Complex exponents are very important for differential equations, so I wanted to be sure to have a quick reference for anyone uncomfortable with the idea. Plus, as an added benefit, this gives an exercise in what it feels like to reason about a differential equation using a phase space, even if none of those words are technically used. As some of you may know, Euler's formula is already covered on this channel, but from a very different perspective whose main motive was to give an excuse to introduce group theory. Hope you enjoy both!
Please make a series on tensors and imaginary numbers because these topics are the least understood among students and misjudged due to their weird names. Also they are really very interesting, Intuitive and fundamental to the universe.
Loved the Euler's formula video. This one too. I'm about to start a course in computing this august. ( I took the advice from your numberphile podcast ) Just want to thank you for these videos the animations are honestly so beautiful to watch. I'm looking forward to the next videos!
I would say that there is simpler proof that also shows the real engineering operation behind that function. We should know that the value of e is related to having $1 dollar in the bank and with 100% interest continuously compounded, then at the end of the year one gets $e = $ 2.7182 This compounding of interest in a numeral $ dollar operation, makes the money grow. If we had to represent the compounding of the value of $ money in the bank, we could use the length of a broomstick, where $1 dollar would be a unit length along with the broomstick, while the continuous compounding at the end of the year would be shown by a length of 2.7182 hence $2.718 But compounding on a broomstick can take place along the length, AND ALSO IN QUADRATURE WITH THE LENGTH OF THE BROOMSTICK, where this would result in a rotation or an orientation of the straight long broomstick. So if we invest in rotating the unit length of the broomstick acting as a rotating vector, rather than elongating it along its length, then e^iA would be the final orientation of the unit broomstick after it terminates it compounding "rotational banking interest" rather than the conventional linear compounding which just changed its length! All we have to do is to draw the vector e^iA where "i" indicates the quadrature direction of the compounding with respect to the length of the unit broomstick length 1, and A is the final compounded angle in radians reached where the unit length of the broomstick will remain a unit length. Hence taking the components of the final rotational compounding e^iA on the two basic axes which we used as a reference, we have e^ iA= cos A +isin A. I would say that the students would appreciate all this OPERATION which describes the real engineering actions behind the " compounding of interest" when the interest is linear or when it is (i. interest angular displacement ) the rotational interest on a unit length of a broomstick, or it could even be a $1 dollar paper sheet, wrapped up like a thin cigarette to represent the unit vector, and the wrapped up thin $ sheet will act as the rotating vector accumulating the rotating angular interest which finally compounds to e^iA. I wager there are a few smiles after readers read all this! One would agree that the philosophy of compounding the $1 dollar paper sheet in a linear manner of its growth in monetary value, or its broomstick length representation, and the rotational angular manner, is exactly the same! where all that engineering actions and reactions are described so accurately in the meaning of the symbols in that expressions. We all need to learn what they mean in engineering actions and reactions and not only the rules of differentiation of the product function used in the video as ...... (the first function)multiplied by (the differential of the second function) plus (the second function) multiplied by (the differential of the first function). I always wonder how many people see the engineering/physical activity in that operation!!!!!!!!!!!!
That's surprising, considering how I was exposed to it was through the MacLaurin series. That answers it immaculately too. As I've matured more in mathematics, still quite infantile, I realized that 3B1B's videos are very visual. Though this video is more approachable, I'm often leaning towards traditional textbook like material with the leap of intuition than his videos that are ambitious visualizations.
I was comfortable with this identiy before, but I had a great “AHA!” moment. This is probably the most concise and surprising way to look at it I’ve ever seen. Thanks a bunch
Think of multiplying by negative one as turning around to face backwards. If you do that twice you will be facing forwards again. That is (-1)*(-1)=1. Now if we ask what number times itself equals negative one, it will be a rotation that when done twice has you facing backwards, either a quarter turn counterclockwise i or a quarter turn clockwise -i. (the complex plane doesn't actually distinguish between clockwise and counterclockwise because that would require it physically be facing in some direction relative to us, and it is an abstract mathematical object, but we aren't abstract and generally draw i above the real number line and -i below, and numbers on the real number line getting larger as you move right)
@@paulfoss5385 So, for real numbers, |R, (-1)^2 = 1, and for complex numbers, |C, (i)^4 = 1. I think it's really cool that that exponent is off by a factor of 2 for |R and |C. I'm trying to think how this generalizes to quaternions (|Q) and maybe octonions (|O) but it's not obvious. |Q also holds that i^2=-1, same as |C, so it's still (i)^4=1. Octonions, well... lol.
theres a really good series from another channel about imaginary numbers that goes into depth about it from the absolute basics, Id recommend giving it a watch
The great thing about math is that you don’t have to take anyone’s word for whether something is right or wrong. You can think through if this explanation makes sense for yourself (spoiler alert, it checks out). To engage with it that way will force you to think critically about what it means to take a derivative of a complex-valued function, which in turn builds good intuitions for complex analysis.
If e^πi=-1 then π=ln(-1) / i Then 2πr= 2r ln (-1) / i = ln (-1^2r) / i Which equals ln (1) / i Which equals 0 Which means that if Euler's identity is true, the circumference of a circle is always 0.
@@squibble311 if you are wondering about anything else, Generally e^ix=cos(x)+isin(x) There's a really easy explanation for this, it has to do with trig
@hiqwertyhi you've forgotten to explain that since a minute is only 60 seconds... 3.14 minutes is in fact about 3:08... And then tell readers to look closely in the bottom right corner of the video at that time... :-B
The idea that multiplying a number by i rotates it 90 degrees into an imaginary axis is probably the most important thing I've ever gone my whole life without knowing.
