What's so special about Euler's number e? | Chapter 5, Essence of calculus
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- Опубликовано: 3 июн 2024
- What is e? And why are exponentials proportional to their own derivatives?
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Timestamps
0:00 - Motivating example
3:57 - Deriving the key proportionality property
7:36 - What is e?
8:48 - Natural logs
11:23 - Writing e^ct is a choice
Corrections:
9:40 - I meant to say "*the derivative of* e to the power of some constant..."
12:30 - What's written as "(1 + r)" should really just be r, by any usual convention for how to write an interest rate.
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Italian: ang
Vietnamese: @ngvutuan2811
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e is the grade I nearly got in calculus.
e was the grade I got in calculus. I didn't have 3b1b.
Exceeds expectations? Nice, mate :)
My ESL teacher used to give out E for Excellent.
@@hexa3389 That is probably what you thought it was for, anyway. :)
@@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.
Grant makes exactly the same point in his TED talk.
That’s why I’m studying math
Math (especially calculus) is indeed a mystery.
Socratic Teaching
Like I always say, mathematics is like a well Euler'd machine.
That's it. That's the joke to end all jokes
rip to people who don't know how euler is actually pronounced and they say - eeewwlaar xD
That is so good ahaha
@@anonymousash6678 sadly I was one of them...
@@T_tintin All good, now you know its pronounced "Oiler" :)
Every child would fall in love with learning if the world had more people like you ♥️
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.
fax
You are just lucky to be born with high intelligence, effort plays a small part compared to your processing and memory
@@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.
If world had more people like you who appreciate others , then it would be far better place👏
Can we just appreciate the time it takes to animate this numbers, graphs and characters ...
He has a slave named Python that he makes do this grunt work.
@@stargazer7644 its called manim actually
@@stargazer7644 At the risk of outraging the feminists, I'm making Julia my slave
@@Hi-6969 no, python is grant's slave and manim is python's slave
@@andyk2181 bruh
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.
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
@@Joe-bb4yi Mama
@@CubeZero nice
@@CubeZero nice
After 6 years in electrical engineering, I finally understood the meaning of e, thanks to this video.
6 years in eee to understand e
@@prun9095 W dp 🗿
@@parallellinesmeetatinfinity What’s that supposed to mean?
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.
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....
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.
The idiots will die out and we will win 😂
@@somxr_738 false. Have you seen Idiocracy?
And then there's the blursed mess that is reddit, and 4chan, the retired hacker group.
how you felt after saying that 🐶👺
@@12-343 very blursed
Instead of saying stuff like "Pi population", just say "Pipulation".
This actually made me laugh out loud thank you
@@CasioAns Instead of saying stuff like "This actually made me laugh out loud, thank you!", just say "This made me LOL, thank you!"
Why would anyone want to say that?
Catface nah man, I think no one should say that.
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!
There goes tomorrows lunch
@@jblangcua2726😂😂
😂😂@@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.
That sounds incredible! What do you work on now?
@@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.
@@jmadratzThaad systems are fucking sick
@@alecmiller5296 Sick in a good or bad way?
@@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 !!!!
+Mohan7 Wow, thank you!
^^^^^^^^^^^^^^^^
Same to me. It removed the mental block about "e" which has been in my brain for years. So beautiful video.
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.
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.
I really like those π creatures...
TheMurderArt Yup, me too. Who wouldn't?
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
I want to eat them. I love Pi.
TheMurderArt Id like to snuggle and live with one. Heck, I'd buy a plusher of those things!
one of them would be the perfect pet
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.
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
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
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.
@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
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.
After years of doing calculus, this video finally made "e" click for me. So stupidly simple, for something so anxiety provoking. Thank you!
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.
Bruh that's exactly how I felt!
Hopefully you didn't become hulk😉. His voice itself creates interests to read.
I'm proud!
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.
Good job bro ! 👌
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 ..........
Good thing you mentioned
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 …
@@hlfan loved it.
bruh... chill
@@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.
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!
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.
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: ruclips.net/video/-dhHrg-KbJ0/видео.html
- The number e explained in depth: ruclips.net/video/DoAbA6rXrwA/видео.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": ruclips.net/video/mvmuCPvRoWQ/видео.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!)
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.
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.
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.
3Blue1Brown how about derivatives of matrices?
3Blue1Brown Why "e" is si important?
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.
The only magic close to the beauty of mathematics is Grant's incredible ability to describe it to laymans such as me
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.
What is e?
Baby don't derive me, don't derive me,
no more
Oh wooow what a great underrated comment
Shouldnt it be baby dont differentiate me don't differentiate me,no more?
