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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)
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.
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.
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
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 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.
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.
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 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 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.
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
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.
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.
"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.
An interesting motivation for Euler's formula is known as the _moving particle argument_ which is a geometric interpretation of what Euler's formula actually means. If we want to explore what eⁱᵗ (where t is assumed to be real) could actually mean without first defining eˣ for a nonreal x we could start with z(t) = eⁱᵗ which maps the real number line onto the complex plane, at least if we can assign a unique complex value to eⁱᵗ for any real t. If we consider the real variable t to represent _time_ and keep in mind that for any real value t a complex number z(t) = x(t) + i·y(t) is represented by the _point_ with coordinates (x(t), y(t)) in the complex plane, then we can see that z(t) = eⁱᵗ describes the position of a point in the complex plane at any time t, which means that this equation can be interpreted as describing the _trajectory_ of a moving point in the complex plane. Now, the next question obviously is _what_ trajectory this equation could represent. The complex number z(t) gives the position of the point at any time t, but if this point changes with changing values of t then we have a moving point which therefore moves with a certain speed at any time t. The horizontal and vertical positions of the moving point at any time t are given by x(t) and y(t) so z(t) = x(t) + i·y(t) can be seen as the _place vector_ of the moving point at time t. Likewise, the _speed_ at which the point moves at any time t is given by the derivatives x'(t) and y'(t) where the first represents the horizontal and the second the vertical velocity at any time t. Note that any movement of a point in the plane is characterized by both _speed_ and _direction_ at any given time t and can therefore be analyzed as a combination of both horizontal and vertical movement at the same time. This means the the derivative z'(t) = x'(t) + i·y'(t) can be seen as a _velocity vector_ which fully characterizes both the direction of the movement and the speed at which the point moves at any time t. So, to get an idea of the trajectory of our moving point in the complex plane, we should investigate the derivative z'(t) = x'(t) + i·y'(t). But how do we find the derivative of z(t) = eⁱᵗ if we don't yet know what eⁱᵗ represents? Well, if we are to assign any meaningful interpretation to eⁱᵗ we should at least start by assuming that the same rules that apply to differentiating f(t) = eᵃᵗ for any real constant a should also apply to eⁱᵗ because i is also a constant. And since for f(t) = eᵃᵗ we have f'(t) = a·eᵃᵗ = a·f(t) this implies that we may expect to have z'(t) = i·eⁱᵗ which we can also write as z'(t) = i·z(t) So, the derivative z'(t) which represents the velocity vector of the moving point in the complex plane at any time t is equal to the place vector z(t) at that same time multiplied by i. Now, multiplication by i has a very simple but interesting geometric representation in the complex plane. If we take any complex number z = x + iy which is represented by the point (x, y) in the complex plane and multiply that number by i we get iz = i(x + iy) = ix + i²y = ix − y = −y + ix which is a number represented by the point (−y, x) in the complex plane. We can easily see that if we rotate the coordinate axes _counterclockwise_ over a right angle around the origin, then the positive real axis ends up on the original positive imaginary axis and the positive imaginary axis ends up on the original negative real axis. Consequently, if we rotate the point (x, y) representing the complex number z = x + iy counterclockwise over a right angle around the origin, then x will be the new vertical position of the point and −y will be the new horizontal position of the point, that is, the new coordinates of the rotated point are then (−y, x) which is the point representing the complex number iz = i(x + iy) = −y + ix. So, _multiplication by i corresponds to a counterclockwise rotation over a right angle around the origin_ in the complex plane. Returning now to our equation z'(t) = i·z(t) which describes the trajectory of our moving point in the complex plane we can see that this means that at any time the velocity vector z'(t) and, therefore, the direction in which our point moves in the complex plane, is _perpendicular_ to the place vector z(t). Now, the direction in which a point moves along a trajectory is simply the direction of the tangent at the point under consideration on the trajectory. So, we are looking at a trajectory of our moving point where the tangent on any