"Complex Numbers are the language of 2D rotation" 7:54 My friend once asked for applications of imaginary numbers. My Dad (an Engineer) said, "They're just for rotation, aren't they?". I couldn't believe that none of my Maths Professors had ever put it that bluntly!
@@regarrzo The best suggestion I'd been able to make was that they showed in the analysis of a system's stability? Imaginary eigenvalues indicated oscillation, if I recall?
@@dylanparker130 I don't really know what is meant by asking for applications. Is your friend looking for an engineering/science perspective or a mathematical perspective? In science and engineering, imaginary numbers can simplify many calculations dealing with perioid things. In mathematics, they are interesting because of their properties alone, e.g. being an algebraically closed field, holomorphic functions being infinitely differentiable, ... I don't really understand what you mean with your comment. What kind of system are you referring to? Linear systems with matrix with imaginary eigenvalues?
The formula f'(x)/f(x) is called the logarithmic derivative, because it is also equal to the derivative of log(f(x)). It can be interpreted as a proportional rate of change. For example, a value that grows by a constant 10% per year has a constant logarithmic derivative, and the original function is an exponential. It is then interesting that this same formula appears for angular speed as well, though I think it makes intuitive sense if you think about it, since angular speed is the scale-invariant form of circular speed. The real part of the formula in the video should also corresponds to the proportional rate of change in the magnitude of f(x), so then we have a complete interpretation of the complex valued f'(x)/f(x) as encoding both the angular velocity and the growth rate of the magnitude.
In this example, we get that the w'= Im[f'(z)/f(z)]. How would you write this out though? The only way i can think to do it is taking the derivative of cos(t)+isin(t) and using that for the numerator, but doesn't that still end up suffering from the discontinuity problem he mentioned?
@@mervynlarrier9424 "How would you write this out though?" Use f = x + iy, then you can rewrite that as (x y' - y x') / (x² + y²). No cos and sin needed, only the function x(t) and y(t).
I hope you'll cover geometric algebra (Clifford Algebra) together with quaternions! Would be fun seeing them related and recover all this geometrical intuition in a single framework.
I've just been independently studying geometric algebra (blame Marc Ten Bosch literally dissing quaternions starting me down that rabbit hole) and Grassmann numbers/algebra (because of spinors in QM) only to find out that they come together at Clifford Algebra, so I should ought to learn about that whole situation next. 😁
Seconded. It's really cool to see how objects satisfying the axioms of quaternions arise out of geometric algebra. Gives them context. By themselves quaternions are rather mysterious and you have to wave your hands a lot to justify using four-dimensional objects to manipulate 3D coordinates.
@@lumipakkanen3510 Quaternions being equivalent to 3D rotors is really not all that a useful insight for practical applications, in fact it only causes confusion. The sooner everyone outside of pure maths forgets about quaternions the better, geometric algebra is a much better framework.
@@viliml2763 True from a fresh perspective. However we now have a history of using quaternions in 3D modeling, so bridging the gap is in order. There are also low-level arguments for using quaternions internally to save a few float multiplications even if the user interface speaks GA. Also remember that quaternions are a geometric algebra in their own right.
Just finished an intro complex analysis class at uni last semester, and I gotta say this is a really good way to explain this stuff. Kind of sad that Cauchy's Integral Formula didn't show up here, especially because it's related to the rotational velocity problem, but I understand why that might be a bit in-depth for a 20 minute video that already needs to spend most of its time explaining the rotational velocity problem.
i'm super rusty on my calculus... but the geometric interpretation afterwards is just *so* intuitive and brilliant! Thank you for making this video, and for all the others. ❤
I've been looking everywhere for uses of complex numbers for the single most important paper on my entire education. It's due in 3 days, and you sir, just saved my life. Awesome video!
Without teachings like this, found both on the internet and in good books, I would not be able study science. I am completely unable to learn by having a bunch of seemingly meaningless information being thrown at my face. Thanks a ton for sharing
While the "mysterious" angle formula arctan is indeed not continuous, the derivative actually is and yields the same result after short calculation: Θ' = (x y' - y x') / (x² + y²) No imaginary numbers needed, but the visual presentation is still worthy of a gold medal
Thank you so much!! As a senior in high school who is looking into studying maths and physics at university, your videos are an invaluable asset for sparking my curiosity and building my intuition for mathematics.
