Why e is e (Calculating Euler’s Number)
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- Опубліковано 9 чер 2024
- In this video, we explore why e (Euler’s number), which appears throughout math and science, in everything from the hydrogen atom, harmonic oscillator, radioactive decay, waves, to Gaussian distributions, compound interest, and all kinds of other things, has the value of 2.71828…
It’s a quick and memorably derivation, and I hope you enjoy! :)
#math #physics
I would say e is even more fundamental than pi. In transcendental number theory, the Lindemann-Weierstrass theorem allows one to deduce "how transcendental" a number is. Recall that transcendental numbers are impossible to create as roots of algebraic equations, but rather can be found through differential equations. Within the theorem, one can basically construct transcendental numbers from the field extension of e^alpha_n (alpha_n is an algebraic number). In other words, every other transcendental number is just a linear combination of "e's" over algebraic numbers. This theorem was used to verify that pi is a transcendental number through Euler's identity. This means that e is the most fundamental transcendental number!
Very interesting observation!
Thanks for posting that comment . In the book Global Scaling by Hartmut Muller the author , on page 16, says that : ".....in this way , the natural exponential function e[exponential x] of the natural argument x=[n series] generates the set of preferred ratios of quantities which provide the lasting stability of real processes and structures regardless of their complexity. This is a very powerful conclusion . " . This could apply to the human or any other heartbeat . Hartmut explains how Euler's number e prevents resonance in planetary orbits due to e 's unique transcendentalism .
pi: continuously changing cyclic things. e: continuously changing exponential things.
e and π are fundamentally linked and one can be written in terms of the other.
@@Unmannedair Would that not apply to all transcendental numbers? It may not be a linear or exponential map, but I'd wager that any transcendental can be mapped (with a continuous function on all of the reals), such that f(T1) = T2, where T and T2 are two arbitrarily chosen transcendentals.
What's interesting is surely not that they are 'linked', but how they are linked (and all transcendentals are linked, by virtue of them being transcendentals, no?), right?
Also: e to the i: continuously changing cyclic things.
Pi:3
e: 3
@@NukeCloudstalker I think you are misunderstanding the point being made. The point being made is that exponential things and cyclic things are actually just the same thing.
I'm currently studying chemical engineering and I had a brief physical chemistry unit in one of my courses where we covered some quantum mechanics. Most things you discuss are only vaguely familiar, and though I'm not entirely sure what's going on I love watching these videos because I am always eager to learn more about these mysterious topics. The animations are beautifully done and I love the casual format.
Dude, I just started making UA-cam videos myself and I’ve been looking a lot for other content creators that make similar stuff to draw inspiration. I just wanted to say I’m really impressed, you make this stuff approachable and fun in a way that I hope to emulate. Great work!
Thanks Lukas, that’s a very kind comment! :) Looking forward to seeing your videos.
e stands for everywhere in maths.
Love the explanations and the beautiful way of showing info.
Your work is magnificent, the humor, the exposition, and the overall enthusiasm is infectious. You motivate me to succeed in mathematics more than any other creator.
Thanks! That means a lot :)
We covered several different ways to derive e while I was doing my math degree but I don't recall this one. It is super great for getting to the number itself! Also, I have gotten a perverse fascination with atoms over the last few years (worked in a lab in industry for about 6 years before my current position, so met lots of chemistry types and found a great appreciation for the subject) and I spent a chunk of last year trying to derive an equation for atoms just using ionization data, the Bohr model, and a spreadsheet.....not because I thought it would actually be useful, but because why not. Anyhow, it gave me serious appreciation for the quantum mech/spherical harmonic approach, in the way that only bumping your head naively against a problem over and over can do. I basically had to teach myself what your hydrogen videos showed using random Google searches, and I am now following this video series with great enthusiasm! Great work on everything! :D
I just wanted to say that I am absolutely in love with the visuals and your way of chewing complicated stuff into the small bites. I was one of the first lucky one two see the video about gravity, and a lot of thing happened! I got into the university, decidet to learn more about complex numbers and was amazed by the elegance and purity of this idea. And after that I read and watched some stuff about quantum mechanics which made me think I understand something.. But this! Oh boy, it’s a gold of the videos that made a click in my brain. Thank you a lot for the work you are doing, it is truly remarkable and inspiring!
Thanks Tim, that’s a very kind comment, and I’m glad you’ve enjoyed these videos! :) Sounds like you’re on a great path with your education.
Great way of approaching this problem! Never thought of setting the given as the derivative of e^x equal to itself then working from there. Most people give the example of compounding interest.
Really great video. I have been having a hard time finding connections between the different definitions of e, but this really provided insight...
One other approach is to use the definition of a derivative. Lim dx -> 0 (e^(x+dx)-e^x)/dx = e^x. From this we conclude that. lim dx--> 0. (e^dx -1)/dx =1. Rearranging we find that. e = lim dx-->0 (1+dx)^1/dx. Or e = lim n -->. Infinity. (1+1/n)^n. Which is the standard definition of e
That’s a great method too! :) I like the elegance of that formula.
