And you not only enlightened me why e^(ix)=cosx + isinx but also why d/dx of cosx and d/dx of sinx are -sinx and cosx INTUITIVELY, so far i only had them memorized. I never knew this great visualisation before! This is gold for a high schooler like me. Please keep doing your amazing work! I like when math is this intuitive. Subbed!
In our math class in uni, the teacher said we had a function A that has all the properties of the sin function, but he didn't tell us. We were talking about the sin function in it's polynomial form, and we only realised it after 3 hours of it being taught
3 роки тому+24
Very well done; content- and animation-wise. My favourite video in the SoME-contest so far.
nothing new to me but still, but it completely deserves a thumbs up, these kinds of animations and explanations are always appreciated, hope you continue with these kinds of videos
Great video! One of the things I don't think gets enough attention when discussing Euler's formula is this deep connection between trigonometric functions and exponential functions. It blew my mind when I realized that exponential functions are periodic on the imaginary axis and while sin and cos grow to infinity.
Excellent presentation. Now, discuss the derivation of Schoedinger’s equation. Your detail could clarify that. Also, you should do a segment on the natural log and complex numbers. Thanks!
Excellent approach; keeping it a higher and conceptual level is the key to understanding the connections between the various mathematical concepts. Getting too lost in the details or just learning only how to calculate in a rote fashion kills understanding in favor of rigor. Both are needed. The traditional education system teaches the number crunching and kills interest in a truly beautiful language (math) by forgetting to connect all of the concepts (1) functions (2)the properties of the all important exponential function (3) derivatives (4) the application to unit vectors and the imaginary dimension that enables rotation (5) the trigonometric connection and (6) the polynomial expression of the same function using a convergent but infinite series (constraining infinity and making it work for us is truly one of the master strokes of mathematics). Then comes applications; electrical engineering and quantum mechanics which are all about waves with an imaginary component and how they sum. True understanding happens by integrating all three levels (1) the mechanics of number crunching which allows us to speak the language (2) the high level conceptual connections between various mathematical topics and approaches which validates the consistency of the language and (3) the application of mathematics as a tool for modeling systems, solving problems, optimizing and evolving systems and Finally there is the mystery that surrounds the fit between the model and the system and the misfit between GR and QM and something deeply hidden. Beauty and mystery, it doesn't get any better. Thanks!
I would argue that the start of the video is back to front in that sines have less to do with triangles and more to do with circles. Strangely, we never seem to be taught it that way round. Sines are "circular functions" and the word itself comes from sinus meaning curve. So, I think we should really start from the unit circle with the "curve height" above the x-axis being called "sine" and it is the sine of the arc length, or angle if you prefer. The related coordinate of that y-axis sine value is the co-sine. That is, it accompanies the sine. From that the trig stuff follows. If you add a tangent to the point then see where that line cuts the y-axis and x-axis we see where the secant (based on secare: to cut) comes from. All the well-known trig relationships then follow using similar triangles.
Very well done video, and excellent explanation. There's another proof for the coincidence of f(x) = e^(ix) and g(x) = cos(x) + i sin(x) for every real x. These two functions both solve the Cauchy problem y' = iy with y(0) = 1. As the solution of this problem is unique, f and g must be equal everywhere.
@@jHan this is your first video!? Beautiful. I'm starting to see this more and more with people who upload these pieces of amazing content for the first time as 3B1B challenge submissions.
At time 06:25, he tells us that: (the derivative ie^(ix) has no real constant changing the function ==> this means that the magnitude of the derivative stays constant at 1); this statement that I wrote between brackets it is not as intuitive as I wish. Further explanation please!
Looking forward to more videos! Thanks for such a lucid explanation and clear animations. Would be great if you could also share your backstory as in what goes behind the scenes to plan and create such a video. That's will make more people curious to explore manim and other tools to create more such open source videos in their domain of interest. Thanks again!
B E A utiful! This reminds me of an 8-part video from Mr. Woo's channel explaining the same thing but he ends it to Euler's identity. Perhaps the next video from you would be explaining the most beautiful equation in the world in such a compact way. +1 from me :D
very good. although, some of the manim latex transitions could be redone to minimize the amount of text that changes. eg @12:24 only the 'cos x' part needs to change, but the whole equation goes through the mangling transition which hides the fact that it's only the real part on the rhs that's changing.
At 9:33 does cosθ = dy/dθ because the triangle with θ at the origin is similar to the triangle with θ on the unit circle? I guess it makes sense if the magnitude of the rate of change is constant like e^ix.