Even though it has been decades I touched or used mathematics. It facinates me to revisit the fundamentals of mathematics for a new perspective just for pure joy and appreciation of mathematics, which I feel I could not do justice a teenage student. Your video very elegantly explains it... Thanks for making such useful videos.
As someone who just completed a secondary school maths curriculum, these videos are perfect since I have just the right amount of prerequisite knowledge to understand what is meant by these videos
I used to want to extend every new thing I learned about complex numbers to quaternions. A few years ago when learning about how quaternions are useful for 3D rotations and more efficient than matrix rotations, I stumbled into geometric algebra. Now I need to know how everything I learn about complex numbers extend to geometric algebras! Fun fact is that complex numbers, quaternions, and vectors, and a bunch or hyper complex number systems are all subalgebras of geometric algebras. Plus other geometric numbers square to 1 and 0 turning circular rotation into hyperbolic rotation or translation. And they operate on any number of dimensions, not just 2 or 3.
WOW!!😍 You increased my affection towards "complex" numbers....though I like to call them "Frisky numbers" ....I personally find them pretty interesting like they play around in the plane like child🥰 keep it up 👍
Thanks very much for making this video. I didn't know that interpretation of multiplication by a complex number! it sounds a lot like the spectral decomposition of a matrix.
I felt a lot better about complex numbers after I took my first complex analysis course. They're really second nature to me now, and I just view them as the plane with a neat multiplication rather than something spooky and mysterious
Great video, it's nice to see a more original video introducing complex numbers rather than regurgitating the rules. I feel like those who like this video would also like "Are Complex Numbers Forced Upon Us? Multiplication in High Dimensions" by James Tanton, it shows their elegance nicely imo
Your videos are really great. Also, I love that you take the time to go through the interpretation of the formulas. This is unfortunately a step that is often missing in math classes. However, it would have been even better if you could have put circular arcs with a point (as is done with vectors) to represent the oriented angles. Also, don't forget to indicate the orientation of the plane, this may help some students. What you could also do is to treat the problem without using complexes and to show at the same time the power of complex numbers so that the viewer can measure the simplification that this brings.
There is simpler way to obtain theta'. Since f = r*exp(i*theta), we know that ln(f) = ln(r) + i*theta differentiating both sides we get f'/f = r'/r + i*theta' and from here it is really easy to solve for theta'.
dude! I wish I would've came across this video before Signals and Systems class, I could've gotten a better grade! dang! It's sooo good, this 20 min video would've made an entire semester easier.
Morphocular, your topics and videos are always so great! Thanks so much for the work you put into them! I can't help but add, for anyone interested in the Riemann Zeta function and its mythical nontrivial zeros and understanding how to find them, the mentions of polar parametric functions and epicycles at the end of the video are incredibly useful. Just take a peek at the Dirichlet Eta Function and its amazing relationship with the Riemann Zeta function. ^_^
i can be rewritten as the product of the x and y basis vector, defined such that xy=-yx, x^2=1, and y^2=1 multiplying vectors by i has the same effect as multiplying a complex number by i for example to rotate 2x+3y a quarter turn, we can do (2x+3y)xy=2xxy+3yxy=2y-3xyy=-3x+2y it gives a nice geometric interpretation of i as a plane (bivector)
Great video. Love to see more on complex numbers, Fourier, epicycles, and quaternions 3D rotations…and General Stokes differential forms if you are into that. Thank you!
OOOOOOOH, I'd love a vide from you on quarternions! I loved the ones from Numberphile and 3b1b, but i think your beautiful visualizations and skill for revealing intuition will be a great addition to the topic
In Geometric Algebra (which is a development of Clifford Algebra), the unit imaginary is given a geometric interpretation that is extremely useful in formulating and solving mathematical problems that arise in a broad range of fields, including quantum mechanics (as well as in high-school-level physics). Our channel is mainly for lower-level users of GA, but some of the members of our associated LinkedIn group are GA experts, and will be happy to direct interested viewers to sources of additional information.
