What's so special about Euler's number e? | Chapter 5, Essence of calculus

[@deepakachu]

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

[@sparrowp2251]

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]

fax

[@fazeblizz5898]

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

[@cadmus2573]

@@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]

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

[@arongil]

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]

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]

Joe

[@CubeZero]

@@Joe-bb4yi Mama

[@Pladinium]

@@CubeZero nice

[@aarohansharma4551]

@@CubeZero nice

[@humbleevidenceaccepter7712]

e is the grade I nearly got in calculus.

[@MagruderSpoots]

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

[@feynstein1004]

Exceeds expectations? Nice, mate :)

[@hexa3389]

My ESL teacher used to give out E for Excellent.

[@TerryJLaRue]

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

[@hexa3389]

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

[@ayushagarwal7631]

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]

Grant makes exactly the same point in his TED talk.

[@FaffyWaffles]

That’s why I’m studying math

[@kmjohnny]

Math (especially calculus) is indeed a mystery.

[@H3XED_OwO]

Socratic Teaching

[@joejitsuway960]

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]

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....

[@suflaj9314]

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

[@stargazer7644]

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

[@Hi-6969]

@@stargazer7644 its called manim actually

[@theroboman727]

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

[@ElytrCSGO]

@@andyk2181 bruh

[@ChimaevMikelson]

😂

[@seanryniker3897]

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

[@TheManWithTheFryingPan]

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

[@ash_ithape]

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

[@pianochannel100]

That is so good ahaha

[@T_tintin]

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

[@ash_ithape]

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

[@vidsangre]

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]

There goes tomorrows lunch

[@Slick6464]

@@jblangcua2726😂😂

[@display9138]

😂😂​@@jblangcua2726

[@ristekarashabanov1879]

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

[@cheesy4384]

6 years in eee to understand e

[@parallellinesmeetatinfinity]

@@prun9095 W dp 🗿

[@Periwinkleaccount]

@@parallellinesmeetatinfinity What’s that supposed to mean?

[@codename-abdullah]

​@@Periwinkleaccountelectrical and electronics engineering

[@Jaishreeram-zq4sc]

Hey bro I have taken electrical engineering recently, is there anything you want to share with me,

[@Squirrel_314]

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]

The idiots will die out and we will win 😂

[@elena6516]

@@somxr_738 false. Have you seen Idiocracy?

[@12-343]

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

[@aidenheffernan7556]

how you felt after saying that 🐶👺

[@parallax256]

@@12-343 very blursed

[@ull893]

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]

+Mohan7 Wow, thank you!

[@mujtabanasir2970]

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

[@ThangTran-qn7vq]

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

[@wiseacredave]

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]

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.

[@3blue1brown]

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. I also give a longer lecture in the lockdown math series: kzbin.infoZxYOEwM6Wbk?si=yjGJazVlVETQQKcn If you'd like another perspective, I'd highly recommend Mathologer's coverage of the topic in these two videos: - e to the pi i for dummies: kzbin.info/www/bejne/Y5XLeaWdYrCVgJI - The number e explained in depth: kzbin.info/www/bejne/eqCkk3Rsp72lraM 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": kzbin.info/www/bejne/o6fQpnaGq7eijbM (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]

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]

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]

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]

3Blue1Brown how about derivatives of matrices?

[@Antonego64]

3Blue1Brown Why "e" is si important?

[@jmadratz]

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]

That sounds incredible! What do you work on now?

[@jmadratz]

@@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]

@@jmadratzThaad systems are fucking sick

[@jmadratz]

@@alecmiller5296 Sick in a good or bad way?

[@alecmiller5296]

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

[@mrmorganmusic]

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.

[@tonycadena47]

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!!!

[@hamiltonianpathondodecahed5236]

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]

Good thing you mentioned

[@marwinhochfelsner]

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]

@@marwinhochfelsner loved it.

[@taiman9423]

bruh... chill

[@Manuel-j3q]

@@marwinhochfelsner 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.

[@jameswise9171]

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

[@math_nerd_guy]

This actually made me laugh out loud thank you

[@big_cuh]

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

[@DoctressCalibrator]

Why would anyone want to say that?

[@squdardt.9719]

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

[@Ryan6.022]

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

[@AustinWLee]

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!

[@BlackRockExecutive]

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.

[@MichelCarroll]

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

[@AssasinMilo]

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]

Bruh that's exactly how I felt!

[@manideepp2229]

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

[@NovaWarrior77]

I'm proud!

[@StrummerDave]

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]

Good job bro ! 👌

[@sharathkumar8422]

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.