After four years of studying electrical engineering in college, no one ever explained it that way until a professor casually mentioned it in a elective I took in my last semester :/ It would have helped so much to learn that earlier
Kudos! This is pedagogy brought to the level of art. I hope your work, together with that of other excellent TH-cam educators, will pave the way for a new generation of teaching.
i've watched other 3blue1brown videos about this but i never really got it until now for some reason. holy crap it's so intuitive now, i don't know what i didn't get it before
THAT WAS SUCH A GREAT EXPLANATION I always struggled with that one, despite the fact you've made quite a few videos about it this was so good though, it makes so much sense now I gotta watch the other ones you've made to see if I can understand them better now
I did not understand why the yellow line (velocity ) was moving faster then the blow line (position ) even though he said the dervative is just equal to to the position at that time,. Can u explain
@@hassanomer7777 It doesn't move faster. If the right side of green line moves at 2n, while left side moves at n, this means its length increases by n. The "speed boost" comes from blue line's end
It's amazing how helpful Grant's use of color is. This was particularly pronounced on his superb Fourier series videos. The old analog chalkboards of my youth did not have this added dimension.
I can’t even stress how important your work is for me. You explain everything in such a beautiful and simple way; you were BORN to teach. Congratulations on all of your videos, sir.
To anyone who wants to think about something deeper with this concept: Think about the function y=(-1)^x. Now this function is discontinuous, it’s not a line at all. It just seems to alternate between y=1 and y=-1 on the xy plane. But what happens if you include the complex numbers, say adding them in a z axis to make a three-dimensional graph. What you’ll find is that the function (-1)^x, or to write it another way e^(i*pi*x), traces out a line in 3d space. A line that is in the shape of a helix, a coil going around the x-axis, and intersecting the xy plane at 1 and -1 at appropriate points.
It is incredible how far you've grown as an educator. Your first take on e to pi*i as your first video on a channel was really confusing to a lot of viewers, me included. But now you are providing an easily conprehensive explanations together with enjoyable visuals on this and many other topics. Thank you for all the work you're putting in your channel
It's Magical ! The way you explain. And I am not even a newcomer on this channel, yet my mind is always blown away by how you brilliantly decompose mathematical concepts to their core. So elegant, so beautiful. This is the way to do it, the proper way to use technology in order to share science and wisdom. So thankful for the Internet, that is allowing us to connect with such quality content. So thankful for you, creators of such quality content. Thank you for what you do ! Peace.
This is the simplest and most intuitive expression for this I have ever seen. Thank you. I'm a grad student in robotics so ODE's and PDE's are my life blood. So, this has always bothered me that the intuition for this concept was so difficult. With such a simple explanation I wonder why I have never seen this before.
Even with your previous videos about e^ipi I couldn't fully grasp WHY it behaves so. Why it has this undeniable link to rotations in the complex plane. As SOON as you brought derivatives into it, and e^x's nature of being its own derivative, it makes absolute perfect sense. Dude, you and blackpenredpen are the math teachers I needed so desperately when I was precalc in high school.
I think this way of teaching "visually" should be introduced in every school in the world... it's amazing how you manage to find the right animation for whatever concept, bravo!
I really like the buildup in this video. You start at derivatives and everything just blends into each other until you're expanding it further. From the effect of positive factors in the exponent, introducing the negative factor and eventually the imaginary factor. Definitely hard to understand if you don't know about the underlying concepts, but once you do, this almost feels like the most perfect way to explain e to the i pi. Great video
Which would instead be 3 minutes and ~8 seconds which doesn't really add up either. 🤷♀️ Not that it matters ultimately. It's still a really good and elegant explanation.
As someone who has been in science long enough to see this identity everywhere, I always hated that the only explanation I ever got was "It works because of Taylor expansion", with the demonstration separating the terms of the Taylor expansion of exp into those of cos and sin. It's just super abstract, like "It works bc of these random formulas that converge to the result after infinite steps". Now I just learned about this way of seeing Euler's formula, was mindblown, did a quick YT search to see if anyone I watched had talked about it before... And of f-ing course 3Blue1Brown did it. I'm kinda mad I never saw this video and also not surprised that you would use this representation that really feels like something you would find on this channel. Such a cool way to see this otherwise very arbitrary-looking identity!
Yeah, when I first saw the definition of e^z = sin z + i cos z It seemed like someone had pulled a fast one. Even after it was shown that it's consistent with Taylor expansion, and so on. If I had started from the place of "multiplying by i is rotating 90 degrees" that might have helped. However, it is also the case that math is consistent and one explanation or proof is equivalent to another.
As an electrical engineer, I learned AC circuit solving through complex exponentials (phasors and impedances), so I'm really excited to see this video. (Also, the pace of the video relative to the usual 15-20 minute vids reminds me of the first video on the channel.)
It would be awesome if you did a series on modern numerical methods. Finite elements (for for elliptic problems) add Runge-Kutta methods for hyoerbolic and parabolic problems, etc... With your unique talent you could make this fascinating topic accesible to millions of talented young people who might chose to pursue a career in applied math. Thanks for all the great work!
You are freaking amazing, do you understand what you just did? you did an almost complete explantion of Euler's formula in few minutes that anyone with a bit of knolowede in caculus can understand. You are the best math channel on TH-cam and one of my favorite channels in general.