Really cool video and well done on the channel explosion, I would really love to see how geometric algebra can explain rotations in not only three but n dimensions, multi vectors ftw
the thumbnail expresses angular velocity as the imaginary part of f'/f, where f is a complex-valued function of time, which does intuitively make sense. but what makes even more sense to me is expressing it as the magnitude of d/dt f/|f|, where f is a vector-valued function of time. this generalizes to n dimensions and doesn't rely on an implicit rotation of reference frame.
Your visuals are excellent and so helpful. Motivation is so important to learning math and you have hit the nail on the head with this video. Thank you!
That's pretty cool! The underlying trick this is based on is called logarithmic differentiation. The central statement of which is: f'(x)/f(x) = d(ln(|f(x)|)/dx. This is easily to see by using the chain rule on the RHS. The real part of the complex logarithm is simply the logarithm of the absolute value of the input, while the imaginary part is precisely the argument. As an interesting aside, if you exponentiate the logarithmic derivative you get something called the geometric derivative, which can also be defined as lim h-->0 [ (f(x*h)/f(x))^(1/h) ] -- so it's like the usual derivative but with each of the operations ratcheted up by one degree. Not really relevant to this application but I find it interesting.
your knowledge and experience help to understand a lot. appreciate a lot. kindly make such beautiful videos. we will also support from our side as much we can as students.
One trick I would say to anyone who can’t figure out how operations with complex numbers work is to just rewrite i as sqrt(-1) and treat is as you would any other square root. It the same thing, but you already have the intuition for working with roots and sometimes the new notation can be intimidating at first and make you forget the main definition of i
Loved your video. I just have started Learning complex numbers in high school and getting to learn so much about it made me mad curious to learn more about it .
I've loved every one of your videos so far, and I'm excited to see where you take the channel in the future! I wish I was in a position where I could join your patreon, perhaps someday. In the meantime, keep up the great work!
A calculation that is just as intuitive (as long as you know vector calculus), and that does not use complex numbers, is taking the norm of the time derivative of the normalized position. By normalizing the position one omits the magnitude, leaving us with only the directional information. Then taking the time derivative of this function gives us an "angular velocity vector", from which we can get the angular velocity by taking its norm. One can then show, using relatively simple vector calculus differentation rules, that this calculation is actually equal to another well-known way to get the angular velocity: that is dividing the cross-radial (or tangential) component of the velocity by the radius (as physicists would do), which is also discussed in the video.
This is a great video but the beginning feels like a personals ad for complex numbers you're like "no no you just don't get him, he's actually a great guy"
I like to think of it like this: We have f(t) = r(t)e^iθ(t). We want θ'(t), but θ is currently up in that exponential, so if we want to get it down, we should use logs. Log of a product is sum of the logs, and the log of an exponential is just the exponent, so that gives us ln(f(t)) = ln(r(t)) + iθ(t). Now we have θ(t), we differentiate to find θ'(t), giving us d/dt ln(f(t)) = f'(t)/f(t) = r'(t) / r(t) + iθ'(t), cause the chain rule still applies. Now we just take the imaginary part to isolate θ'(t), giving us Im(f'(t)/f(t)) = θ'(t).
21:03 wait wait wait, are you also the guy that runs the UA-cam channel Serpentine Integral? Those videos helped me so much when I was learning double integration
Sir, your explanations are pretty, really nice. You explain in a very clear way. I can imagine how long it takes for you to produce a video like this. Congratulations Friend, for your effort. I am an observer [economist] from Brazil! I do not know where you are!
The steps from 11:40 to 14:40 can be done much faster: Take the ln of the formula at 11:40, giving ln(f) = ln(r) + i theta. Then take the derivative, giving f'/f = r'/r + i theta'. Then take the imaginary part on both sides, and you have the result.
Here is a little simpler approach for d/dt \theta(t) = Im{ f’(t) / f(t)}. Since f(t) = r(t) exp(i \theta(t) ), wehave log (f(t)) = log(r(t)) + i \theta(t). The derivative of log(f(t)) gives the expression for d/dt \theta
Very nice. Although I have used complex numbers a whole lot, I find this explanation quite enlightening. The presentation is wonderful. In my mind, the complex numbers are just the same as the 2D-vector space R^2, on which a particular multiplication is defined. Then, nothing is really imaginary and i^2 = -1 is just a short hand representation of the multiplication of these vectors. Then, nothing imaginary is left and we are not in fact stating that some strange number multiplied by itself is equal to minus 1. I had to figure this out in order to make sense of complex numbers and soon after that I found out that lots of other people had figured this out before me. But, for some strange reason noone ever explained it to me in this manner.