point along the trajectory is perpendicular to the line segment from the origin to that point. But this can only be a _circle_ centered at the origin, because a tangent to a circle is always perpendicular to the radius from the centre of the circle to the point of tangency. So, we have now established that z(t) = eⁱᵗ maps the real number line onto a circle centered at the origin in the complex plane, but what more can we tell about this circle? Well, first of all, if we take t = 0 we have z(0) = e⁰ = 1 so the trajectory of our moving point passes through the point (1, 0) in the complex plane which is the point representing the number 1 + 0i = 1. So, the trajectory of the moving point is the _unit circle_ which is the circle centered at the origin with radius one. Also, since the velocity vector z'(t) is rotated counterclockwise over a right angle relative to the place vector z(t) we can tell that our moving point traverses the unit circle counterclockwise. But what about the speed at which our moving point traverses the unit circle? Well, since z(t) is a complex number represented by a point on the unit circle for any real value of t we have |z(t)| = 1 for any real t. But since z'(t) = i·z(t) this means that we also have |z'(t)| = |i·z(t)| = |i|·|z(t)| = 1·1 = 1. So, the magnitude or absolute value |z'(t)| of the velocity vector is always 1, which means that our moving point travels around the unit circle at a uniform speed of 1 unit per unit of time. Since the moving point passes the point (1, 0) on the unit circle at time t = 0 and since the circumference of the unit circle is 2π and since the point traverses the unit circle at a uniform speed 1 this means that the moving point passes the point (1, 0) at any time t = k·2π = 2kπ for all integer values of k. By now, this should all start to sound very familiar if you remember the unit circle definitions of the sine and cosine. In fact, if we have a point that traverses the unit circle counterclockwise at unit speed and which passes the point (1, 0) on the unit circle at time t = 0 then the unit circle definitions of the sine and cosine imply that we have x(t) = cos t and y(t) = sin t. Since the point (x(t), y(t)) represents the complex number z(t) = x(t) + i·y(t) this means that we have z(t) = cos t + i·sin t and since we started with z(t) = eⁱᵗ we can now conclude that for any real value of t we have eⁱᵗ = cos t + i·sin t which is of course Euler's formula.
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
Btw at 9:59, I just realised you can differentiate the Taylor series of sin(x) to instantly find the Taylor series of cos(x) instead of having to do the whole method again
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.
e^i(theta), and just plug in pi. Oh, and to prove Euler’s identity, do the Mac. Series of e^ix (just plug in ix for all places that are x). Compare that to the Mac. series of sinx and cosx. You’ll find that it has exactly all the terms in the cosx Mac. series, and all the terms in the sinx, but multiplied by i
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 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
Gosh maths is fascinating. I wish I’d worked harder at school. I remember Mr Laycock at the polytechnical central London showing us that equation and remarking on how interesting it was.
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 ! ❤❤🎉😅
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!
I'm in calculus right now and I understand how d/dx of x^2 is 2x and of sin is cos, that makes enough sense to me, but I've always been curious of how you'd derive it from those other ways of writing the derivative so thank you!
I don't feel that way. This video clearly explained a lot of things within less than 15 minutes without just proving the identity. He could have just shown the Taylor Series without proving it. There are many more examples. It is true that a little knowledge of calculus is needed, yet it was explained really well.
Perhaps it's just me, but all of this reminds of analog to digital conversion in electronics. This seems sort of like the mathematical equivalent to an op amp.
![The most beautiful equation in math, explained visually [Euler’s Formula]](http://i.ytimg.com/vi/f8CXG7dS-D0/mqdefault.jpg)
![The most beautiful equation in math, explained visually [Euler’s Formula]](/img/tr.png)
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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 😅
e^yipee
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.
The animation, the explanation and the way you covered so many topics in one video is just amazing.
Step 1: cry
???
@@FaizThe-kc6dk because this shit hard as fuck😭😭
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.
as a teenager sad that such resources were not available in your time, None of my teachers explain such stuff, RUclips is a life saver
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.
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
@@joey5305yes , exactly, but this video is still god level .
@@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 is the most beautiful thing i have ever seen. Amazing explanation. Thank you!
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
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.