UA-cam text is hard to understand and pain for eyes to so is there any alternate method through which I can understand what you written Here
@@Oms-xk2zb sure , i rewrote the equations on paint and i saved the image on drive , here is the drive link of the image : drive.google.com/file/d/1p8Jw9L5OTkJ3Dx-_q272ilK6MPwZx5Xs/view?usp=sharing
It would be also cool to use this polynomial representation, to show why e^(ix) is the same as cos(x)+isin(x).
Beautiful!
Great video!
Thanks! :)
this guy is so underrated. Too bad I can't understand any of these videos (besides this one)
you have to start somewhere. keep watching
@@jamescollier3 Thanks, I'm starting to learn calculus in my free time. I've learned the rules for differentiation and a few integration tricks. I'm using Khanacademy right now to get a more rigorous approach to learning it instead of just random questions in my head lol.
Thank you, sir for sharing this knowledge. I am from bio background, but still i understood the whole derivation. Finally, i now know the origins of e and its value. [DiowE]
Thanks for watching, and I’m glad you enjoyed the video! :)
There’s a page in Isaac Newton’s notebooks where he does this calculation out by hand with his quill pen to about 15 places - most of the fractions are repeating decimals, so it’s not hard - and somewhere else he apologizes for wasting time doing such a thing, but he didn’t have anything else on his mind at the time.
In the Haskell programming language you can define a function like 'integrate :: Double -> [Double] -> [Double]' for power series, then define 'let expSeries = integrate 1 expSeries' and actually obtain the power series of 'exp' by means of lazy evaluation.
I'm a Grade 11 student from South Asia. It hasn't even been a month since we learned that d/dx(e^x)=e^x and trust me the way our teacher led us through that differentiation blew our minds at the end. Our teacher, a person driven by religion, mathematics and philosophy calls e, along with pi and infinity, the symbols of God. He also loves Euler like you wouldn't believe lol. Whatever the case, it was truly a highlight of my school life. It's like discovering negative numbers for the first time.
Your videos tackle pretty advanced topics mathematically so I'm glad to I see I'm reaching levels where I can understand your approach to a problem aka you didn't use anything to describe something like e which appeared totally alien to me. Thank you and keep up the good work!
I remenber to read R.Penrose about exponential and derivatives on the road of reality. One of the main conclusions said like it was the only real function from the survival point of view, the others are simply destroyed after a while ;-)
Love the videos. May I ask how you make your animations, like what software/programming language you use? Thanks!
Thanks! :) I use Python. Matplotlib for 2D things, and plotly for 3D.
@@RichBehiel Interesting, I was looking at using Manim for a video I'm attempting to make (just a overarching introduction to quantum field theory) but I've been having a rough time figuring out how to animate, let alone in 3D. Is there some sort of reference material or course on the methods you use?
That would be really cool, and I’m looking forward to seeing that video!
3D is inherently challenging. I try to avoid it, unless it really adds something. Many physical principles can be communicated effectively in 2+1 dimensions. And I’ve found that 1+1 dimensions, spacetime diagrams in particular, are actually confusingly low-dimensional. So it’s useful to do 2D plots that evolve over time, which you can make by calling plt.contourf in a “while t < tMax” loop and grabbing the frame in each loop iteration.
I’ve been meaning to post some example codes, just haven’t gotten around to it yet. Part of the problem is that my animation codes are all spaghetti code, since I just write them quick and dirty to get the animation and move on. It would be more helpful to slow down and write them neatly, but it seems I’m always in a rush these days.
I just love tungsten cubes.
It's weird to me that adding a load of rational numbers together gives a number that is trancendental. Does that work for any infinite series of rationals or just special ones? What needs to happen to make them special? o.o
e has more depth to it. It produces sin/cos as its Re/Im. e^ix does rotation. e^(a*d/dx) does shifting. There's way too many things that have e inside of them to stick to that single property.
Sure, but those properties can all be derived from the fact that d/dx[e^x] = e^x.
As we saw in the video, we can derive the full MacLaurin series for e^x from the fact that it’s equal to its own derivative. That series defines everything there is about e^x. For example, look at the maclaurin series for sine and cosine and you’ll see that each has half the terms of e^ix, with the sin(x) terms needing a multiple of i. You can derive Euler’s identify from that. It has to do with the cyclic nature of the derivatives of sin and cos.
For rotations, this is due to the fact that d/dx[e^ix] = i*e^ix which comes from the chain rule. If the rate of change is i times the function, and i rotates a complex number by 90 degrees, then you get circular motion - when the value is 1, the direction of motion is i, when the value is i, the direction of motion is -1, etc. That’s why e^ix rotates.
@@RichBehiel I just watched the video a second time, and it just shows e to be a solution to that equation. Sorry, I have no idea what I meant when I wrote my previous comment.
All good! :)
To me: 'e' has never really meant 'euler's number', and instead was a stand-in for 'eigenfunction' (of the derivative operator).