This video is your pure gold! It will be very very VERY useful for the work that I'm doing... Thanks you a lot for publishing it :) I would just like to warn you regarding the last statement you made about quaternions... As you said, that unit complex numbers describe 2D rotations. This is true, because unit quaternions map the U(1) algebra, which is isomorphic to SO(2) which in turn describes the regular 2D rotations. However, the unit quaternions do NOT map the 3D rotations. This is an historical misconception, dating back to the actual "invention" of questions. Unit quaternions map the SU(2) algebra, which is NOT perfectly isomorphic to the SO(3) algebra of regular 3D rotations. For example, there is the famous "problem" of the required 4*pi rotation (instead of 2*pi) to restore the correct sign. So, unit quaternions do NOT map 3D rotations. The two algebras are intrinsically different. The fundamental representation (2) of the group SU(2) describes "spinors", while the fundamental representation (3) of SO(3) describes "vectors". They are NOT the same object. Just like bosons are not fermions and/or matter is not radiation (two topics in Physics which are somehow related to the groups/algebras above). Apart from that, thank you again for this marvelous video :)
this hwas awesome, i didn't know that you could find angular velocity like this. i hope for another great video explaning quarternions and maybe also a video on others like the split-complex numbers and tessarines
As a CG person, I would LOVE to see a really good description of quaternion rotation, and how to picture it. (So I can put better 'drivers' on the rotation in the animation curves)
"Complex Numbers are the language of 2D rotation" 7:54
My friend once asked for applications of imaginary numbers. My Dad (an Engineer) said, "They're just for rotation, aren't they?". I couldn't believe that none of my Maths Professors had ever put it that bluntly!
Probably because it's not true in a mathematics context that complex numbers are just for rotation
@@regarrzo The best suggestion I'd been able to make was that they showed in the analysis of a system's stability? Imaginary eigenvalues indicated oscillation, if I recall?
@@dylanparker130 I don't really know what is meant by asking for applications. Is your friend looking for an engineering/science perspective or a mathematical perspective?
In science and engineering, imaginary numbers can simplify many calculations dealing with perioid things. In mathematics, they are interesting because of their properties alone, e.g. being an algebraically closed field, holomorphic functions being infinitely differentiable, ...
I don't really understand what you mean with your comment. What kind of system are you referring to? Linear systems with matrix with imaginary eigenvalues?
@@regarrzo I was referring to systems with equilibria whose stability can be studied through the eigenvalues of an associated Jacobian Matrix.
@@dylanparker130 Ahh, then I understand. Thanks for clearing it up!
I would love to see a video on quaternions
Me too! I love them, but I don't understand the polar-form.
same because i don’t understand them at all
Probably a series of videos
Also would like to see a video on quaternions
Strange we'd get this video before quaternions, given how widespread they are in applications.
the animations are so clean that I almost forgot I was watching a math video I was so mesmerized 😭
Complex Analysis is such an interesting field, and I think everyone would love to see more on this topic. Great video!
I concur. Let's analyze this complex subject.
I found a really excellent lecture playlist that covers the most important parts. ua-cam.com/play/PLMrJAkhIeNNQBRslPb7I0yTnES981R8Cg.html
Hi, Morph. This is a really great video!
I also appreciate how you include well-written captions. Not every math channel does that.
The formula f'(x)/f(x) is called the logarithmic derivative, because it is also equal to the derivative of log(f(x)). It can be interpreted as a proportional rate of change. For example, a value that grows by a constant 10% per year has a constant logarithmic derivative, and the original function is an exponential. It is then interesting that this same formula appears for angular speed as well, though I think it makes intuitive sense if you think about it, since angular speed is the scale-invariant form of circular speed. The real part of the formula in the video should also corresponds to the proportional rate of change in the magnitude of f(x), so then we have a complete interpretation of the complex valued f'(x)/f(x) as encoding both the angular velocity and the growth rate of the magnitude.
wow
In this example, we get that the w'= Im[f'(z)/f(z)]. How would you write this out though? The only way i can think to do it is taking the derivative of cos(t)+isin(t) and using that for the numerator, but doesn't that still end up suffering from the discontinuity problem he mentioned?