Now I understand everything in those math videos that I'm addicted to 😂😂😂(especially the cos(x) + isin(x) part)
Thanks!
-1
First time I m seeing this much effort to explain Euler identity. Well done
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.
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 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.
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.
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
This is how one should sound when explaining mathematics, it is disturbing that other channels on youtube use too much emotions in their tone.
When you blink in math class: 3:30
This video was so good and i was so immersed that i forgot it was about e^ipi
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 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.
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
I suggest you google for convergent series and divergent series, which will explain your question
Euler's identity using calculus, trig and Tylor. Impressive. Your video covers that ridiculous expression quite elegantly. Awesome work.
Your videos are really great for learning these things even made more sense than the Stewart calculus book.
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
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!
me who understood nothing,but still found it cool 😂
not alone there, bud.
>click video
>insanely strong accent
>go back
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.
Didn't know Euler was chill like that
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.
This way, you can take logarithms of negative numbers:
ln(−x) = ln(x) + ln(−1) = ln(x) + iπ
lg(−x) = ln(−x) / ln(10) = (ln(x) + iπ) / ln(10)
Thank you for explanation of one of the most beautiful equation in so beautiful way.
The best explanation man
I understood everything you really explain all the context and didn't went direct to the taylor series 😅
You know it when you wanna like the video but you accidentally dislike it and then it says feedback shared with creator? TwT
I liked it now. ^w^
"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.
@@dorukayhanwastaken I know, but that message is sounds so sad.
Thank you!
An interesting motivation for Euler's formula is known as the _moving particle argument_ which is a geometric interpretation of what Euler's formula actually means. If we want to explore what eⁱᵗ (where t is assumed to be real) could actually mean without first defining eˣ for a nonreal x we could start with
z(t) = eⁱᵗ
which maps the real number line onto the complex plane, at least if we can assign a unique complex value to eⁱᵗ for any real t. If we consider the real variable t to represent _time_ and keep in mind that for any real value t a complex number
z(t) = x(t) + i·y(t)
is represented by the _point_ with coordinates (x(t), y(t)) in the complex plane, then we can see that z(t) = eⁱᵗ describes the position of a point in the complex plane at any time t, which means that this equation can be interpreted as describing the _trajectory_ of a moving point in the complex plane.
Now, the next question obviously is _what_ trajectory this equation could represent. The complex number z(t) gives the position of the point at any time t, but if this point changes with changing values of t then we have a moving point which therefore moves with a certain speed at any time t.
The horizontal and vertical positions of the moving point at any time t are given by x(t) and y(t) so z(t) = x(t) + i·y(t) can be seen as the _place vector_ of the moving point at time t.
Likewise, the _speed_ at which the point moves at any time t is given by the derivatives x'(t) and y'(t) where the first represents the horizontal and the second the vertical velocity at any time t. Note that any movement of a point in the plane is characterized by both _speed_ and _direction_ at any given time t and can therefore be analyzed as a combination of both horizontal and vertical movement at the same time. This means the the derivative z'(t) = x'(t) + i·y'(t) can be seen as a _velocity vector_ which fully characterizes both the direction of the movement and the speed at which the point moves at any time t.
So, to get an idea of the trajectory of our moving point in the complex plane, we should investigate the derivative z'(t) = x'(t) + i·y'(t). But how do we find the derivative of z(t) = eⁱᵗ if we don't yet know what eⁱᵗ represents? Well, if we are to assign any meaningful interpretation to eⁱᵗ we should at least start by assuming that the same rules that apply to differentiating f(t) = eᵃᵗ for any real constant a should also apply to eⁱᵗ because i is also a constant. And since for f(t) = eᵃᵗ we have f'(t) = a·eᵃᵗ = a·f(t) this implies that we may expect to have
z'(t) = i·eⁱᵗ
which we can also write as
z'(t) = i·z(t)
So, the derivative z'(t) which represents the velocity vector of the moving point in the complex plane at any time t is equal to the place vector z(t) at that same time multiplied by i.
