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This is absolutely incredible. I just finished Calculus II, and when we talked about Taylor Series they were explained very poorly. This not only made me completely understand how Taylor series are formed, but also what e^ipi really means (something I’ve been wondering for a very long time). People like you make the world of learning a better place. Thank you
It was a great explanation, but be honest if this was the first time you saw any of these concepts, you wouldn't understand any of this based on this video alone. It takes time and exposure with these subjects to really grasp what is taught. My calc 2 teacher explained this. My differential equations teacher explained this. And while I was in the class, it didn't make a lot of sense. But trust me they explained it just as well if not far more in depth in class. Again, it's a great video but I wish people wouldn't blame teachers for students not immediately grasping confusing concepts. Now maybe you didn't have the best teacher but in my experience it's not the teacher's fault. (It's not the students either)
This might be the most concise (and perhaps the prettiest) explanation of euler's identity that I've ever seen. I love how you show each derivation (pun intended) step by step and using first principles, really shows that you're not just listing off an arbitrary set of rules but actually understand what each of them mean.
Actually, if you take the definition e^x=lim_{n\to\infty}(1+x/n)^n, a much more satisfactory and intuitively appealing explanation of e^{i\pi}=-1 can be given. The Taylor series explanation has zero intuitive appeal; it works, but it does not show why the identity is really true. Just google for e^i pi youtube , and probably the first thing that comes up is a non-Taylor series explanation. Mathologer gives such an explanation, but there may be also others.
You did an amazing job explaining It! I tried many times to understand the Euler's identity, but I couldn't because other people didn't do what you did. You explained everything from the basics, and It really helped a lot.
Dude, this is the beauty of math that an engineer like me won’t truly be able to experience. We learn a lot of stuff without proper explanation so it’s nice to see a video like this
Oh my gosh. That equation was a wonder to me when I was a teenager and very good at math. Now I am a senior citizen and have understood it for the first time. Thank you for your excellent explanation.
it's like magic... I first saw this equation in the video "Math vs Animation". Since I am just in 11th grade, I know nothing about complex numbers or eulers number e. So I never understood what those complex equations mean. But now when you wrote e^ipi = cosx + isinx I was shocked, because I remember this equation from that video. This is so well done, thank you so much for the explanation!
@@realsstudios8153bruh I literally did I'm now one grade higher and studying vector calc and a tiny bit of number theory and I'm looking to move forward to abstract algebra (this is my mom's acc I use it to watch math and geography vids)
this has been my favorite equation since i read it on a math book, but now is the first time i actually understood the process of it. most useful 14 minutes of math in my life
This is insane, there's about 5 different concepts in this video that I understood for the first time despite years of studying Maths and having to take these things for granted. Amazing video, thank you.
We can go further by writing the identity as e^ipi = -1 and from that write ipi = ln(-1) then dividing both sides by I we get pi = ln(-1) /i or pi = ln(-1) / square root of-1 which is a new definition of pi. I am sorry my keyboard doesn’t have much in the way of mathematical notation.
Well, yes and no. Tl;dr while π is one solution to the equation x = ln(-1)/i, that does not work as a definition of π. L;r When you’ve got imaginary numbers in the exponent you’ve got to be very careful when manipulating values because you may end up with an equation that has multiple solutions. In fact, logarithms kind of break on the complex plane for this very reason. While Euler’s is true, “iπ” is not the only the only value that gives -1 when you take its power of e. Looking back at the explanation you may notice that e^(i3π) also equals -1, as e^(i3π) = cos(3π)+i*sin(3π) = -1+0i. In fact for any integer n, (2n+1)π is a solution to the equation x=ln(-1). The fact that logarithms are multivalued in the complex plane means that doing algebra on them starts to fall apart, so you can’t totally trust the exponent/logarithm rules that you learned for the real numbers. For instance, you’ll notice that by manipulating your equation a little more, using classic real valued log rules, you may arrive at the conclusion that π=0: Start with π = ln(-1)/i. Now in math we don’t like to have square roots in the denominator, including i, so let’s turn that into -i*ln(-1) by multiplying the fraction by i/i. Next, we can move part of that scalar inside the logarithm like so: π = -i*ln(-1) = i*ln(-1^-1) = i*ln(-1) There is only one number x on the entire complex plane such that x = -x, 0. Thus, given that π = -i*ln(-1) = i*ln(-1) = -π, π must be equal to 0. The reason we can arrive at this false conclusion is because while π IS a solution to x = ln(π)/i, -π is ALSO a solution. So is -3π, -5π, 3π, 856203757π, and every other odd product of pi. You can see this clearly in the fact that -1 is equal to itself raised to any odd power, thus we can pull any odd number we want out of the expression ln(-1).