@@mervynlarrier9424 "How would you write this out though?"
Use f = x + iy, then you can rewrite that as (x y' - y x') / (x² + y²). No cos and sin needed, only the function x(t) and y(t).
I hope you'll cover geometric algebra (Clifford Algebra) together with quaternions! Would be fun seeing them related and recover all this geometrical intuition in a single framework.
I've just been independently studying geometric algebra (blame Marc Ten Bosch literally dissing quaternions starting me down that rabbit hole) and Grassmann numbers/algebra (because of spinors in QM) only to find out that they come together at Clifford Algebra, so I should ought to learn about that whole situation next. 😁
Seconded. It's really cool to see how objects satisfying the axioms of quaternions arise out of geometric algebra. Gives them context. By themselves quaternions are rather mysterious and you have to wave your hands a lot to justify using four-dimensional objects to manipulate 3D coordinates.
@@lumipakkanen3510
Quaternions being equivalent to 3D rotors is really not all that a useful insight for practical applications, in fact it only causes confusion.
The sooner everyone outside of pure maths forgets about quaternions the better, geometric algebra is a much better framework.
@@viliml2763 True from a fresh perspective. However we now have a history of using quaternions in 3D modeling, so bridging the gap is in order. There are also low-level arguments for using quaternions internally to save a few float multiplications even if the user interface speaks GA. Also remember that quaternions are a geometric algebra in their own right.
Just finished an intro complex analysis class at uni last semester, and I gotta say this is a really good way to explain this stuff. Kind of sad that Cauchy's Integral Formula didn't show up here, especially because it's related to the rotational velocity problem, but I understand why that might be a bit in-depth for a 20 minute video that already needs to spend most of its time explaining the rotational velocity problem.
i'm super rusty on my calculus... but the geometric interpretation afterwards is just *so* intuitive and brilliant! Thank you for making this video, and for all the others. ❤
I've been looking everywhere for uses of complex numbers for the single most important paper on my entire education. It's due in 3 days, and you sir, just saved my life. Awesome video!
I hope that paper went well.
Without teachings like this, found both on the internet and in good books, I would not be able study science. I am completely unable to learn by having a bunch of seemingly meaningless information being thrown at my face.
Thanks a ton for sharing
While the "mysterious" angle formula arctan is indeed not continuous, the derivative actually is and yields the same result after short calculation:
Θ' = (x y' - y x') / (x² + y²)
No imaginary numbers needed, but the visual presentation is still worthy of a gold medal
Um what about theta = 0?
for Θ=0 => y=0, regardless from which side you approach the x-axis
so Θ' = y' / x
which is the correct result
6:12 Wikipedia gives an interesting proof of Euler's formula via the Taylor expansions of e^x, sin(x) and cos(x)
Thank you so much!! As a senior in high school who is looking into studying maths and physics at university, your videos are an invaluable asset for sparking my curiosity and building my intuition for mathematics.
I just want to say that your videos are one of the most interesting thing in math UA-cam.
Hand down the best explanation of complex arithmetic I’ve ever seen! Thanks for the video!
19:21 I would love to see a video about quaternions from you in the future! I loved this one!
love how you put the background in a dimmed yellow, so my eyes won't get tired
Even though it has been decades I touched or used mathematics. It facinates me to revisit the fundamentals of mathematics for a new perspective just for pure joy and appreciation of mathematics, which I feel I could not do justice a teenage student.
Your video very elegantly explains it... Thanks for making such useful videos.
As someone who just completed a secondary school maths curriculum, these videos are perfect since I have just the right amount of prerequisite knowledge to understand what is meant by these videos
I used to want to extend every new thing I learned about complex numbers to quaternions. A few years ago when learning about how quaternions are useful for 3D rotations and more efficient than matrix rotations, I stumbled into geometric algebra.
Now I need to know how everything I learn about complex numbers extend to geometric algebras!