Now, multiplication by i has a very simple but interesting geometric representation in the complex plane. If we take any complex number z = x + iy which is represented by the point (x, y) in the complex plane and multiply that number by i we get iz = i(x + iy) = ix + i²y = ix − y = −y + ix which is a number represented by the point (−y, x) in the complex plane.
We can easily see that if we rotate the coordinate axes _counterclockwise_ over a right angle around the origin, then the positive real axis ends up on the original positive imaginary axis and the positive imaginary axis ends up on the original negative real axis. Consequently, if we rotate the point (x, y) representing the complex number z = x + iy counterclockwise over a right angle around the origin, then x will be the new vertical position of the point and −y will be the new horizontal position of the point, that is, the new coordinates of the rotated point are then (−y, x) which is the point representing the complex number iz = i(x + iy) = −y + ix. So, _multiplication by i corresponds to a counterclockwise rotation over a right angle around the origin_ in the complex plane.
Returning now to our equation
z'(t) = i·z(t)
which describes the trajectory of our moving point in the complex plane we can see that this means that at any time the velocity vector z'(t) and, therefore, the direction in which our point moves in the complex plane, is _perpendicular_ to the place vector z(t). Now, the direction in which a point moves along a trajectory is simply the direction of the tangent at the point under consideration on the trajectory.
So, we are looking at a trajectory of our moving point where the tangent on any point along the trajectory is perpendicular to the line segment from the origin to that point. But this can only be a _circle_ centered at the origin, because a tangent to a circle is always perpendicular to the radius from the centre of the circle to the point of tangency.
So, we have now established that z(t) = eⁱᵗ maps the real number line onto a circle centered at the origin in the complex plane, but what more can we tell about this circle? Well, first of all, if we take t = 0 we have z(0) = e⁰ = 1 so the trajectory of our moving point passes through the point (1, 0) in the complex plane which is the point representing the number 1 + 0i = 1. So, the trajectory of the moving point is the _unit circle_ which is the circle centered at the origin with radius one. Also, since the velocity vector z'(t) is rotated counterclockwise over a right angle relative to the place vector z(t) we can tell that our moving point traverses the unit circle counterclockwise.
But what about the speed at which our moving point traverses the unit circle? Well, since z(t) is a complex number represented by a point on the unit circle for any real value of t we have |z(t)| = 1 for any real t. But since z'(t) = i·z(t) this means that we also have |z'(t)| = |i·z(t)| = |i|·|z(t)| = 1·1 = 1. So, the magnitude or absolute value |z'(t)| of the velocity vector is always 1, which means that our moving point travels around the unit circle at a uniform speed of 1 unit per unit of time. Since the moving point passes the point (1, 0) on the unit circle at time t = 0 and since the circumference of the unit circle is 2π and since the point traverses the unit circle at a uniform speed 1 this means that the moving point passes the point (1, 0) at any time t = k·2π = 2kπ for all integer values of k.
By now, this should all start to sound very familiar if you remember the unit circle definitions of the sine and cosine. In fact, if we have a point that traverses the unit circle counterclockwise at unit speed and which passes the point (1, 0) on the unit circle at time t = 0 then the unit circle definitions of the sine and cosine imply that we have x(t) = cos t and y(t) = sin t. Since the point (x(t), y(t)) represents the complex number z(t) = x(t) + i·y(t) this means that we have
z(t) = cos t + i·sin t
and since we started with
z(t) = eⁱᵗ
we can now conclude that for any real value of t we have
eⁱᵗ = cos t + i·sin t
which is of course Euler's formula.
Algebraic visualization is too cool!
But geometric visualization is an awesome!
Good work!
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
this is the best explanation of the Euler identity i have ever Seen
Btw at 9:59, I just realised you can differentiate the Taylor series of sin(x) to instantly find the Taylor series of cos(x) instead of having to do the whole method again
The most beautiful equation is 3 sausages, 2 slices of bacon, 2 fried eggs and 2 slices of bread and butter on my plate.
I don’t know much about Taylor series as I was never formally taught but this made perfect sense. Nice
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.
That's an excellent video!!! Congratulations!!! Everything was very clear!!!
I barely made it thru calc 1 but this was very easy to follow!