[@TheMurderArt]

I really like those π creatures...

[@shugaku2461]

TheMurderArt Yup, me too. Who wouldn't?

[@Tsskyx]

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]

I want to eat them. I love Pi.

[@shugaku2461]

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

[@TheMurderArt]

one of them would be the perfect pet

[@micaduce509]

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.

[@herewegoagain-p3p]

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...

[@thinkanddo2352]

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!

[@sgnMark]

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!

[@jonasrla]

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]

Ow cool. I'll remember that

[@martinkunev]

He says that at 8:42

[@jonasrla]

@@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]

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

[@mahdiyousef4516]

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🤓

[@thecountoftuscany9493]

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]

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]

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]

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]

​@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

[@doremysheep7864]

Sadly schools don't want us to be intuitive. Some (maybe most? Maybe all? Idk) schools just want us to shut up and do as we're told. Memorize rather than learn.

[@thetntsheep4075]

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

[@oscarfriberg7661]

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]

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.

[@24steelerfan24]

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]

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

[@kolrabi]

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]

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

[@amor76]

@@blankiecat9302 That's the joke

[@cbw_]

r/wooosh

[@Kualinar]

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

[@boysurfs]

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

[@DMarci2001]

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.

[@poppyflorist]

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...

[@giantneuralnetwork]

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

[@oneRyanJoseph]

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

[@tronmend219]

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!!!!

[@adrianguadalupemunozjaureg5425]

Thanks a million for the knowledge. I am teaching my first calculus class and this series has inspired me, taught me, enlightened me and motivated me to the be the best teacher I can be. Again, I can not thank you enough. Love from Mexico. Peace!

[@beefandpotatoes6525]

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.

[@osmium3132]

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!

[@marceloavila87]

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]

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.

[@JorgetePanete]

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

[@rimelgana1656]

Oh wooow what a great underrated comment

[@muhammadqasim7056]

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

[@muhammadqasim7056]

Derivation is different

[@MarcoBarbato]

😂😂😂

[@Zaraki35711]

Muhammad Qasim Dilawari u

[@Akira_2201]

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!

[@astronomy-channel]

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!

[@michaelbruce4987]

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

[@TheForresthu]

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.

[@jonathanlimm7221]

That is a depressingly good interest rate.

[@mikehaskel8455]

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

[@nameguy101]

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]

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]

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]

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

[@migueldiaz-acevedo8211]

I'm 33 years old and have worked in and around science for most of my adult life and this video just changed the way I see so many things. This helped me gain so much intuition in an instant (~ 14 mins)

[@arlenn729]

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 KZbin, 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!

[@jaysonl3685]

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)

[@yxlxfxf]

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]

Nice!

[@BarYamin]

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

[@MrPetoria33]

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

[@Hobbit096]

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]

so, how did your exam go

[@MrEerwin]

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

[@paulfrindle7144]

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 :-)

[@MuriloHGomes-vd6zt]

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

[@HsinTsungChu]

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]

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

[@doriancostley9075]

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!

[@Ms.Mamabean]

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.

[@HaIPeHaOP]

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 :)

[@Brainiac_124]

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

[@TheLuckySpades]

RIP Infinite Series

[@Fermion.]

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?._

[@natashajacquelineTauro]

Thank you for making math so intuitive and beautiful!

[@kaivalyajoshi354]

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

[@Ben-sk7ey]

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.

[@유리진주-w8l]

I am a Korean high school student. I learn this contents in my school. Your video is so cool. Thank you for this video, I could understand math more easily.

[@davidcooper4327]

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

[@SirusDas]

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

[@alt-f1108]

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.

[@ianthebadguy]

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!

[@greyATK]

I'm binge watching the calculus playlist, but this video in particular made me sit up and be wowed at the explanation to this derivative. My teachers make us copy and memorize these, which I hate because I have a hard time memorizing, but hearing the explanation behind them is helping me understand, and consequently learn them better. thanks for the video!

[@Shivag1604]

Never learned exponentials in such an intitutive way. Amazing

[@TheAgamemnon911]

You said you wanted a geometric interpretation? How about working with the fact that exponentials are self-similar in the way that you can either scale them on the y (representing a different starting value) or move them on the x (representing a shift in time) and still get the same result graph. I don't know if that would help or hinder understanding, though...

[@3blue1brown]

Hmmm, there could be something here. You could show how taking a tiny step to the right is, in fact, equivalent to multiplying by some constant. I think the upshot would be the same, which is that you could conclude that an exponential a^t is proportional to itself, with some mystery proportionality constant, but there could be something nice to seeing that with respect to horizontal slides being equivalent to vertical scaling.