Thank you sir. This is the most elegant and understandable explanation I've seen, which ties these important concepts together. It is truly amazing that Euler and others understood these things some 3 centuries ago, yet we still struggle with them today.
in my opinion, the general formula e^ix = cos x + i sin x is more beautiful since it directly shows that e^ix makes a circle on the complex plane, and the one with pi just says that halfway around the circle it's -1. but with 2pi (or tau) it's 1, and with pi/2 (or tau/4) it's i, and with 3pi/2 (or 3tau/4) it's -1, which are beautiful in themselves as well, so i think all points should be included.
in my opinion, 1 + 1 = 2 is still the most beautiful equation ever because all of math is based off of this seemingly simple equation that cannot be proved.
@@theunstoppable0357 Sure, but the book containing it is a little slow to get started. It only reaches this part of the plot when it's about 2/3 of the way in.
This was awesome! I’m an electrical engineering major and never really understood how imaginary numbers fit into calculus until this video. The explanation and derivations were very concise and helpful. Thanks!
I salute you 🙏🙏🙏 god of mathematics . You explained me derivatives , pi , i , e e^x = cosx + i sinx and Euler's identity itself in a single video better than any teacher I have seen and far far better than my school teachers . 🙏🙏🙏🙏🙏 Respect
This video is incredible. It is so thought-out and well put-together anyone would be able to understand this. I truly wish one day I can be like you, helping other people learn. Thanks a bunch.
e^iπ = -1 is a much more beautiful equation in my opinion. Not only is there no problem with having negative numbers in equations, but it gets the meaning of e^iπ across much better: -1 is the number π radians around the unit circle. Rearranging it not only obscures the entire point of e^iθ, but it also makes light of the significance of negative numbers as a whole.
Euler's real achievement is a function identity e^(ix)=cos x + i*sin x, not the above-mentioned numerical curiosity (e^(i*pi)+1=0) that stems from the identity.
I have watched countless hours in countless genres of videos on RU-vid for over 10 years, but this is easily one of the best videos I have ever watched. Brilliant work!
I am literally astonished by how much clear the explanation is, of each and every concept of mathematics expalined in the video. Please never stop making videos.
so this guy actually explained radians, calculus and other stuff in 13 minute video. he explained them well enough to actually understand the subject matter. you sir are a legend. so damn rare to find content like this. hats off man
This is just so lovely. Seeing so many concepts in math explained so quickly, feeling like I could understand all of this without any previous knowledge, because of how well this was explained. If only all of math would be explained to me like this. Really cool to see all these different concepts play together aswell, they don't look like they should make any sense, but math can be just this beautiful
I'm gonna be perfectly honest, I dodn't understand maybe 20% of the things discussed in this video, but I understood it just well enough that it was satisfying to see it come together in the end and know I wasn't completely lost.
The incredible beauty of the equation explained! As a physicist, I only ever considered it through the real-imaginary phase relation, but never considered the derivation! Thank you!
I was not expecting much, but honestly, this is the best explaination I've ever heard. The only thing I would add is to make the deffinition of sin more clear.
Okay, I got to ex^iπ = -1 on my own, and I know what pi, exponents and roots are, but man-oh-man was I totally not keeping up with the formulae once you got into complex numbers and derivatives. Like, I was watching you explain, watching you simplify, and was completely trusting your math because I realized quickly that I was truly out of my depth. That's absolutely NOT a knock on you! It's all on me. This is calculus waaay beyond my skill level. But I still find it fascinating and this video makes me want to understand these principles better. I WANT to be able to practically apply them to manipulate equations this artfully. I envy anyone who comes by these skills easier than I do.
