The Most Beautiful Equation
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- Опубліковано 19 тра 2024
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Euler's Identity is one of the most popular math equations. In this video you'll learn what it really means.
Chapters:
00:00 Intro
00:33 Pi
01:28 i
02:07 Derivative
10:00 e
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yes
e^iπ + 1 = 0 so e^iπ = -1
Amazing. Too many concepts 😮
I find this more beautiful: e to the power of i tau equals unity.
It is the most awful equation, e^i.pi =-1. You are idiots, the Tau equation is the most beautiful.
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 comes from Euler identity
e^xi=cos(x)+isin(x)
x=pi
e^pii=cos(pi)+isin(pi)
e^pii=-1
@@user-lu6yg3vk9zwe watched the video thanks 👍
I agree, its a great video... I just finished 8th grade and I'm half lost but interested nonetheless 😅
If only my math teacher explained this so well.
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)
The problem then is the system. For expecting us to get this shit in in one hour
Ikr when my teacher told me this I only understood 1 word, but when he is telling me the same thing I understand at least 2
... aint that the truth ...@@dagan5698
This is your math teacher and I'm failing you in the upcoming exams.
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.
Step 1: cry
Derivative, Taylor seires, sin cosine, iota , pie all in one video!! Incredible
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.
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.
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
The animation, the explanation and the way you covered so many topics in one video is just amazing.
A really good explanation. I have heard of solving Euler's Identity with the Taylor series before, but you really went into detail explaining how the Taylor series works, how to turn sin, cos, and e^x into Taylor series, and finally bringing it all home equating the Taylor series of e^x to that of sin and cos. All in 13 minutes. Impressive.
I feel scared for understanding everything in this video.
Same bro 😢
This channel is too underrated, this was explained flawlessly
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!
Euler's form is there in complex numbers in class 11th. You will know if you are preparing for JEE.
@@joey5305 "i sTuDiEd cALcULuS iN gRaDe 4" -🤓
Fr
First time I m seeing this much effort to explain Euler identity. Well done
me who understood nothing,but still found it cool 😂
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.
it can't be proved because that's the definition of 2. 2 is just shorthand for 1 + 1
There is actually a proof for this
There's actually a video on Half as Interesting titled "The 360-page proof that 1 + 1 = 2"
@@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.
That’s kinda like proving that the colour purple is green and blue. U can’t prove it, it’s just the definition of purple.
This is the most beautiful thing i have ever seen. Amazing explanation. Thank you!
You explain so much about complex numbers in this little video, which never been told in any maths class. Thats brilliant!
That was the best explanation of the derivative limit I've ever seen!
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.
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
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).
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.
Now I understand everything in those math videos that I'm addicted to 😂😂😂(especially the cos(x) + isin(x) part)
Thanks!
-1
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 UA-cam, gave a like and subscribed for more of these vids
This video was so good and i was so immersed that i forgot it was about e^ipi
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
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
You got me hooked from the start itself, your explanation was very interesting! You just earned a new sub from me
Your videos are really great for learning these things even made more sense than the Stewart calculus book.
This is how one should sound when explaining mathematics, it is disturbing that other channels on youtube use too much emotions in their tone.
This is the best video prooving Euler's identity on youtube that I've seen so far. Brilliant video
Absolutely incredible video. One of the most beautiful videos i have seen to get the explanation of the most beautiful equation. I loved how you went from basics to the derivation. Amazing ! ❤❤🎉😅
Very elegant explanation and the process behind the formula
Squared ❌ Squirt ✅
Circle❌ cericle✅
Euler's identity using calculus, trig and Tylor. Impressive. Your video covers that ridiculous expression quite elegantly. Awesome work.
Beautiful explanation, just the right level and speed, thanks!
the way it all pieces together at the end is crazy.
The best explanation man
I understood everything you really explain all the context and didn't went direct to the taylor series 😅
That was really well explained. Thanks, makes me understand it a bit more
One of the best maths video on UA-cam I have ever watched
Thankyou very much
It is indeed beautiful equation and explanation! Thank you so much, Sir🙏
I'm a math teacher and i found this video extremely helpful and inspirational, thanks!
This has been the most comprehensove video on this topic I have ever watched
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
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!
what a marvelously constructed video. Truly beautiful and incredibly informative. Bravo
I have never found anybody explain this thing this well
Thank you for this video, you explain things so beautifully intuitive
Awesomely well explained 👏👏👏👏🙏🙏🙏…….. especially the animations for calculus…. Loved it so much that I went and subscribed to the whole channel …..❤❤❤
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.
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!
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.
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
Whole Derivative lesson 😮 incredible thank you ❤
This is impressive man, tks for the video!
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)
Brilliantly explained. ♡
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.
I learned so much in math from this video. Thank you very much!
this is the best explanation of the Euler identity i have ever Seen
thankyou so much , you really made my day . this is simply beautiful
Excellant, what i was looking for since last 25 years approx.
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.
Top notch quality content delivered beautifully.. Do continue uploading..❤ loved it..
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! ❤
You, my friend, are excellent at explaining !!!!
It was absolutely amazing
I finally learned the Euler identity
beautiful explanation
Beautifully done my guy👍🏻
Thank you soo much for this video!!!!!!!
SOOOOO Beautiful. I am literally out of my head!!!!!
Dude just ❤❤❤ it please do one for fourier transforms dude 🙏🙏
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.
UR SUCH A CHAD BROO...SIMPLEST WAY TO UNDERSTAND THIS MAN
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.
This is a truly beautiful video the showing the magic of math
Great job. Excellent explanation.
i think that was the most beautiful video i have ever seen i literally have goosebumps
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.
Excellent explanation!
Fabulous explanation
What a great explanation😊👍
Hi Digital Genius, really great video!
Can I ask you with what kind of software you have created it?
Greetings.
You’re awesome bro!! Great job 👏🏽
Algebraic visualization is too cool!
But geometric visualization is an awesome!
Good work!
You explained *The Most Beautiful Equation* in *The Most Beautiful Way !!!*
🤗👏👏👏👍
2:04 Making a counterclockwise circle and putting i's powers where they would be on the complex plane was deliberate, wasn't it?
I LEARNED SO MUCH ABOUT TRIG AND DERIVATIVES FROM THIS I KNOW WHAT THEY MEAN THANK YOUUUUUUU🙏🙏🙏🙏🙏🙏
I had been wondering how the rules of derivatives was made. Lucky me who found this beautiful explanation.
This was so beautiful it made me cry
I clicked to know about the identity, and I was carried over the whole Calculus I&II course in like 10 mins💀
Great explanation ❤
Thank you so much
How you prepare such videos
What programs used
&Thx❤
Great work, I really hope your channel blows up
Beautiful 😍❤️ thank you
Your explanation is too clean bro😊😊