Fun fact is that complex numbers, quaternions, and vectors, and a bunch or hyper complex number systems are all subalgebras of geometric algebras.
Plus other geometric numbers square to 1 and 0 turning circular rotation into hyperbolic rotation or translation. And they operate on any number of dimensions, not just 2 or 3.
Great video! As a physics student with a passion for maths, this was really interesting and useful to watch.
WOW!!😍 You increased my affection towards "complex" numbers....though I like to call them "Frisky numbers" ....I personally find them pretty interesting like they play around in the plane like child🥰
keep it up 👍
First time I fully understood this topic. One of the most useful vídeos for me in internet.
I do look forward to quaternions also. Your unique viewpoint helped me to see more. Thks
I love complex numbers!
Same
Have seggsss
@@ujjawalk6780 Should I have sex with complex numbers? I think it's going to take forever because there are so many of them.
I love undertime slopper!
@@FunnyAndCleverHandle What's that?`Is that a guy on Tiktok?
Thanks very much for making this video. I didn't know that interpretation of multiplication by a complex number! it sounds a lot like the spectral decomposition of a matrix.
Wow! Top notch content. Cannot wait to watch the quaternion video.
took a couple minutes, but i got it, and its absolutely elegant af
and intuitive
Thanks for making the background audio stand back a little and not dominate your voiceover
I felt a lot better about complex numbers after I took my first complex analysis course. They're really second nature to me now, and I just view them as the plane with a neat multiplication rather than something spooky and mysterious
Great video, it's nice to see a more original video introducing complex numbers rather than regurgitating the rules. I feel like those who like this video would also like "Are Complex Numbers Forced Upon Us? Multiplication in High Dimensions" by James Tanton, it shows their elegance nicely imo
Your videos are really great.
Also, I love that you take the time to go through the interpretation of the formulas. This is unfortunately a step that is often missing in math classes.
However, it would have been even better if you could have put circular arcs with a point (as is done with vectors) to represent the oriented angles.
Also, don't forget to indicate the orientation of the plane, this may help some students.
What you could also do is to treat the problem without using complexes and to show at the same time the power of complex numbers so that the viewer can measure the simplification that this brings.
You’re as intelligent as you’re kind to us, it’s pleasure to be part of the journey of this channel
There is simpler way to obtain theta'.
Since f = r*exp(i*theta),
we know that ln(f) = ln(r) + i*theta
differentiating both sides we get
f'/f = r'/r + i*theta' and from here it is really easy to solve for theta'.
Love your videos! First time catching one on release
dude! I wish I would've came across this video before Signals and Systems class, I could've gotten a better grade! dang! It's sooo good, this 20 min video would've made an entire semester easier.
Morphocular, your topics and videos are always so great! Thanks so much for the work you put into them!
I can't help but add, for anyone interested in the Riemann Zeta function and its mythical nontrivial zeros and understanding how to find them, the mentions of polar parametric functions and epicycles at the end of the video are incredibly useful. Just take a peek at the Dirichlet Eta Function and its amazing relationship with the Riemann Zeta function. ^_^
i can be rewritten as the product of the x and y basis vector, defined such that xy=-yx, x^2=1, and y^2=1
multiplying vectors by i has the same effect as multiplying a complex number by i
for example to rotate 2x+3y a quarter turn, we can do (2x+3y)xy=2xxy+3yxy=2y-3xyy=-3x+2y
it gives a nice geometric interpretation of i as a plane (bivector)
I finally understand this video!! Dope
I loved that step at 12:00 - genius!
Great video. Love to see more on complex numbers, Fourier, epicycles, and quaternions 3D rotations…and General Stokes differential forms if you are into that. Thank you!
Could checkout 3blue1brown's video on quaternions
This video could not have been more timely for me, thank you Morphocular :D
Thanks! Please do the quaternion time derivative.
I like this video
Makes me excited to learn more about it in my next semester
please don't stop making these videos
OOOOOOOH, I'd love a vide from you on quarternions! I loved the ones from Numberphile and 3b1b, but i think your beautiful visualizations and skill for revealing intuition will be a great addition to the topic
Amazing
your channel is so underrated
Thank you very much for reaching!