I am glad people like 3Blue1Brown exist on TH-cam! I could learn all day and not have to pay $50K per year at college/university, which is now nothing but venue to buy a certificate that you have a BA/BS!
As a chap who dropped Maths as soon as possible, I thought I’d watch to see if even I could understand it in about 3 minutes. I am actually pleasantly surprised. Seeming I never did differentiation I don’t understand the significance of what I’ve learnt, but it did make me smirk that i gets its own entire spatial dimension.
Such an intuitive explanation of a formula that has always seemed so abstract to me. I will now always have this concrete image in my head when I think of it. Thank you :)
I thought at the end you'd show how, if you project the circular motion in the complex plane onto the real axis, you see simple harmonic motion. This could help high school students to relate better. Btw I absolutely love this videos! Short but original and always make your concepts click no matter how much you think you know on the topic. Keep inspiring 3Blue1Brown!
The last bit, when you pointed out that e^iπ does not mean multiplying e by itself iπ times (even though this would have been the case if the exponent had been an integer, like e^3 = e*e*e) made me realize that it is the same thing that happens when you find an analytic continuation for a series. For instance, ζ(-1) = -1/12, but this equality does not mean that 1+2+3+...=-1/12, since the series for the Riemann Zeta function is only defined for numbers greater than 1. I had never realized before that even with simple arithmetic, such as powers, you can already get a feeling for the rule "you cannot always interpret new things using your old knowledge", because sometimes a blind substitution can be misleading.
I was researching e^ix= cos(x) + i*sin(x) after watching Lockdown math and wondered why there seemingly 'just happened to be' a neat connection between two quite different things: circles and e. From what I understood on Wikipedia and forums, there was an explanation that involved the Maclaurin series of e, sin, & cos, and whilst it is a proof, it didn't satisfy my question. But then this video made the connection obvious (with the help of another 3b1b vid about e and calculus) and I can now say that I understand e^iπ=1 :)
I've been arguing for a while that we should use physical models to teach math. You can represent addition and multiplication physically by superimposing 2 rulers on top of each other. Addition is a sliding action and multiplication is a scaling action, which you can represent by pulling them apart from each other while keeping the 0's overlapping and moving the 1 over whatever you're trying to scale up by. This makes complex numbers easier, since addition and multiplication work exactly the same way. This also makes it easier to go from only the nonnegative integers to the whole set of integers, since you can ask a question like "3+what=2" and it's not surprising that your incomplete number line has *something* that would answer that question, and seeing the structure laid out for you makes the negative numbers easier to use.
thats why the dude made the Manim library. you can easily make these math animations as a teacher, and shit works easily. it uses python too, so its hella easy.
@@ericvauwee4923 fr tho these videos are kinda nice to watch but for the exams and rigorosity in university they don’t really help tbh. Boomers now pretending we can watch 3 TH-cam videos and get our engineering degree lol
first i thought a random video about e pi. why would i care, saw that often enough. then i noticed 3blue1brown and thought "im sure i dont know enough yet" 'click'
Watching this again after just having done some problems involving resonant circuits, where Tau is used as a time constant, I just had the revelation that this is actually not unrelated to this Tau = 2*pi idea! Freakin beautiful. You help more people with these videos than you could ever know!
This makes it so clear, it's astonishing. I had to immediately re-watch it just because it's so elegant. It also makes it obvious why eⁱᵗ=cos t + i sin t.
I never liked e to the i pi, mostly because it seemed everyone was just so amazed over a statement. you could just accept it and integrate it into your math, or you could not. what I am more interested in is the proof, and what it means, but when a while ago several different youtubers started making different explanations for why e to the i pi equals -1, I was never quite satisfied. They always seemed to make some jump in comprehension. This jump is the same one that you just smoothed over beautifully, and I am in awe of how this problem that has been bugging me for like 2 years now has been explained in such a short period of time. Still I have more questions like if the only way mathematicians can explain or compute complex exponents is through e or if there is some more generalized formula or explanation for complex exponents. Also I wonder if there is an explicit definition for what exponents mean in relationship to the complex plane, beyond e. This promise of another video excites me, because it might answer these questions where others did not, but even if it doesn't, I remain in awe of your skills in explanation, as well as very grateful for your videos, because I don't know where I would be if it wasn't for your essence of algebra and calculus series.
Euler's Formula is the generalized version for complex exponents. e^ix = cos x + i*sin x. Basically think of x as an angle in complex space with the same logic as discussed in the video. It's all very magical how so many separate things end up relating.
Never before have I understood a video about this identity, but the ONLY time I fell behind was your position vs. velocity notation at one point. You should be _incredibly_ proud.
@Thomas Wilkinson I agree with the point about time having a "standard" unit. Though I'd clarify that the unit is essentially the number 1 on the number line. That is, a derivative of 2 with respect to t is a speed of 2 ones per one. The distance "one" on the number line is just the natural unit. So I'd say the unit is actually important here, as you can't measure distance in R^1 without the distinguished distance "one" (or some other distinguished distance) to relate it to.
I was taught that e^it = cost + isint but I couldn't for the life of me work out why. Hours on wikipedia led me nowhere. But this explained it brilliantly in 3 minutes.
One example: Taylor series. If you compare the Taylor series of e^x, cos(x) and sin(x) and insert it to x then you'll notice that e^(it) looks like cos(it) and i * sin(it) added together (granted, it has to be taken with a grain of salt given that it's an infinite sum but you still can see why that's true).
