please keep up the great work.

So amazing... your videos are all beautiful arts. Love your videos. You are one of the most amazing youtube channels I ever seen. Hope for more beautiful and amazing videos :)

After hours of reflecting on this lesson, I think I finally understand it! Everything clicks and it feels so satisfying! Thank you Grant, for showing us the beauty of mathematics :)

@singh2702

2 жыл бұрын

So did you get that rate of change can be instantaneously changed? which is impossible.

This is another great video, because when we were introduced to the constant of 'e' many decades ago - we were told that it was just another constant to remember by rote, just like Pi. So there was I was wondering 'why' and getting no explanation at all. If you asked the tutor 'why' they didn't know! So instead of learning the math syllabus, I was wondering about 'why' and getting left behind. In the end after several decades I stumbled on the 'why' bit all on my own, in some kind of 'brain burp' in my 40's, I'd worked it out for myself!. Just imagine how much better it would have been if I'd understood in the first place? Now this is what you are doing; filling in the gaps in education that many of us have suffered for decades. I'm now 72 and retired, but this this is priceless stuff for younger generations. Thank you so very much :-)

This is amazing! I have studied math on and off my whole life, mostly for fun, but never realized this special quality of e. So cool!

Speaking about the (2^dt-1)/dt constant, if we substitute 2^dt-1=u, then dt=log_2 (u+1), which makes the expression become u/log_2 (1+u), using logarithm properties we can rewrite it as 1/log_2 *(1+u)^1/u* . As dt->0, 2^dt-1 also goes to 0, therefore u->0. This here explains another commonly used definition of the number e as lim n->0 of (1+n)^1/n

@3blue1brown

7 жыл бұрын

Nice!

@BarYamin

7 жыл бұрын

Very nice. I tried getting to that limit value a few times before, never saw this substitution.

@MrPetoria33

7 жыл бұрын

Famfly Or letting m = 1/n, you get the alternative form seen in some other videos: lim m->Inf of (1 + 1/m)^m

You reinvented math for me. 3 years of high school left nothing but fear and darkness in my brain when it comes to maths. Now it starts to see maths again, visually, naturally.

Thanks a lot. I really like how you tie everything together in such a fun and intuitive way :)

Woooow!! This series is eye opening. I've never been pleased with explanations of Euler's number e until now. My life is complete. :)

Alongside PBS's Infinite Series, Numberphile and Mathologer, I believe you are the best KZread maths channel. Bravo! :-)

@TheLuckySpades

5 жыл бұрын

RIP Infinite Series

@Fermion.

4 жыл бұрын

Even now, in mid 2019, I still stalk those channels for new content. BTW: _Why'd you do it Infinite Series? Doge ponders - So smart; much brain; marketing?._

Never learned exponentials in such an intitutive way. Amazing

I love your videos because you make us reflect about the deepness of the purpose of some calculus or numbers. Keep up the good work!

Again, so insightful. What a great teacher and a great team producing this series!

I wish to go back in time in 11th Grade and learn this again in the classroom with this much deep understanding!

I am 32 yr old. all my life I used to blindly use "e" in my calculations without knowing what it is. And thanks to you I let out a long "OOooooohhhhhh!!!!!!" today after watching your video. I wish our schools taught the way you do instead of just mugging up as it s.

This was very helpful. Thank you so much 3Blue1Brown!

This is a better demonstration than the usual one of continuous compounding and it explains how the term "natural" logarithms comes about.

Aargh! I was doing ok until now! I'm an old git (62) so not grasping stuff as I might have a few decades ago. Just not 'grokking' the e thing yet. I get that it perfectly describes compound interest, and I know it forms a function that is its own derivative. I may have lost it when we got to natural logs (only ever had the base-10 kind at school!) Oh well - watch again tomorrow, and see if it's clearer :)

@pointlesslylukesplainingpo1200

3 жыл бұрын

why you tryna learn this stuff at 62 lol

@tolep

3 жыл бұрын

@@pointlesslylukesplainingpo1200 to make sure he isn't dead. That's how we feel at certain age (I'm 47)

it probably says something about one's nerdiness and major when one immediately recognizes 0.693... to be ln(2).. still, after seeing mathologer's second e video and having my math teacher explain bit by bit the relation of the exponential to complex numbers after calc, i genuinely didn't expect to still be amazed to see new things about this constant, and i'm incredibly glad to see you cover these topics so beautifully :o i genuinely can't wait for you explaining the taylor series, as expanding functions into it always seemed like black magic hijinks, and no textbook ever satisfied me in this matter....