In Geometric Algebra (which is a development of Clifford Algebra), the unit imaginary is given a geometric interpretation that is extremely useful in formulating and solving mathematical problems that arise in a broad range of fields, including quantum mechanics (as well as in high-school-level physics). Our channel is mainly for lower-level users of GA, but some of the members of our associated LinkedIn group are GA experts, and will be happy to direct interested viewers to sources of additional information.
I saw the last part of your video with the future topics list. Please do the calculus of variations. There aren’t enough good videos on the topic.
Really cool video and well done on the channel explosion, I would really love to see how geometric algebra can explain rotations in not only three but n dimensions, multi vectors ftw
the thumbnail expresses angular velocity as the imaginary part of f'/f, where f is a complex-valued function of time, which does intuitively make sense. but what makes even more sense to me is expressing it as the magnitude of d/dt f/|f|, where f is a vector-valued function of time. this generalizes to n dimensions and doesn't rely on an implicit rotation of reference frame.
The new thumbnail is much better
best video on complex number for understanding its practical use! best
Your visuals are excellent and so helpful. Motivation is so important to learning math and you have hit the nail on the head with this video. Thank you!
Thank you. Hope to see your quaternion video.
Excellent video on this topic, this also explains how rotation matrix works in computer graphics
Thank you.
Genius. Thank you. I find a great value in your videos
Love these videos. So easy to understand and very informative. Can't wait for more to come
That's pretty cool! The underlying trick this is based on is called logarithmic differentiation. The central statement of which is: f'(x)/f(x) = d(ln(|f(x)|)/dx. This is easily to see by using the chain rule on the RHS.
The real part of the complex logarithm is simply the logarithm of the absolute value of the input, while the imaginary part is precisely the argument.
As an interesting aside, if you exponentiate the logarithmic derivative you get something called the geometric derivative, which can also be defined as lim h-->0 [ (f(x*h)/f(x))^(1/h) ] -- so it's like the usual derivative but with each of the operations ratcheted up by one degree. Not really relevant to this application but I find it interesting.
your knowledge and experience help to understand a lot. appreciate a lot. kindly make such beautiful videos. we will also support from our side as much we can as students.
One trick I would say to anyone who can’t figure out how operations with complex numbers work is to just rewrite i as sqrt(-1) and treat is as you would any other square root. It the same thing, but you already have the intuition for working with roots and sometimes the new notation can be intimidating at first and make you forget the main definition of i
Stop making me excited for learning calc! Just one more year before it begins. Also love the animations and how these topics always tie up in the end
Loved your video. I just have started Learning complex numbers in high school and getting to learn so much about it made me mad curious to learn more about it .
I think I found one of my new favorite math fields
I've loved every one of your videos so far, and I'm excited to see where you take the channel in the future! I wish I was in a position where I could join your patreon, perhaps someday. In the meantime, keep up the great work!
Amazing. Simply amazing and elegant presentation of this mathematical field. Keep on the excellent work!
Great video as always.
Also, was the "angle" at 1:30 an intended pun?
very nice video, hope to see some explanation about quaternions
Please make a video (series?) on calculus of variations. This is a wide open hole that hasn’t really been covered yet on UA-cam to my knowledge
A calculation that is just as intuitive (as long as you know vector calculus), and that does not use complex numbers, is taking the norm of the time derivative of the normalized position. By normalizing the position one omits the magnitude, leaving us with only the directional information. Then taking the time derivative of this function gives us an "angular velocity vector", from which we can get the angular velocity by taking its norm. One can then show, using relatively simple vector calculus differentation rules, that this calculation is actually equal to another well-known way to get the angular velocity: that is dividing the cross-radial (or tangential) component of the velocity by the radius (as physicists would do), which is also discussed in the video.
This is so beautiful , thank you so much! I am so grateful
This is a great video but the beginning feels like a personals ad for complex numbers you're like "no no you just don't get him, he's actually a great guy"
very inspiring video. Thank you for the masterpiece.