A think the first explanation needs at least to understand curves in space and their derivatives (vector fields), but the second only needs basic differential calculus, so the second is a better approach i think for explain it. I like the fact, using linear algebra, that the exponential function is the eigenvector of the differential operator for or eigenvalue, and then a second-degree differential operator has as eigenvector the trig. functions with eigenvalue = -1, so the trig. function must be a linear combination of exps; then the fact that the linear operator is degree two, so the eigenvalue of that operator corresponde to the square of the eigenvalue of the first-degree operator, tales that the eigenvalue of the linear eq. D^2(y) = -y it's just "i", and then your initial conditions dictate the linear combination of exponential functions. That result requieres to know linear algebra and calculus, but for me it's the less "magical" because you are not matching what it seems pears and apples, or just pluging "i" in exp because someone was curious.
Very clear speaking and graphics. Only derivatives of e^(ix) are discrete so at that point your proof is wrong. You should show that derivatives can be uniquely extended to fractional derivatives. Then that the extension is smooth. Then that fractional derivatives of e^(ix) don't change absolute value of the function. Then you finish the proof showing that e^(ix) = cos x +i sin x
The statement that derivatives can be uniquely extended to fractional derivatives is incorrect. In fact, it can be rigorously proven that over any vector space, a linear operator which is surjective and has nontrivial kernel cannot be fractionalized, even if you disregard the ambiguities that inherently emerge with fractional exponentiation in the scalar field.
Great video! I just finished watching the first part, of the geometrical approach, and got most of the proof intuitively, but there is one thing that still doesn't work out in my mind. Can someone please elaborate on why it is the case that the 90-degree angel of the derivative creates a circular pattern in the complex plane?
Yeah, that moment at 6:00 and further is somewhat without any proofs. Blah-blah and voila - exp(ix) just is a circle. Not clear why at all. Not clear why derivative is there etc. Need to watch other channels.
Truly amazing discovery - what is even more amazing is the human brain who invented the magical J = SQRT(-1) and found its correlation to trigonometry. It is all hidden inside our brains, and manufactured by the Universe !
A nice and well thought out video, with nice explanations for why e^(ix)=cos(x)+i*sin(x). There is one thing that bugs me though, which would be that you without explanation use the power rule in order to find the derivative of e^(ix). This is nice, and totally ok to do, but it is not obvious that the derivative of complex numbers is well defined, exist, and have the chain rule. That is since the complex numbers can represent 2d-space, while the "normal" derivative is usually defined from a small change in 1d-space. I would not expect a full explanation of this here, but a comment would have been nice. Still, if you expect e^(ix) to show circular motion in advance, one could say that the motion would still be one dimensional, and therefore be able to give meaning to the derivative, but that would not be as rigourus as I think you wanted this video to be. I'm just rambling on at this point, but this is really just a minor thing to bring up, and I think that the rest of the video explained everything in a consistant and nice manner.
Yeah, I should've been a bit more rigorous. Complex differentiation does follow the chain, product, and quotient rules, and e^z is complex-differentiable in the entire complex plane, so we can simply use the chain rule. Perhaps proofs and deeper dive of these concepts could be a video for the future!
You are misunderstanding the concept. At no point are you actually required to take the derivative over a complex domain. Only the codomain is complex, the domain is still the set of real numbers.
absolutely amazing I do have one tiny thing at 12:27 and 12:30 the animation is kind of messy and I had to replay that part to see what happened no complains tho... also cool video
![The most beautiful equation in math, explained visually [Euler’s Formula]](http://i.ytimg.com/vi/f8CXG7dS-D0/mqdefault.jpg)
Math and Physics are art and they are needed to perform by an artist. That was really beautiful.
Superlative. Best teachers are on UA-cam!
And you not only enlightened me why e^(ix)=cosx + isinx but also why d/dx of cosx and d/dx of sinx are -sinx and cosx INTUITIVELY, so far i only had them memorized. I never knew this great visualisation before! This is gold for a high schooler like me.
Please keep doing your amazing work! I like when math is this intuitive. Subbed!
In our math class in uni, the teacher said we had a function A that has all the properties of the sin function, but he didn't tell us. We were talking about the sin function in it's polynomial form, and we only realised it after 3 hours of it being taught
Very well done; content- and animation-wise. My favourite video in the SoME-contest so far.