[@paianjen]

This page has a nice visual representation of e, betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/

[@hypercent]

"an exponential a^t is proportional to itself" -> Every function is proportional to itself, with proportionality constant 1 [f(x)=1*f(x)]. On a related note, in the video at 9:38 you say that e^ct=c*e^ct which is obviously only true for c=1. Of course I know what you mean, it's written on the screen after all, but some people might still get confused by how you say it. Nevertheless, I just love your videos. Thank you very much for making them.

[@shyrealist]

I think the visualisation was there... if you count the number of little pi pupils that are growing you find that the number is the same as the number of fully grown pi pupils. That shows the proportional relationship. Thinking about it now the fact that you need to visualise is the continuous rate of growth in order to get to the next 'generation', hmmm... Is the animation growing them at an (incorrect) constant rate?

[@BytebroUK]

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]

why you tryna learn this stuff at 62 lol

[@tolep]

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

[@akash3478]

If you couldn't follow what he was talking about at 9:28 ,here it is. Now look at the frame of 9:28 Let 3t be equal to g d(e^g)/dg=e^g as we know derivative of e^g should be itself, right? Now let's multiply dg on both sides We get , d(e^g)=e^g(dg) Let's put the value of g which we had supposed earlier. d(e^(3t))=e^(3t)(d(3t)) Derivative of 3t would be 3 ofcourse Finally , d(e^(3t))=(3)e^(3t) And this does not work for only 3 but any constant like it . Therefore, d(e^(nt))=(n)e^(nt) For n to be any constant. Hope this helped. Lol I'm so idle.

[@kaiohenrique7633]

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

[@saarlevy9891]

I'm so sad that I had already memorized all of these facts, right now they are just fun ways to understand the subjects. But if I would have learned those facts for the first time, my mind would have been blown.

[@RajShekhar-jy2zi]

@@want-diversecontent3887 you're learning calculus at 10? Cool .

[@RajShekhar-jy2zi]

@@want-diversecontent3887 in India it is taught in grade 11. If you are understanding then what's the problem in learning it .

[@SSM24_]

Honestly, there's a ton of value in going through these videos already being familiar with the things he's talking about - that feeling of "wait, _that's_ why that's a thing?!" is still amazing.

[@Vibranium375]

@@want-diversecontent3887 You are never early or late in learning something add long as you have the prerequisite knowledge and skills and are interested

[@gamingbutnotreally6077]

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

[@slimpbisquitte3942]

oh my god. My teacher just told us today to look 'what is e' up in the internet because we will learn e tomorrow. that a nice timing. i love you

[@Pianodog]

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.

[@this_is_mayhem]

This is one of the most mind blowing things i've ever heard. Really astonishing how some abstract number e could naturally elegant describe real world processes

[@abcrtzyn]

I'm happy to know that pi creatures are asexual

[@Ekvitarius]

How?

[@lukz9437]

@@Ekvitarius their population can grow even if only one exists 1:10

[@onepangaean3018]

They're mathematical amoeba

[@Cashman9111]

@@lukz9437 i suppose even if a tiny fraction of 1 exists ;)

[@chonchjohnch]

I thought you meant asexual in the “narcissistic teenager” sense and I was extremely confused for a moment

[@mu.iskanderkrayem8391]

I hope you'll make a video about Taylor series and power series

[@AuroraNora3]

Alex Kr He said in Chapter 1 he would do that in Chapter 10

[@munkithabid3434]

8:14 little pi creature is actually wondering about its creation

[@juliak5980]

Lol

[@darkblaze1594]

Just like we do!

[@vesix__]

I've studied mathematics at a university level, and this video gave me a better intuition of e than I've ever heard before, mind blown!

[@monton206]

The graphics are beautiful in this video. What a relaxing, aesthetic way to explain such a concept. The vibe of this video made it really easy for me to understand what e is and understanding math does not usually come easy to me.

[@omrialkabetz5602]

An interesting fact is that the number e only seems strange to us because we choose to write it in base 10 ! If we choose a different (Mixed radix) numeral system, it seems much more natural. For instance, in the factorial number system the number e is written as: 10.011111111111... recurring. This is due to the series expression of the number e as the sum 1/0! + 1/1! + 1/2! ... There is also a very nice visual representation of the factorial base as a branching tree on the number line, in which each branch represents a different digit, and the radix (number of branches) increases with each new digit. It makes you wonder whether there is a natural base in which pi is written in a simple manner. More information about the Factorial number system: en.wikipedia.org/wiki/Factorial_number_system And a discussion about the visual representation of factorials: math.stackexchange.com/questions/1383150/visualizing-the-factorial

[@Kirqos]

i don't know much about the factorial number system, but surely e is not just "weird" because of the base 10. no matter what base, e will always be irrational (even transcendental).