@@MarioFanGamer659 I saw that explanation yesterday when I had the same question, and whilst it is a proof, it still had me wondering why there 'just happened to be' a connection between two apparently quite different things - circles (& thus sin/cos) and e. But then this video (with help from another by 3b1b about e and calculus) made the connection obvious, and I can now say that I understand e^iπ=1. That's why I love this channel :)
Complex exponents are very important for differential equations, so I wanted to be sure to have a quick reference for anyone uncomfortable with the idea. Plus, as an added benefit, this gives an exercise in what it feels like to reason about a differential equation using a phase space, even if none of those words are technically used.
As some of you may know, Euler's formula is already covered on this channel, but from a very different perspective whose main motive was to give an excuse to introduce group theory. Hope you enjoy both!
Please make a series on tensors and imaginary numbers because these topics are the least understood among students and misjudged due to their weird names. Also they are really very interesting, Intuitive and fundamental to the universe.
I am not feeling well😗😗😌😌
Loved the Euler's formula video. This one too.
I'm about to start a course in computing this august. ( I took the advice from your numberphile podcast ) Just want to thank you for these videos the animations are honestly so beautiful to watch. I'm looking forward to the next videos!
@Wolfgang Kleinschmit did you see his Essence of Calculus series?
Last I checked, 3.14=/=4.08
This is by far the most intuitive explanation of this identity I've ever seen!
CozmicK G I would check out the explanation with the differential equation y’’ + y = 0
@@johnlesslie1963 where is this at? i can’t find it
KRUH
I would say that there is simpler proof that also shows the real engineering operation behind that function.
We should know that the value of e is related to having $1 dollar in the bank and with 100% interest continuously compounded, then at the end of the year one gets $e = $ 2.7182
This compounding of interest in a numeral $ dollar operation, makes the money grow. If we had to represent the compounding of the value of $ money in the bank, we could use the length of a broomstick, where $1 dollar would be a unit length along with the broomstick, while the continuous compounding at the end of the year would be shown by a length of 2.7182 hence $2.718
But compounding on a broomstick can take place along the length, AND ALSO IN QUADRATURE WITH THE LENGTH OF THE BROOMSTICK, where this would result in a rotation or an orientation of the straight long broomstick. So if we invest in rotating the unit length of the broomstick acting as a rotating vector, rather than elongating it along its length, then e^iA would be the final orientation of the unit broomstick after it terminates it compounding "rotational banking interest" rather than the conventional linear compounding which just changed its length!
All we have to do is to draw the vector e^iA where "i" indicates the quadrature direction of the compounding with respect to the length of the unit broomstick length 1, and A is the final compounded angle in radians reached where the unit length of the broomstick will remain a unit length.
Hence taking the components of the final rotational compounding e^iA on the two basic axes which we used as a reference, we have e^ iA= cos A +isin A.
I would say that the students would appreciate all this OPERATION which describes the real engineering actions behind the " compounding of interest" when the interest is linear or when it is (i. interest angular displacement ) the rotational interest on a unit length of a broomstick, or it could even be a $1 dollar paper sheet, wrapped up like a thin cigarette to represent the unit vector, and the wrapped up thin $ sheet will act as the rotating vector accumulating the rotating angular interest which finally compounds to e^iA.
I wager there are a few smiles after readers read all this! One would agree that the philosophy of compounding the $1 dollar paper sheet in a linear manner of its growth in monetary value, or its broomstick length representation, and the rotational angular manner, is exactly the same! where all that engineering actions and reactions are described so accurately in the meaning of the symbols in that expressions. We all need to learn what they mean in engineering actions and reactions and not only the rules of differentiation of the product function used in the video as ...... (the first function)multiplied by (the differential of the second function) plus (the second function) multiplied by (the differential of the first function). I always wonder how many people see the engineering/physical activity in that operation!!!!!!!!!!!!
Ok
I have known that e^i*pi = -1 for many years.
This video was the moment I understood that fact. Thank you.
That's surprising, considering how I was exposed to it was through the MacLaurin series. That answers it immaculately too. As I've matured more in mathematics, still quite infantile, I realized that 3B1B's videos are very visual. Though this video is more approachable, I'm often leaning towards traditional textbook like material with the leap of intuition than his videos that are ambitious visualizations.
im having the same experience
I was comfortable with this identiy before, but I had a great “AHA!” moment. This is probably the most concise and surprising way to look at it I’ve ever seen. Thanks a bunch
No wonder the Euler’s formula in Alan Becker’s “Animation Vs. Math” is given sentience, it’s all about position and movement. It can do so much.
"Multiplying by i has the effect of rotating numbers 90°" - Both lost and mind blown.
Think of multiplying by negative one as turning around to face backwards. If you do that twice you will be facing forwards again. That is (-1)*(-1)=1. Now if we ask what number times itself equals negative one, it will be a rotation that when done twice has you facing backwards, either a quarter turn counterclockwise i or a quarter turn clockwise -i. (the complex plane doesn't actually distinguish between clockwise and counterclockwise because that would require it physically be facing in some direction relative to us, and it is an abstract mathematical object, but we aren't abstract and generally draw i above the real number line and -i below, and numbers on the real number line getting larger as you move right)
@@paulfoss5385
So, for real numbers, |R, (-1)^2 = 1, and for complex numbers, |C, (i)^4 = 1. I think it's really cool that that exponent is off by a factor of 2 for |R and |C. I'm trying to think how this generalizes to quaternions (|Q) and maybe octonions (|O) but it's not obvious. |Q also holds that i^2=-1, same as |C, so it's still (i)^4=1. Octonions, well... lol.