Damn, how did you enlighten me with all this in only 13 minutes!
Very underrated channel, youre so good at explaining, and you even give examples.
These educational videos made with Manim are spawning everywhere lately. And I couldn't be more grateful!
Videos like these are now my best way to learn mathematics. Thanks so much. More elbow-grease to your efforts. 👏
First time for me that someone described it so simply and obviously.
You put all of my thoughts about euler's Formula into a beautiful video great job
You've made it possible for me and I'm sure many many others to now visualise these relationships and connect the dots. Thank you so much.
You are doing a great job unpacking so much in this video.
I need to watch it again to absorb more if it.
Best explanation of all vids on the internet and straight forward
nothing new to me but still, but it completely deserves a thumbs up, these kinds of animations and explanations are always appreciated, hope you continue with these kinds of videos
Great explanation. This I think is the essential insight of the 2 years of study I’ve just completed reduced to 15mins. Thank you.
Superb video, a work of art. Super easy to follow - you guide us well through these topics. Thank you.
Great job! You made everything super clear and added some insight along the way - the best combination :) BTW - clever channel name!
This video was concise and to the point. Clear information bundled up tight.
Amazing video, Physics major and we use this all the time, now I have a much more intuitive understanding of the Euler's formula
here before this channel blows up
same
Me too.
I'm subscriber #89
#116 here.
#1081
Great video! One of the things I don't think gets enough attention when discussing Euler's formula is this deep connection between trigonometric functions and exponential functions. It blew my mind when I realized that exponential functions are periodic on the imaginary axis and while sin and cos grow to infinity.
This is some high quality material right here. I'm looking forward for your video on Fourier transform.
Wow! Thank you so much for this extremely helpful video!!
thank you!
this is the first time i see a good intuitive motivation for Euler's formula _beside_ using the Taylor expansion and that always bugged me.
Outstanding!!!!!! clearly comprehensive
Magnificent.... expect something more like this
please make more content. Very high-quality sciences. Thanks a lot
incredibly good explanation. Every high school and university should show this video
Beautiful man, just beautiful, I like how you started with basics
The simple way you explain this, combined with the beautiful narration is just...
Even 10th grade me could understand this!!
Beautifully explained, connecting all the dots reaching the eureka moment. Thank you so much. Have subscribed to your channel immediately 😁
Best video on this topic I've seen.
You deserve more subscribers, amazing explanation loved it.
Thank you for the information.
I had the same question when i saw it recently for the first time at school. Thanks for the video :)
Absolutely stellar. I can't thank you enough for this video. Looking forward to watching new stuff.
i spent a day trying to prove this with the taylor series, still my most enjoyable day in terms of math
Excellent presentation. Now, discuss the derivation of Schoedinger’s equation. Your detail could clarify that. Also, you should do a segment on the natural log and complex numbers. Thanks!
With quality as high as this I thought you'd have over a million subscribers! Really, well done bro! Remember me when you make it big haha XD
Thoughtful, beautiful and insightful...keep going because this is road not taken in the math world...and of course thanks...
Amazingly done. Explanation and visualization were very well presented. Great job!
Awesome video! I've always been intrigued by the conection between trig functions and complex numbers. I really enjoyed your explanations.
Well done👌👌👏.....12:08 side point yo be noted!!
Excellent approach; keeping it a higher and conceptual level is the key to understanding the connections between the various mathematical concepts. Getting too lost in the details or just learning only how to calculate in a rote fashion kills understanding in favor of rigor. Both are needed.
The traditional education system teaches the number crunching and kills interest in a truly beautiful language (math) by forgetting to connect all of the concepts (1) functions (2)the properties of the all important exponential function (3) derivatives (4) the application to unit vectors and the imaginary dimension that enables rotation (5) the trigonometric connection and (6) the polynomial expression of the same function using a convergent but infinite series (constraining infinity and making it work for us is truly one of the master strokes of mathematics).
Then comes applications; electrical engineering and quantum mechanics which are all about waves with an imaginary component and how they sum.
True understanding happens by integrating all three levels (1) the mechanics of number crunching which allows us to speak the language (2) the high level conceptual connections between various mathematical topics and approaches which validates the consistency of the language and (3) the application of mathematics as a tool for modeling systems, solving problems, optimizing and evolving systems and
Finally there is the mystery that surrounds the fit between the model and the system and the misfit between GR and QM and something deeply hidden. Beauty and mystery, it doesn't get any better. Thanks!