I like to think of it like this:
We have f(t) = r(t)e^iθ(t). We want θ'(t), but θ is currently up in that exponential, so if we want to get it down, we should use logs. Log of a product is sum of the logs, and the log of an exponential is just the exponent, so that gives us ln(f(t)) = ln(r(t)) + iθ(t). Now we have θ(t), we differentiate to find θ'(t), giving us d/dt ln(f(t)) = f'(t)/f(t) = r'(t) / r(t) + iθ'(t), cause the chain rule still applies. Now we just take the imaginary part to isolate θ'(t), giving us Im(f'(t)/f(t)) = θ'(t).
Nice
Beautiful, lucid. Similar to another math explaner in format, but without the affectation and twee.
I'll love it if you ever make a video like this on quaternions!
This is how a FM demodutor is often implemented. Many times some filtering is included as well.
21:03 wait wait wait, are you also the guy that runs the UA-cam channel Serpentine Integral? Those videos helped me so much when I was learning double integration
Sir, your explanations are pretty, really nice. You explain in a very clear way. I can imagine how long it takes for you to produce a video like this. Congratulations Friend, for your effort. I am an observer [economist] from Brazil! I do not know where you are!
Can't wait for a video on Quaternions
The steps from 11:40 to 14:40 can be done much faster: Take the ln of the formula at 11:40, giving ln(f) = ln(r) + i theta. Then take the derivative, giving f'/f = r'/r + i theta'. Then take the imaginary part on both sides, and you have the result.
Videos left: hyperbolic numbers
quaternions
biquaternions
octonions
split-octonions
sedenions
trigintaduonions
and dual numbers.
cool
I'm was in awe the whole time 😭
Here is a little simpler approach for d/dt \theta(t) = Im{ f’(t) / f(t)}. Since f(t) = r(t) exp(i \theta(t) ), wehave log (f(t)) = log(r(t)) + i \theta(t). The derivative of log(f(t)) gives the expression for d/dt \theta
loved it dude!
keep it up
greetings from brazil
Please continue. Great things from small.
Very nice. Although I have used complex numbers a whole lot, I find this explanation quite enlightening. The presentation is wonderful.
In my mind, the complex numbers are just the same as the 2D-vector space R^2, on which a particular multiplication is defined. Then, nothing is really imaginary and i^2 = -1 is just a short hand representation of the multiplication of these vectors. Then, nothing imaginary is left and we are not in fact stating that some strange number multiplied by itself is equal to minus 1.
I had to figure this out in order to make sense of complex numbers and soon after that I found out that lots of other people had figured this out before me. But, for some strange reason noone ever explained it to me in this manner.
This video is your pure gold!
It will be very very VERY useful for the work that I'm doing...
Thanks you a lot for publishing it :)
I would just like to warn you regarding the last statement you made about quaternions...
As you said, that unit complex numbers describe 2D rotations. This is true, because unit quaternions map the U(1) algebra, which is isomorphic to SO(2) which in turn describes the regular 2D rotations.
However, the unit quaternions do NOT map the 3D rotations. This is an historical misconception, dating back to the actual "invention" of questions. Unit quaternions map the SU(2) algebra, which is NOT perfectly isomorphic to the SO(3) algebra of regular 3D rotations.
For example, there is the famous "problem" of the required 4*pi rotation (instead of 2*pi) to restore the correct sign.
So, unit quaternions do NOT map 3D rotations. The two algebras are intrinsically different.
The fundamental representation (2) of the group SU(2) describes "spinors", while the fundamental representation (3) of SO(3) describes "vectors". They are NOT the same object. Just like bosons are not fermions and/or matter is not radiation (two topics in Physics which are somehow related to the groups/algebras above).
Apart from that, thank you again for this marvelous video :)
16:50 actually blew my mind
I cant believe it was that easy to get the curvature of a 2d parametric line
Amazing work, I don't know what to say. I really, really appreciate it.
Great video! The awesome visualizations helped me understand complex numbers a lot more 😃
this hwas awesome, i didn't know that you could find angular velocity like this.
i hope for another great video explaning quarternions and maybe also a video on others like the split-complex numbers and tessarines
Loved and Subscribed from India
An easier way to derive the identity in 13:35 is to directly compute f' using the product rule and divide through by f
top tier math content
As a CG person, I would LOVE to see a really good description of quaternion rotation, and how to picture it. (So I can put better 'drivers' on the rotation in the animation curves)
Awesome video, neat explanation