I think he has a video about group theory that explains more about this rotation.
theres a really good series from another channel about imaginary numbers that goes into depth about it from the absolute basics, Id recommend giving it a watch
@@sdoman7215 WelchLabs is probably the channel you're thinking of.
Oh my god that makes so much sense.
Why was this never explained to me like this?
because it's wrong...
@@umair5602 Expand?
@@umair5602 go on
The great thing about math is that you don’t have to take anyone’s word for whether something is right or wrong. You can think through if this explanation makes sense for yourself (spoiler alert, it checks out). To engage with it that way will force you to think critically about what it means to take a derivative of a complex-valued function, which in turn builds good intuitions for complex analysis.
If e^πi=-1 then
π=ln(-1) / i
Then 2πr= 2r ln (-1) / i
= ln (-1^2r) / i
Which equals ln (1) / i
Which equals 0
Which means that if Euler's identity is true, the circumference of a circle is always 0.
*uses τ*
Ah, i see you are a man of culture aswell
I too was very pleased to see that.
You mean why not 2*pi?
isn't pi cultural lmaoo
*Uses both π and τ*
What a fantastic way of avoiding an argument!
Pi is better for pretty much every application
2:32 Good thing you initially drew more than 4 arrows to illustrate this point XD
r/angryupvote
people uploading lectures, and videos like this on youtube are literal gods
the e^iπ part comes right at 3.14 but also the e^i𝜏 part comes right at 3:14
nice
*i* wonder whats e^iφ
@@squibble311 if you are wondering about anything else,
Generally e^ix=cos(x)+isin(x)
There's a really easy explanation for this, it has to do with trig
@hiqwertyhi you've forgotten to explain that since a minute is only 60 seconds... 3.14 minutes is in fact about 3:08... And then tell readers to look closely in the bottom right corner of the video at that time... :-B
wow, actually amazing
3:14 is not π minutes into the video.
The idea that multiplying a number by i rotates it 90 degrees into an imaginary axis is probably the most important thing I've ever gone my whole life without knowing.
Wow, how did anyone understand maths before 3b1b
@@ASLUHLUHC3 They teach you polar coordinate multiplication literally in 11th Grade or so
After four years of studying electrical engineering in college, no one ever explained it that way until a professor casually mentioned it in a elective I took in my last semester :/ It would have helped so much to learn that earlier
"We're sorry, the number you have dialed is imaginary. Please rotate your phone 90 degrees and try again."
@@FranziskavonKarma They just say that cis(a)*cis(b) = cis(a+b). They don't go beyond that
gotta click fast when 3blue1brown posts
3B1B: *uploads *
Me: I have the fastest hand in the west.
QUICKLY!
Why?
Translation: Look how smart I am everybody! I watch 3B1B!
What's the rush? This guy takes time to make quality videos. It's not like there's one every day.
The only 4min video that actually led me to understand the dinamic behind Euler's identity. Well explained! Congratulations!
Kudos! This is pedagogy brought to the level of art. I hope your work, together with that of other excellent TH-cam educators, will pave the way for a new generation of teaching.
i've watched other 3blue1brown videos about this but i never really got it until now for some reason. holy crap it's so intuitive now, i don't know what i didn't get it before
His channel is pretty magical! But so is yours, so.
Xidnaf I loved your videos , I hope your doing well
That warm fuzzy feeling you get when one of your favorite youtubers watches the videos of another :))
Please make a new video
Oh hey Xidnaf, didn't expect you here. We miss you!
THAT WAS SUCH A GREAT EXPLANATION
I always struggled with that one, despite the fact you've made quite a few videos about it
this was so good though, it makes so much sense now
I gotta watch the other ones you've made to see if I can understand them better now
I did not understand why the yellow line (velocity ) was moving faster then the blow line (position ) even though he said the dervative is just equal to to the position at that time,.
Can u explain
@@hassanomer7777 it's not yellow it's green
@@hassanomer7777 It doesn't move faster. If the right side of green line moves at 2n, while left side moves at n, this means its length increases by n. The "speed boost" comes from blue line's end
It's amazing how helpful Grant's use of color is. This was particularly pronounced on his superb Fourier series videos. The old analog chalkboards of my youth did not have this added dimension.
"I'll spare that detail for the next video"
Almost 2 years passed, he's finally done it 😭😭😭
Bro, which one is that next video you are
referring to?
@@aarjith2580 The latest one. He hadn't touched the differential equation for 2 years.
@@chizhang2765 oh so the one about probability that he's yet to make is still a pretty long way away huh
stop getting so emotional
@@Fire_Axus stop being a dumbass
I can’t even stress how important your work is for me. You explain everything in such a beautiful and simple way; you were BORN to teach. Congratulations on all of your videos, sir.
To anyone who wants to think about something deeper with this concept:
Think about the function y=(-1)^x. Now this function is discontinuous, it’s not a line at all. It just seems to alternate between y=1 and y=-1 on the xy plane. But what happens if you include the complex numbers, say adding them in a z axis to make a three-dimensional graph. What you’ll find is that the function (-1)^x, or to write it another way e^(i*pi*x), traces out a line in 3d space. A line that is in the shape of a helix, a coil going around the x-axis, and intersecting the xy plane at 1 and -1 at appropriate points.
Hilfigertout woah.
And not just any helix, sinusoidal at that!
This is what I call high quality commenting.
@@nuralimedeu "woah" - Neo.
Wouldnt it be two helixes mirrored since the function is ambiguous on complex numbers?