Amazing
For years 😁 revolving around youtube to find simple explanation
Finally you are 🌺🌺
That was magical...simple and elegant!
Great explanation, very clear train of thought. I wish all my teachers would be like you
I would argue that the start of the video is back to front in that sines have less to do with triangles and more to do with circles. Strangely, we never seem to be taught it that way round. Sines are "circular functions" and the word itself comes from sinus meaning curve.
So, I think we should really start from the unit circle with the "curve height" above the x-axis being called "sine" and it is the sine of the arc length, or angle if you prefer. The related coordinate of that y-axis sine value is the co-sine. That is, it accompanies the sine. From that the trig stuff follows.
If you add a tangent to the point then see where that line cuts the y-axis and x-axis we see where the secant (based on secare: to cut) comes from. All the well-known trig relationships then follow using similar triangles.
Holy smokes!!! This is amazing!!! I don't really follow the first one but for the Taylor series one, that's unreal!!!
It's the first time I can figure out what the Euler equation means! And that means a lot for me!!!
Very well done video, and excellent explanation. There's another proof for the coincidence of f(x) = e^(ix) and g(x) = cos(x) + i sin(x) for every real x. These two functions both solve the Cauchy problem y' = iy with y(0) = 1. As the solution of this problem is unique, f and g must be equal everywhere.
Absolutely amazing for your first video!
Question: How long did it take for you to learn Manim?
It took me maybe a month to get the basics down, but it may take more or less depending on what you want to animate.
@@jHan this is your first video!? Beautiful. I'm starting to see this more and more with people who upload these pieces of amazing content for the first time as 3B1B challenge submissions.
At time 06:25, he tells us that: (the derivative ie^(ix) has no real constant changing the function ==> this means that the magnitude of the derivative stays constant at 1); this statement that I wrote between brackets it is not as intuitive as I wish. Further explanation please!
This video made Euler's identity the clearest to me, how do you not have more than 50 subscribers?
Looking forward to more videos! Thanks for such a lucid explanation and clear animations. Would be great if you could also share your backstory as in what goes behind the scenes to plan and create such a video. That's will make more people curious to explore manim and other tools to create more such open source videos in their domain of interest. Thanks again!
Very good! You really answered my question about that relationship and the usefulness of complex functions.
Great job -- subscribed, and looking forward to more!
I liked it a LOT!
Very nice channel name :)
Great video, best I've seen on this topic
Hello, is there any email/discord to reach out to you?
Amazing animation and explanation! You have a new subscriber :)
B E A utiful! This reminds me of an 8-part video from Mr. Woo's channel explaining the same thing but he ends it to Euler's identity. Perhaps the next video from you would be explaining the most beautiful equation in the world in such a compact way.
+1 from me :D
Amazing. More video like that please
very good. although, some of the manim latex transitions could be redone to minimize the amount of text that changes. eg @12:24 only the 'cos x' part needs to change, but the whole equation goes through the mangling transition which hides the fact that it's only the real part on the rhs that's changing.
Great Job 👍👌. I needed this explanation.
Best explanation I've heard yet
Just a shortcut ua-cam.com/video/56BpfqpR7Ko/v-deo.html
Wish "imaginary numbers" were just called "lateral numbers"
Geometric numbers
Circular numbers
Spatial numbers
Whole numbers
Orbital numbers
Spherical numbers
Angular Numbers
Rotational numbers
???
I started gasping and lighting screaming when i saw the end of the taylor series proof. Im bewildered
That proof is wrong. Taylor series are valid for real integers, not marvel universe numbers like i*x.
@pelasgeuspelasgeus4634 *bigger gasp*
@@Founder2721 meaning?
I love this formula..its so beautiful !!
Dont know nothing about maths but i had this in recomended, guess your getting blessed by the algorithm. Looks interesting tho
Thanks for your great work 👍
11:53 -12:07 it's all coming together! 🤯
Trigonometry, calculus, complex numbers, EVERYTHING is in this video😭
beautifully explained
At 9:33 does cosθ = dy/dθ because the triangle with θ at the origin is similar to the triangle with θ on the unit circle? I guess it makes sense if the magnitude of the rate of change is constant like e^ix.
great explanation!
A think the first explanation needs at least to understand curves in space and their derivatives (vector fields), but the second only needs basic differential calculus, so the second is a better approach i think for explain it.