I'm gonna use this video to introduce calculus to any person beginning their journey with a engineering degree, I think it'll be perfect to show many things to be teached, specially calculus 1 and 4 (idk how it is for you guys, but 4 is ODEs and PDEs)
I'm an engineering dropout now pursuing a degree in the humanities, personally I wouldn't change it for the world but the beauty of mathematics is something I will always appreciate. I am glad I took all those calculus classes just for that fact alone, and this is a wonderful video.
"Feedback shared with creator" means the uploader sees the dislike count, not that they get a notification or something. It's a pleasantry meant to hide the fact that the dislike button is dead.
I have a question, since Taylor's series is infinite, even if the values being added are very small for larger powers, shouldn't the sum still equal infinity given that we are adding an infinite no. Of times? Even if Ii were to add 1*10^-100 to itself, it would still be infinity given enough time right? So if we talk about infinite series, how can we approximate the value to anything less than infinity
Great Video! Really enjoyed how you went step by step, but in order the understand everything you need some knowledge of calculus, stillt great and „short“ video
This is the easiest video to understand, it truly fragments every piece to fully explain for the main topic. Wow, I'm already getting excited for college. The best explanation I have ever watched in RU-vid, gave a like and subscribed for more of these vids
Very good breakdown indeed. Always thought of the derivative of trig functions as shifting in phase by π/2, perhaps got too lazy to think of all the algebra and imposed limits.
Yo in 8th grade when we learned sin and cos I asked if it was possible to make sin x with an ifinitely long polynomial when I looked at the graph in Desmos and the teacher told me you couldn’t but now we have Taylor series in calculus.
Idk if i my sub counts, but this video blew up my mind, and it was the first one i watched of this channel! I HAD to show appreciation and ig the best i could do was subscribe and leave a comment.... Keep up bro! You are doing a great job, i have always connected with maths best, but i never understood Euler's identity much except the formulas related to it, but you have explained it so well including its origin and need! Im just flabbergasted! Thanks again! Love maths! ❤
I swear to god that my honest reaction when you got to the result was just "get outta here" haha. Amazing video. I never understood why people always say that this is the most beautiful equation until now.
For some reason I keep hearing people saying that "Euler identity is famous because it links fundamental constants in math", and to me this seems such an upside down statement. Euler's identity is famous and important because it describes an important process, and not because it has "fundamental constants". It is very easy to create formulas with these fundamental constants. Second, these constant are "fundamental" because they describe elements in this important process. So saying that the identity is famous because the fundamental constants misses the whole idea behind it. In any case, the important process is that in product of complex numbers, their lengths multiply, but the angles add up. So if we only look only at angles, we just transformed multiplication to addition (which usually is much easier to work with). This process is none other than the standard exponent rule exp(a+b)=exp(a)*exp(b). Because when you go over an entire circle, you return to the place where you started with, you expect to have a solution to exp(z)=1, and you can "call" this solution 2PI*i (the i is there because we only look at angles). We could have just as well started with the "fundamental constant" e^2, and then the solution would have been just PI*i. Saying that this formula is true because all sorts of derivatives and Taylor series of sines and cosines, while might be a valid proof, it still misses the whole reason why it is important.
as a high school freshmen who doesn't know calculus, this video made sense. I don't know how you did it but damn was that an amazing video! Really though, thank you
I use these type of videos as background noise, because I don't pay attention to them and when i do, I don't understand, some day I'll watch this again and try and understand it!
This was the best math video i have ever seen in my life. It had some easy stuff that wasn't need to be explained but in my opinion that what made it great
As a precalc dude, I was still interested in Taylor series and decided to jumble the formula around a bit, so I calculated it for sin, cos, e^x and I was so close to reaching this without knowing😭 The dude that first tried out the i multiplication approach is a genius