It is incredible how far you've grown as an educator. Your first take on e to pi*i as your first video on a channel was really confusing to a lot of viewers, me included. But now you are providing an easily conprehensive explanations together with enjoyable visuals on this and many other topics. Thank you for all the work you're putting in your channel
U know he's awesome when ur math professor teaches u using his videos.
It's Magical !
The way you explain.
And I am not even a newcomer on this channel, yet my mind is always blown away by how you brilliantly decompose mathematical concepts to their core. So elegant, so beautiful.
This is the way to do it, the proper way to use technology in order to share science and wisdom.
So thankful for the Internet, that is allowing us to connect with such quality content.
So thankful for you, creators of such quality content.
Thank you for what you do !
Peace.
Interestingly, it's "creator" (singular). Just one guy does this
@@piman7319 Sure, but 3B1B is not the only great content creator on the Internet ;-)
You deserve a Nobel Prize for this video, such an intuitive explanation.
This is the simplest and most intuitive expression for this I have ever seen. Thank you. I'm a grad student in robotics so ODE's and PDE's are my life blood. So, this has always bothered me that the intuition for this concept was so difficult. With such a simple explanation I wonder why I have never seen this before.
Even with your previous videos about e^ipi I couldn't fully grasp WHY it behaves so. Why it has this undeniable link to rotations in the complex plane. As SOON as you brought derivatives into it, and e^x's nature of being its own derivative, it makes absolute perfect sense. Dude, you and blackpenredpen are the math teachers I needed so desperately when I was precalc in high school.
That is by far the best, and most intuitive explanation of Euler's formula I have ever seen. Thanks!
I think this way of teaching "visually" should be introduced in every school in the world... it's amazing how you manage to find the right animation for whatever concept, bravo!
I am constantly amazed at your ability to so elegantly show the beauty in mathematics. This was simply incredible.
I really like the buildup in this video. You start at derivatives and everything just blends into each other until you're expanding it further. From the effect of positive factors in the exponent, introducing the negative factor and eventually the imaginary factor. Definitely hard to understand if you don't know about the underlying concepts, but once you do, this almost feels like the most perfect way to explain e to the i pi. Great video
YeS! Last time i watched this i didn't know about e, now we have been learning it in school and i understood the introduction way better. 🙂
Holy shit I love this. So beautiful and intuitive. Already understood Euler’s formula and more or less why, but this really made me feel it in my gut
I was afraid that by 3.14 minutes he meant 3 minutes and 14 seconds, which is, of course, inaccurate. I'm glad that didn't happen!
Grant's too smart for that ;P
Which would instead be 3 minutes and ~8 seconds which doesn't really add up either. 🤷♀️ Not that it matters ultimately. It's still a really good and elegant explanation.
@@Gameboygenius look at the timestamp 3:08 in the bottom right hand corner
4 min 8 sec ≈ 4.13 min 🤔
@@conrada5 I missed that. So ok, fine I guess, but I feel that the explanation was not entirely finished at that point.
The simple explanation of why this works just blew my mind.
As someone who has been in science long enough to see this identity everywhere, I always hated that the only explanation I ever got was "It works because of Taylor expansion", with the demonstration separating the terms of the Taylor expansion of exp into those of cos and sin. It's just super abstract, like "It works bc of these random formulas that converge to the result after infinite steps".
Now I just learned about this way of seeing Euler's formula, was mindblown, did a quick YT search to see if anyone I watched had talked about it before...
And of f-ing course 3Blue1Brown did it. I'm kinda mad I never saw this video and also not surprised that you would use this representation that really feels like something you would find on this channel.
Such a cool way to see this otherwise very arbitrary-looking identity!
Yeah, when I first saw the definition of e^z = sin z + i cos z It seemed like someone had pulled a fast one. Even after it was shown that it's consistent with Taylor expansion, and so on. If I had started from the place of "multiplying by i is rotating 90 degrees" that might have helped. However, it is also the case that math is consistent and one explanation or proof is equivalent to another.
Explained my confusion about Euler s formula so elegantly! Thank you!
Which one?
I'm not first
and I'm not last
but when 3b1b uploads
I click fast.
I Meaaaan you were last.
@@fgvcosmic6752 have some patience
Someone summoned my name?
Roses are red
@@WackyAmoebatrons omfg
I wish I had this for my students a quarter century ago!!! Beautiful :-)
Can't wait for the full decomposition video! This was fantastic (many thanks from an electrical engineering student!)
One of the most beautiful mathematical relations explained in the most beautiful, insightful, and intuitive way. KUDOS !!!
Amazing video. I've never had it explained this way to me. I've always just understood E^jX is equivalent to the unit circle and accepted it as so.
As an electrical engineer, I learned AC circuit solving through complex exponentials (phasors and impedances), so I'm really excited to see this video.
(Also, the pace of the video relative to the usual 15-20 minute vids reminds me of the first video on the channel.)
You have the best visuals for any channel on TH-cam I've ever seen. Outstanding work!
First time I saw this explanation without requiring calculus. Neat.
After a year or so of using it, I can now officially say I fully understand Euler's Formula. Bless up. 🙌
Kurzgesagt AND 3Blue1Brown uploaded today! This is fantastic!
Also, I can't neglect to say, very elegant proof and brilliant explanation!
They should do a collab
It would be awesome if you did a series on modern numerical methods. Finite elements (for for elliptic problems) add Runge-Kutta methods for hyoerbolic and parabolic problems, etc... With your unique talent you could make this fascinating topic accesible to millions of talented young people who might chose to pursue a career in applied math. Thanks for all the great work!