I like the fact, using linear algebra, that the exponential function is the eigenvector of the differential operator for or eigenvalue, and then a second-degree differential operator has as eigenvector the trig. functions with eigenvalue = -1, so the trig. function must be a linear combination of exps; then the fact that the linear operator is degree two, so the eigenvalue of that operator corresponde to the square of the eigenvalue of the first-degree operator, tales that the eigenvalue of the linear eq. D^2(y) = -y it's just "i", and then your initial conditions dictate the linear combination of exponential functions. That result requieres to know linear algebra and calculus, but for me it's the less "magical" because you are not matching what it seems pears and apples, or just pluging "i" in exp because someone was curious.
Very clear speaking and graphics. Only derivatives of e^(ix) are discrete so at that point your proof is wrong. You should show that derivatives can be uniquely extended to fractional derivatives. Then that the extension is smooth. Then that fractional derivatives of e^(ix) don't change absolute value of the function. Then you finish the proof showing that e^(ix) = cos x +i sin x
The statement that derivatives can be uniquely extended to fractional derivatives is incorrect. In fact, it can be rigorously proven that over any vector space, a linear operator which is surjective and has nontrivial kernel cannot be fractionalized, even if you disregard the ambiguities that inherently emerge with fractional exponentiation in the scalar field.
Good quality content man! A bit fast but people can pause if they need a moment to think.
I've been playing with the Lorentz Factor. e^(i*arctan(i*v/c))=(-v/c+1)/sqrt(-v^2/c^2+1) which is γ*(1-v/c).
I love using (x+y) instead of just x in my Taylor series. You gotta double the number next to the factorial to keep it good
ngl thought this a was a 3blue 1 brown video then i saw the channel name keep up the good work
Hey great video! I'm studying Electrical engineering and this was very interesting for my signals course.
I like how Heaviside's Pi and Lambda function are named by symbols that look like the shape of the signal.
Absolutely amazing 🤗
Excellent presentation. vow !!
Great video! I just finished watching the first part, of the geometrical approach, and got most of the proof intuitively, but there is one thing that still doesn't work out in my mind. Can someone please elaborate on why it is the case that the 90-degree angel of the derivative creates a circular pattern in the complex plane?
Yeah, that moment at 6:00 and further is somewhat without any proofs. Blah-blah and voila - exp(ix) just is a circle. Not clear why at all. Not clear why derivative is there etc. Need to watch other channels.
really well done!!!!
Thnks for uploading such a great video ❤💞😊
Hey, love this lesson!
Now i can create more complex fractals than ever, thanks!!!
amazing man! subscribed
Great video!
Truly amazing discovery - what is even more amazing is the human brain who invented the magical J = SQRT(-1) and found its correlation to trigonometry. It is all hidden inside our brains, and manufactured by the Universe !
I looked at the thumbnail and thought it was a 3b1b video
Edit: read the description, now I know why
Also edit: this video was very beautifully made
Wow excellent explanation. Could you please 🙏 make videos on Vector Geometry
U know what. You should make more of it.
Very good video, thank you!
A nice and well thought out video, with nice explanations for why e^(ix)=cos(x)+i*sin(x).
There is one thing that bugs me though, which would be that you without explanation use the power rule in order to find the derivative of e^(ix). This is nice, and totally ok to do, but it is not obvious that the derivative of complex numbers is well defined, exist, and have the chain rule. That is since the complex numbers can represent 2d-space, while the "normal" derivative is usually defined from a small change in 1d-space. I would not expect a full explanation of this here, but a comment would have been nice.
Still, if you expect e^(ix) to show circular motion in advance, one could say that the motion would still be one dimensional, and therefore be able to give meaning to the derivative, but that would not be as rigourus as I think you wanted this video to be.
I'm just rambling on at this point, but this is really just a minor thing to bring up, and I think that the rest of the video explained everything in a consistant and nice manner.
Yeah, I should've been a bit more rigorous. Complex differentiation does follow the chain, product, and quotient rules, and e^z is complex-differentiable in the entire complex plane, so we can simply use the chain rule. Perhaps proofs and deeper dive of these concepts could be a video for the future!
You are misunderstanding the concept. At no point are you actually required to take the derivative over a complex domain. Only the codomain is complex, the domain is still the set of real numbers.
Explication merveilleusement claire
absolutely amazing I do have one tiny thing
at 12:27 and 12:30 the animation is kind of messy and I had to replay that part to see what happened
no complains tho... also cool video
Thank you. The dangle has an angle. 👍