This has been the most comprehensove video on this topic I have ever watched
UR SUCH A CHAD BROO...SIMPLEST WAY TO UNDERSTAND THIS MAN
e^i(theta), and just plug in pi. Oh, and to prove Euler’s identity, do the Mac. Series of e^ix (just plug in ix for all places that are x). Compare that to the Mac. series of sinx and cosx. You’ll find that it has exactly all the terms in the cosx Mac. series, and all the terms in the sinx, but multiplied by i
I love to add the minus sign to this equation too, because that one is missing in my opinion:
exp (-pi*I) +1 =0
is equally true
This is the best video prooving Euler's identity on youtube that I've seen so far. Brilliant video
You explain very good, although I understand only basic english
Beautiful explanation, just the right level and speed, thanks!
Probably the best Maths video I've ever seen!
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 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
Truly the one of the few Digital geniuses of youtube
You explained *The Most Beautiful Equation* in *The Most Beautiful Way !!!*
🤗👏👏👏👍
This video is a masterpiece in basic calculus
Holy shit this video just summarized all the concepts you learned in Calculus!!!
I learned more about the definition of derivatives than I did in 3 semesters of Calculus ( Calculus 1through 3)
I'm impressed that this video actually goes from ground zero to the proof of Euler's identity in just 13min❤
I've never fallen in love with math more than now.
The effort put into this video is immaculate
Well, I'm not sure if I "understand" the formula, but the proof sure is beautiful.
This was explained excellently. However, the derivative of sin(x) using first principle could be done more easily
Gosh maths is fascinating. I wish I’d worked harder at school. I remember Mr Laycock at the polytechnical central London showing us that equation and remarking on how interesting it was.
Mr lay what??!?!!
Thank you so much
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 ! ❤❤🎉😅
Your explanation is too clean bro😊😊
I had been wondering how the rules of derivatives was made. Lucky me who found this beautiful explanation.
Best proof of euler's identity, pure algebraic proof is always the best proof
I learned so much in math from this video. Thank you very much!
I just a 16yo getting slightly interested in maths and I wanted to see what oilers identity was.
I did not expect this 😭
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 so beautiful it made me cry
I'm in calculus right now and I understand how d/dx of x^2 is 2x and of sin is cos, that makes enough sense to me, but I've always been curious of how you'd derive it from those other ways of writing the derivative so thank you!
Excellant, what i was looking for since last 25 years approx.
That was really well explained. Thanks, makes me understand it a bit more
This is impressive man, tks for the video!
the best video that I have watched in such a long time
beautiful explanation
Who needs college when you have guys like this!
what a marvelously constructed video. Truly beautiful and incredibly informative. Bravo
Thank you for this video, you explain things so beautifully intuitive
Top notch quality content delivered beautifully.. Do continue uploading..❤ loved it..
Awesome. I finally understand it.
Best calculus video ever 🎉
Beautifully done my guy👍🏻
I LEARNED SO MUCH ABOUT TRIG AND DERIVATIVES FROM THIS I KNOW WHAT THEY MEAN THANK YOUUUUUUU🙏🙏🙏🙏🙏🙏
best video I ever watched
Very beautiful. Thank you
This video is beautiful! Like an art form!
Awesomely well explained 👏👏👏👏🙏🙏🙏…….. especially the animations for calculus…. Loved it so much that I went and subscribed to the whole channel …..❤❤❤
It’s not an equation but an identity.
I identify as e^(iπ)+1=0
It has no variables so it’s both
No clue but the explanation is so well i understand it
What a great explanation😊👍
Maybe it's just me, but I feel like this explanation is too fast for majority of the audience.
Just slow the speed
I don't feel that way. This video clearly explained a lot of things within less than 15 minutes without just proving the identity. He could have just shown the Taylor Series without proving it. There are many more examples.
It is true that a little knowledge of calculus is needed, yet it was explained really well.
He explained everything from basics. I don't think it's fast!
Perhaps it's just me, but all of this reminds of analog to digital conversion in electronics. This seems sort of like the mathematical equivalent to an op amp.