You are freaking amazing, do you understand what you just did? you did an almost complete explantion of Euler's formula in few minutes that anyone with a bit of knolowede in caculus can understand.
You are the best math channel on TH-cam and one of my favorite channels in general.
I am glad people like 3Blue1Brown exist on TH-cam! I could learn all day and not have to pay $50K per year at college/university, which is now nothing but venue to buy a certificate that you have a BA/BS!
As a chap who dropped Maths as soon as possible, I thought I’d watch to see if even I could understand it in about 3 minutes.
I am actually pleasantly surprised. Seeming I never did differentiation I don’t understand the significance of what I’ve learnt, but it did make me smirk that i gets its own entire spatial dimension.
Watching 3b1b makes me feel like I am unlocking the unused 90% of my brain.
Bad move buddy.
So you believe the founder of Scientology?
The idea that we use only 10% of our brain is nonsense, and has no basis in science. The fact that a lot of people still believe this baffles me.
Exactly
haha exactly
3Blue1Brown can you do a video about the 7 millenium problems?
huge demand from my side
He'll prolly solve them ez lol
he did 1 on the riemman hypothesis
Then he'd accidentally solve them, and where's the fun in that?
I actually just come here after watching "Animation vs Math" 😅
This is the best explanation yet. Congrats
Such an intuitive explanation of a formula that has always seemed so abstract to me. I will now always have this concrete image in my head when I think of it. Thank you :)
I love how it dings at 3:08
I really apprecaite the 3.14 minutes being used correctly here, some people think 3.14 minutes would be the same as 3:14 and that's obviously false!
why
But the length of the video is 4:08 and 3.14 minutes should be 3:08
@@jorge8596 look at bottom right at 3:08
I thought at the end you'd show how, if you project the circular motion in the complex plane onto the real axis, you see simple harmonic motion. This could help high school students to relate better.
Btw I absolutely love this videos! Short but original and always make your concepts click no matter how much you think you know on the topic. Keep inspiring 3Blue1Brown!
Underrated comment
This without a doubt is the best explanation I've seen of this yet.
This series deserves the youtube subscription
The last bit, when you pointed out that e^iπ does not mean multiplying e by itself iπ times (even though this would have been the case if the exponent had been an integer, like e^3 = e*e*e) made me realize that it is the same thing that happens when you find an analytic continuation for a series. For instance, ζ(-1) = -1/12, but this equality does not mean that 1+2+3+...=-1/12, since the series for the Riemann Zeta function is only defined for numbers greater than 1. I had never realized before that even with simple arithmetic, such as powers, you can already get a feeling for the rule "you cannot always interpret new things using your old knowledge", because sometimes a blind substitution can be misleading.
That cleared up a whole lot more
ζ(e^iπ) = -1/12
never realized this before
That makes sense simply because the multiple of an imaginary is still an imaginary in the form ki. e isn't powered to anything in the real axis.
Is there a love button on TH-cam? There should be. A like is not enough.
Patreon is your love button
There's three. There's "subscribe", "share", and the link to Patreon.
no need to kiss his ass, you won't get a creator heart
I was researching e^ix= cos(x) + i*sin(x) after watching Lockdown math and wondered why there seemingly 'just happened to be' a neat connection between two quite different things: circles and e.
From what I understood on Wikipedia and forums, there was an explanation that involved the Maclaurin series of e, sin, & cos, and whilst it is a proof, it didn't satisfy my question. But then this video made the connection obvious (with the help of another 3b1b vid about e and calculus) and I can now say that I understand e^iπ=1 :)
This is beautiful! Has to be the best way to show e to the i pi = -1 that I've ever seen! Thank you!
This Chanel deserves all respect as possible
I've been arguing for a while that we should use physical models to teach math. You can represent addition and multiplication physically by superimposing 2 rulers on top of each other. Addition is a sliding action and multiplication is a scaling action, which you can represent by pulling them apart from each other while keeping the 0's overlapping and moving the 1 over whatever you're trying to scale up by. This makes complex numbers easier, since addition and multiplication work exactly the same way. This also makes it easier to go from only the nonnegative integers to the whole set of integers, since you can ask a question like "3+what=2" and it's not surprising that your incomplete number line has *something* that would answer that question, and seeing the structure laid out for you makes the negative numbers easier to use.
thats why the dude made the Manim library. you can easily make these math animations as a teacher, and shit works easily.
it uses python too, so its hella easy.
i wish i had this channel when i was doing engineering. kids these days spoilt af
I'm 12 and I tried to ignore this comment
Wow you are such a miracle being able to do engieering even without animations wow such impressive
@mario mario same
Ok boomer
@@ericvauwee4923 fr tho these videos are kinda nice to watch but for the exams and rigorosity in university they don’t really help tbh. Boomers now pretending we can watch 3 TH-cam videos and get our engineering degree lol
You are an incredible instructor. Thanks for what you do.
when you turned it 90 degrees, I was completely blown away. incredible stuff
Its always so fascinating when you explains it from another perspective
this is lovely to watch and listen to. thank you
Kudos on calculating 3.14 minutes correctly.
No. It was actually 3.08 minutes
@@Anonymous-df8itno, it was 3.14 minutes, which is 3:08
@@Nebulisuzer Should I delete my initial reply and this reply?
Just wondering who came for Alan Becker
I did.
Present!