@iankrasnow5383

7 жыл бұрын

I know everyone in my class hated power series (and they never seem to come up in any calculus classes after you learn them), but I really liked learning about the concept at least, if not the actual algorithm for generating them. Why? Because the concept of the Taylor series answered a question I had been thinking of. That is, can you design a function which approximates any shape, or do you just get these function shapes which seem fairly limited and random? With power series, you learn that polynomial functions can be used to approximate the shape of exponentials, or trigonometric functions, but the idea could probably be expanded to work for any shape which is a continuous, differentiable function. In that sense, the concept is really quite beautiful. Even more-so since the series approximations for exponentials and sine functions are such simple patterns.

@sadboihours

7 жыл бұрын

ah, that's already so much more than anyone ever cared to tell us in class, even though this is exactly why the taylor series looks and behaves the way it does :") after looking into it myself, i''m actually shocked how little is said about taylor series besides just how to calculate them (which barely anyone in my class really grasped at first, either way), as it's so useful in approximating, around a certain point, ugly functions into nice, easy to analyse polynomials, and because of this plays such an important role in physics. even though being able to calculate things like sinx or e^x pretty much by hand if you're ever stranded on a deserted island without a calculator, or deriving euler's formula seem to be fantastic enough reasons to know what power series are about and not to hate them passionately.

@dlevi67

7 жыл бұрын

+RepriseOfxq010 That your eyesight converges slowly, bilaterally and asymptotically. In other words, my friend, you are blessed with a highly insightful form of mathematical strabismus.

@thatonedevastatingleek380

5 жыл бұрын

I just knew it cause .693, ln2 comes up in the half life equations in chemistry

This video was amazing! Don't know why I took so much to watch this. For me, was one of the most clarified and concise explanation about exponential derivatives.

omg so all this time, I didn’t understand where the derivative of a^x came from but this video explains it so well!! it makes so much sense now

One of the first things I thought about when I learned derivatives is how exponentials fit in, since their slope is based on themselves

The animation is so good and tidy! I like it!

@honeycomblord9384

4 жыл бұрын

Could this kind of comment be called “Like++”? But yeah, this channel has excellent animation.

Just one word : Amazing..The way you have explained one of the most basic yet mysterious topics, is awesome..Wish you were my maths teacher during school days..

Your videos makes the world better. Thank you for this beautiful work!

I love your videos but I wish there was a channel like yours but dedicated to Physics instead of Math. This is also why your video about the Brachistochrone is my favourite one.

@praveensingh4525

5 жыл бұрын

Physics videos by Eugene is pretty decent.

Markus Persson was a contributor to this video (13:22)? Good guy Notch!

@linuxguy1199

7 жыл бұрын

He contributes every video

@SharKCS11

7 жыл бұрын

Oh, interesting. This is the first time I actually noticed the contributor's list haha. Definitely a channel worth supporting!

@ErkaaJ

7 жыл бұрын

Markus Persson is a very ordinary Swedish name though.

@vampyricon7026

7 жыл бұрын

+

@jonny4233

7 жыл бұрын

He posted on reddit... www.reddit.com/r/math/comments/5b6klv/who_cares_about_topology_inscribed_rectangle/

I feel so lucky to stumble upon this series. I’m professional engineer and never really had a TRUE understanding of calculus before watching these videos, I mean, I could calculate it… but this is different level, beautiful :)

Awesome explanation!!. I've never watched such a good explanation of this functions and its relationship with DE. KUDOS!!

The more videos I watch of this series, I feel like getting exposed to some sacred knowledge that our teachers avoid so carefully.