[@omrialkabetz5602]

e is irrational, no doubt about that (either way it can't be written as a fraction). The factorial number system isn't a standard base, because the radix isn't constant. For example, when we write down the number 15 in decimal, the first digit to the right denotes the number to be multiplied by one, while the second digit to the right denotes the number to be multiplied by ten, and so on. On the factorial number system, the first digit to the right denotes the number to be multiplied by 1!, the second digit by 2!, the third digit by 3!, and so on (any integer could be written this way). Just as you could represent any number on the number line with a binary tree (where each branch denotes the next digit in binary), is it also possible to draw such a tree for the factorial number system (where the number of branches increases by 1 with each following digit). There is also a rather natural way to extend the factorial base to all natural numbers. For instance, the number 0.0123 in the factorial base represents the number: 0*1! + 0*(1/1!) + 1*(1/2!) + 2*(1/3!) + 3*(1/4!) = 23/24 (just as in decimal it denotes 0*10 + 0*(1/10) + 1*(1/100) + 2*(1/1000) + 3*(1/10000) = 123/1000). And the number 10.01111111111... is equal to: 1*2!+0*1!+0*(1/1!)+1*(1/2!)+1*(1/3!)+1*(1/4!) ... = e

[@Kirqos]

interesting. thanks

[@GM-th5jj]

50% of Engineering mathematics makes sense now... I had never quite understood where e came from in all those equations about rates of decay and time constants. Thank you!

[@Aikman94]

IKR! This is by far the best explanation of the exponential I have found.

[@Oliwarder]

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

[@honeycomblord9384]

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

[@st3althyone]

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.

[@krishnaraj3989]

Wow. I have no words left. This man deserves a noble Prize for teaching.

[@exhalerwolf1272]

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

[@SharKCS11]

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

[@linuxguy1199]

He contributes every video

[@SharKCS11]

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

[@ErkaaJ]

Markus Persson is a very ordinary Swedish name though.

[@vampyricon7026]

+

[@jonny4233]

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

[@gradow]

A very nice trick to use on that last numerical part on "figuring out" the derivative of 2^t is to note that since 1=2^0 we can rewrite (2^{dt}-1)/dt as (2^{dt}-2^0)/dt then we can see that this constant is simply the derivative of 2^t at t=0. This tells us that not only the rate of growth of a^x is proportional to itself, but also that it's actually itself times the rate of growth at the start of the process (t=0).

[@yxlxfxf]

we can see the same thing using the formula in the video d(2^t)/dt=2^t*const, substituting t=0 makes the 2^t term become 1.

[@leadnitrate2194]

The comments under this video are fascinating, in that they seem to want to help the next viewer as much as the video itself. Thanks for a lovely insight.

[@darkdelphin834]

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!

[@ShermanSitter]

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!

[@kirtidapandya2918]

I'm -1/12 th

[@alexhancu969]

But there are not an infinite amount of viewers nor humans so HAH you can't be the - 1/12 th

[@kirtidapandya2918]

Ovidiu Hancu ...I'm living in parallel universe..so in my universe there r infinite no of them .....and Mona Lisa is painted by Shakespeare.. HAH..

[@jjtt]

*Σ_n=1^∞ (n) != -1/12*

[@AuroraNora3]

Juan T You mean -1/12

[@kirtidapandya2918]

Hoo Dini ....yaa

[@ASLUHLUHC3]

Pausing this at 10:38 (during my second time watching) and feeling it all click was SO GODDAMN SATISFYING

[@telecorpse1957]

I'm pausing at the very same moment to a mind-burning feeling of it not quite clicking. Not satisfying at all, but definitely humbling. In a good way I think :)

[@WilliametcCook]

1:22 "t squared" *2^2 t happens to also be 2, which does make it t squared, but that's not how the function works.

[@S8EdgyVA]

Omg, never noticed it said “t^2” before, maybe he mispronounced 2 as t, they ARE similar sounding

[@greegorygrimlee5487]

Baby Pi creatures eyes at around 1:00 to 1:05 is the kind of subtle thing that proves the attention to detail in 3b1b videos. Thank you so much for working so hard at this, it's a wonderful vehicle for your excellent communication of complex ideas. xox

[@StrummerDave]

Brilliant. I wish my math profs in university had known how to teach. You explained in a few minutes what 4 professors failed to explain in 2 years.