Derivation is different
😂😂😂
Muhammad Qasim Dilawari u
That is a depressingly good interest rate.
Either that or it's a time-lapse video over many decades!
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
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.
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.
This is the greatest comment I've ever seen on youtube
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!!!
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.
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. ;)
But x is the wildcard that we dont know of thats is whole purpus it can be anything
@@blankiecat9302 That's the joke
r/wooosh
The first are constants. X, Y, Z,... are variables. It can be a variable that you change, or that you find.
Björn P. If we found x, I bet we could finally find Y. Like goshdamn!
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.
same, just as in his videos when I simplify the numbers/variables disappear.
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.
This is an amazingly clear explanation of not only the 'what' and 'how', but also the ever elusive 'why'. Thank you!
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.
Ow cool. I'll remember that
He says that at 8:42
@@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.
Yeah, that goes exactly because the value and the slope have the same value, its just a corollary based on that fact
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🤓
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!
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...
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!
Yes! So excited :-) love the series so far, you're an inspiration.
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!
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!
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.
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 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 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...
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.
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.
In fact all we require is differentiability at x=0 and you can prove f(x) = Ce^x for real values C.
I wish to go back in time in 11th Grade and learn this again in the classroom with this much deep understanding!
This is my favorite math less I have ever had. I find this so fascinating.
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)
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
Nice!
Very nice.
I tried getting to that limit value a few times before, never saw this substitution.
Famfly Or letting m = 1/n, you get the alternative form seen in some other videos: lim m->Inf of (1 + 1/m)^m
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!!!!
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 :)
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.
These videos have helped me more through calculus than two year's worth of my teachers explaining it to me.
Thank you @3Blue1Brown
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!
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'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!
I love your channel. I'm very familiar with the topics you present but you explain them wonderfully and I always learn something new.
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
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.
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 RUclips, 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!
I am absolutely in love with this series! Thank you for these beautiful videos.
You are the best! I cannot imagine Calculus without your help! Thank you.
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.
Thank you Grant and thank you to 3blue1brown. The insights you provide on your video pertaining to mathematical concepts are truly appreciated.
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.
Your lucidity to teach is admirable! I'm learning so much calculus with this serie, very grateful.
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
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!
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.
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.
Never learned exponentials in such an intitutive way. Amazing
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!
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 for making math so intuitive and beautiful!
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 is a better demonstration than the usual one of continuous compounding and it explains how the term "natural" logarithms comes about.
Alongside PBS's Infinite Series, Numberphile and Mathologer, I believe you are the best RUclips maths channel. Bravo! :-)
RIP Infinite Series
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?._
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.
so, how did your exam go
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 amazing! I have studied math on and off my whole life, mostly for fun, but never realized this special quality of e. So cool!
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 :)
So did you get that rate of change can be instantaneously changed? which is impossible.
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.
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.
This is one of the rare channels that actually taught me something
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....
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.
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.
+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.
I just knew it cause .693, ln2 comes up in the half life equations in chemistry
Markus Persson was a contributor to this video (13:22)?
Good guy Notch!
He contributes every video
Oh, interesting. This is the first time I actually noticed the contributor's list haha. Definitely a channel worth supporting!
Markus Persson is a very ordinary Swedish name though.
+
He posted on reddit... www.reddit.com/r/math/comments/5b6klv/who_cares_about_topology_inscribed_rectangle/
Woooow!! This series is eye opening. I've never been pleased with explanations of Euler's number e until now. My life is complete. :)
Wow. I have no words left. This man deserves a noble Prize for teaching.
One of the first things I thought about when I learned derivatives is how exponentials fit in, since their slope is based on themselves
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 :)
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..
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.
I hope you'll make a video about Taylor series and power series
Alex Kr He said in Chapter 1 he would do that in Chapter 10
The animation is so good and tidy! I like it!
Could this kind of comment be called “Like++”?
But yeah, this channel has excellent animation.
What an elegant and beautiful explanation of e! Kudos to 3B1B!!!
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
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 :)
why you tryna learn this stuff at 62 lol
@@pointlesslylukesplainingpo1200 to make sure he isn't dead. That's how we feel at certain age (I'm 47)
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.
@@want-diversecontent3887 you're learning calculus at 10? Cool .
@@want-diversecontent3887 in India it is taught in grade 11. If you are understanding then what's the problem in learning it .
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.
@@want-diversecontent3887 You are never early or late in learning something add long as you have the prerequisite knowledge and skills and are interested
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
Again, so insightful. What a great teacher and a great team producing this series!
8:14 little pi creature is actually wondering about its creation
Lol
Just like we do!