Hi
exactly
Something that looks so daunting and complicated as eulers identity is actually pretty simple and elegant when broken down like this
The feeling of getting closer to the ultimate truths of our universe. Simply enlightening. Thank you so much.
first i thought a random video about e pi. why would i care, saw that often enough. then i noticed 3blue1brown and thought "im sure i dont know enough yet" 'click'
Brilliant. Absolutely brilliant. This is excellent teaching, as we've come to expect from your channel. Thank you!
how ins-'π'-ring
As-'τ'-nding
'τ'tally agree with that τ is as'τ'nding τ say otherwise is des'π'caple.
The Major That was awful
it's not a good science/math video unless someone makes awesome puns about the concepts involved
(-1)^1/2 186282mps what you did there.
Watching this again after just having done some problems involving resonant circuits, where Tau is used as a time constant, I just had the revelation that this is actually not unrelated to this Tau = 2*pi idea! Freakin beautiful.
You help more people with these videos than you could ever know!
the best explanation so far
Dang the idea of 'e' is truly blowing my mind!
e deserve more love than pi
E
Ice Medic!
Good job 3Blue1Brown...here's a pie 🥧
I see you're pandering to the tau-folk
Ew
π folk unite!
My little grad
shilling for BIG TAU
CS people on the sidelines using both while eating popcorn
This makes it so clear, it's astonishing. I had to immediately re-watch it just because it's so elegant. It also makes it obvious why eⁱᵗ=cos t + i sin t.
What a wonderful breakdown of that pesky identity that stumped my visual undersanding for so long. Incredible video, incredible channel!
Amazing! I never thought that there would be such a simple explanation to such a complicated equation. You are the best maths teacher!
I never liked e to the i pi, mostly because it seemed everyone was just so amazed over a statement. you could just accept it and integrate it into your math, or you could not. what I am more interested in is the proof, and what it means, but when a while ago several different youtubers started making different explanations for why e to the i pi equals -1, I was never quite satisfied. They always seemed to make some jump in comprehension. This jump is the same one that you just smoothed over beautifully, and I am in awe of how this problem that has been bugging me for like 2 years now has been explained in such a short period of time.
Still I have more questions like if the only way mathematicians can explain or compute complex exponents is through e or if there is some more generalized formula or explanation for complex exponents. Also I wonder if there is an explicit definition for what exponents mean in relationship to the complex plane, beyond e. This promise of another video excites me, because it might answer these questions where others did not, but even if it doesn't, I remain in awe of your skills in explanation, as well as very grateful for your videos, because I don't know where I would be if it wasn't for your essence of algebra and calculus series.
Euler's Formula is the generalized version for complex exponents. e^ix = cos x + i*sin x. Basically think of x as an angle in complex space with the same logic as discussed in the video. It's all very magical how so many separate things end up relating.
Who else got this recommended after seeing animation vs math?
I wish man i came here in the hopes of not failing mathematics
Never before have I understood a video about this identity, but the ONLY time I fell behind was your position vs. velocity notation at one point. You should be _incredibly_ proud.
This took my math lecturer about 30 minutes to explain, but you've done it in just 4 minutes!
Why am i coming to this after watching Animation vs Math?
Alan becker brought all of us here!! 😅
gave an out loud "holy shit" when the derivative was rotated by 90 degrees - that was really amazing
Been watching lots of videos showing how to prove that e^πi is equal -1 and this is far the most easy to understand explanation i ever had.
Finally, a video about this beautiful, elegant proof!
Orange learned this in seconds
Even though I know what he means, I wince a little whenever he says "Your velocity is twice your position"
Divided by one second... ^^
@Thomas Wilkinson They're not "just numbers", they're vectors, and since velocity is change in position per unit time, the units have to be different
@@PharoahJardin More like.. An arbitrary but constant time interval of your choice
Coordinates are implicitly measures of distance, which means you also need to include time
@Thomas Wilkinson I agree with the point about time having a "standard" unit. Though I'd clarify that the unit is essentially the number 1 on the number line. That is, a derivative of 2 with respect to t is a speed of 2 ones per one. The distance "one" on the number line is just the natural unit. So I'd say the unit is actually important here, as you can't measure distance in R^1 without the distinguished distance "one" (or some other distinguished distance) to relate it to.
Older video but just had to say that this was beautiful. Thank you 3blue1brown , this is actually truly amazing
I was taught that e^it = cost + isint but I couldn't for the life of me work out why. Hours on wikipedia led me nowhere. But this explained it brilliantly in 3 minutes.
One example: Taylor series. If you compare the Taylor series of e^x, cos(x) and sin(x) and insert it to x then you'll notice that e^(it) looks like cos(it) and i * sin(it) added together (granted, it has to be taken with a grain of salt given that it's an infinite sum but you still can see why that's true).
@@MarioFanGamer659 I saw that explanation yesterday when I had the same question, and whilst it is a proof, it still had me wondering why there 'just happened to be' a connection between two apparently quite different things - circles (& thus sin/cos) and e.
But then this video (with help from another by 3b1b about e and calculus) made the connection obvious, and I can now say that I understand e^iπ=1.
That's why I love this channel :)
Shouldn't this video have been released on 6/28? (Or 3/14?)
Or 22/7? Close enough anyway
@@juan3141 👍
Or 7/22, seeing as 22/7 is the second rational approximation of π, after 3.
(3, 22/7, 333/106, 355/113…)
Or 31/4, written in DD/MM/YYYY format instead of YYYY/MM/DD. Because surely you wouldn't use the absurd MM/DD/YYYY format, right? Right?!
@@GRBtutorials LAND OF THE FREE