Whether that corresponds to geometric transformations such as hyberbolic rotations, regular rotations (about the origin), or even translations (in projective geometry, these are 'rotations' about the vanishing point/point-at-infinity).
Nonsense!
Nifty
Amazing video
Thanks, glad you enjoyed it! :)
yes
e can be 1 too?
thanks. but i still keep a feeling that truth is out there. we're overlooking something fundamental in math.
Two point seven Jackson Jackson...
45 90 45!
But why did we put x = 1?
e = e^1
@@RichBehiel oh ok thanks.
But.. does "e" really have a prominent role in the smallest atom in the universe? Its kind of just there because you put it there because it was convenient, no? I mean, your "e^{i En t / hbar}" term is really just "C^{i t}" for C="e^{En hbar}". Your mathematical training just taught you that maybe to keep the calculus simple you should stick with "e" instead of "C".
Even more, why have "C^{i t}", for real C and real t? It could have just as well been written as "A*i^t" for different constant A and t. After all i^x is just e^{x * i * pi/2}. Or why not just "A^t" for complex A and real t?
Good point! The math could have been reformulated without using e per se. But the simplest way of writing it involves using e.
If you look at the first C in your comment, e^{En hbar}, that C would be just as natural as e, as long as En and hbar were arbitrary constants which we may as well put in the exponential. But the energy eigenvalue and hbar both have real significance in other contexts, not just as things in an exponential. So then when constructing C, one would still have to wonder what’s the deal with this number e.
@@RichBehiel so rewrite "e^{i En t / hbar}" as "i^{2 En t / (pi hbar)" and switch hbar to h to get "i^{4 En t / h}" which has the constants you care about, no e, and hey, even the pi went away...
Ok, that’s messing with my head. I’ll need some time to digest that. You’ve got me rethinking a lot of things about e 😅
@@RichBehiel hah, your schrodinger equation videos messed with my head first. (thank you).
@Tim: Great ideas, got me thinking quite a bit. :) One problem I see with your approach is that i is not only equal to e^(i pi/2), but also so e^(i 5pi/2), e^(-i 3pi/2) and so on. So writing i^{4 En t / h}, you would always additionally have to specify that one has to use i = e^(i pi/2) and not one of the infinitely many other possibilities.
Why a not equal to zero
Because when x = 0, all the terms except a cancel out, so we have e^0 = a, and a nonzero number raised to the zeroth power is 1.
But ... out of all places, in the physics of waves, the value of e as 2.71828... is NOT RELEVANT at all! Since you have an i in the exponent, it comes down to sin and cos.
And if you do not have i in the exponent, it all comes down to sinh and cosh, as cosh is the even part of e^x and sinh is the odd part and cosh(x) +sinh(x) = e^x
@@ingiford175 Yes, so? There is an i.
oh shit i may actually be first for once
Not only are you first, but this is actually my first “first” comment! The ultimate first.
@@RichBehiel oh wow i'm honored lol! your videos have been so helpful for self-teaching and improving my intuition, really appreciate all your work!!
Respectos to the First of the Firsts.
Listen, I get that this is cool and all but I do find it kind of funny that you’d think a lot of people who watched your hydrogen atom videos wouldn’t know how to do this. Good standalone video still, no hate at all, just looks a little funny
Yeah, one of the challenges of speaking to a diverse audience is knowing what people know and don’t know. I’ll be doing harder stuff soon, next few vids will be on relativistic QM, building up to applying the Dirac equation to hydrogen. In the meantime, just wanted to a quick standalone thing on e since I noticed it was in all of the equations, and it’s one of my favorite calculations because of the way it feels like we get something for nothing. There are also a lot of people with advanced degrees in STEM who just took e for granted and never saw how it was derived.
Anyway, over time I’ll try to become more calibrated to what people want to see. My main goal with this channel is to explore the nature of the electron, using all the tools of modern science, while bringing as many people along for the ride as possible. I might have the occasional short videos that are tangential to that goal, if I think it’s a neat calculation or something. But I respect everyone’s time and I hope I’m not putting totally irrelevant stuff out there. So thanks for your feedback! :)
@@RichBehiel I don’t have any formal higher education background but I have taught myself a lot of math and loved physics in school. I’m about to go to college. I was able to follow your first hydrogen atom video very well, it’s a great video. I did my best with the second video, and it also looks like a great video, but you lost me, it got very very crazy for me. I wanted to ask, what’s the difference between a Hamiltonian and a Lagrangian? I know Lagrangian mechanics and the Hamiltonian operator as you explained it looks almost identical in principle
There are a lot of similarities between the Lagrangian and the Hamiltonian. It’s too much to write up in a comment, but this website does a great job of explaining the similarities and differences:
profoundphysics.com/lagrangian-vs-hamiltonian-mechanics/#:~:text=The%20main%20difference%20between%20these,total%20energy%20of%20a%20system.
@@RichBehiel That’s a great resource, thank you
that's why I never appreciated math. a lot of it feels like a cheap trick
Same as 27296296 and so on