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Complex Numbers Have More Uses Than You Think
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- Опубліковано 15 сер 2024
- Complex numbers are often seen as a mysterious or "advanced" number system mainly used for solving similarly mysterious or "advanced" problems. But really, once you get used to them, they're really an elegant and (ironically) simple mathematical tool with application to more down-to-earth problems besides Quantum Mechanics or advanced Differential Equations or something. Let's see what these numbers can do!
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=Chapters=
0:00 - Intro
1:45 - Complex number basics
3:31 - Interpreting complex number multiplication
7:55 - Angular velocity
10:42 - Calculating angular velocity using complex numbers
15:18 - Interpreting the formula
17:59 - More uses of complex numbers
19:48 - Special announcement!
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CREDITS
The music tracks used in this video are (in order of first appearance): "Dream Escape", "Checkmate", "Orient", "Rubix Cube", "Frozen in Love"
The track "Rubix Cube" comes courtesy of Audionautix.com
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The animations in this video were mostly made with a homemade Python library called "Morpho". If you want to play with it, you can find it here:
github.com/mor...
"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!
the animations are so clean that I almost forgot I was watching a math video I was so mesmerized 😭
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.
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
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?
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.
Hi, Morph. This is a really great video!
I also appreciate how you include well-written captions. Not every math channel does that.
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
Hand down the best explanation of complex arithmetic I’ve ever seen! Thanks for the video!
I just want to say that your videos are one of the most interesting thing in math UA-cam.
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
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?
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.
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. ^_^
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.
19:21 I would love to see a video about quaternions from you in the future! I loved this one!
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 👍
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, 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
I do look forward to quaternions also. Your unique viewpoint helped me to see more. Thks
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.
took a couple minutes, but i got it, and its absolutely elegant af
and intuitive
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.
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
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
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
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
Love these videos. So easy to understand and very informative. Can't wait for more to come
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!
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)
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.
First time I fully understood this topic. One of the most useful vídeos for me in internet.
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
love how you put the background in a dimmed yellow, so my eyes won't get tired
Thank you. Hope to see your quaternion video.
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!
I like this video
Makes me excited to learn more about it in my next semester
Love your videos! First time catching one on release
Videos left: hyperbolic numbers
quaternions
biquaternions
octonions
split-octonions
sedenions
trigintaduonions
and dual numbers.
cool
This video could not have been more timely for me, thank you Morphocular :D
Thanks! Please do the quaternion time derivative.
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
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.
Thanks for making the background audio stand back a little and not dominate your voiceover
Wow! Top notch content. Cannot wait to watch the quaternion video.
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.
Amazing
your channel is so underrated
Genius. Thank you. I find a great value in your videos
You’re as intelligent as you’re kind to us, it’s pleasure to be part of the journey of this channel
Amazing. Simply amazing and elegant presentation of this mathematical field. Keep on the excellent work!
I cant believe it was that easy to get the curvature of a 2d parametric line
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.
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"
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.
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
"Multiplying by a complex number rotates and scales a point; as opposed to real number multiplication which only scales."
Real numbers also rotate and scale with multiplication. The rotation is just taught by rote in a way that obscures what it is, as famously dramatized by Edward James Olmos in Stand and Deliver. "A negative times a negative equals a positive. Say it!"
best video on complex number for understanding its practical use! best
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.
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
Great video as always.
Also, was the "angle" at 1:30 an intended pun?
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
just a quick question: When you take the imaginary part of both sides to find the angular velocity, doesn't that imply that the term r'(t)/r(t)=0 when we take the real part? But that is clearly not true since the radius is constantly changing. What am I missing?
When taking the real part of the right-hand side, the real part of f'(t)/f(t) isn't necessarily zero. Instead, you get that it's actually equal to the real part of r'(t)/(t), and the two cancel out
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'.
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 video, hope to see some explanation about quaternions
loved it dude!
keep it up
greetings from brazil
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
I loved that step at 12:00 - genius!
The new thumbnail is much better
Excellent video on this topic, this also explains how rotation matrix works in computer graphics
Thank you.
Great video! The awesome visualizations helped me understand complex numbers a lot more 😃
This is how a FM demodutor is often implemented. Many times some filtering is included as well.
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.
I'm was in awe the whole time 😭
I think this formula needs to simplified a little bit further - omitting the "(t)" in front of all of these variables showing they are actually functions:
θ' = Im[(x' + iy')/(x + iy)], but this requires us to divide complex numbers in their rectangular (non-polar) form, which, if you don't want to use trig functions, requires one of those pieces of high school algebra I never quite remember. After you do, you get:
θ' = (xy' + x'y)/(x^2 + y^2), removing all evidence that you used complex numbers to get to this.
This is so beautiful , thank you so much! I am so grateful
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!
I think I would just use what they study as physics (according to some sources I have but won't tell you) to find that you should compute the cross product (see below) of the linear velocity and the position and divide by the squared distance. The cross product can be thought of as the dot product with one of them rotated 90° or the only non0 coordinate of it if we imagine them as vectors in a coordinate plane in 3D. The rotation is what I think the generalised vector product of n-1 vectors is for 2D vectors; both the rotated vector and the cross product used here are cases of my generalisation, which for m n-dimensional vectors gives a nxnx...xn (n-m n's) tensor that (for n-m>1), if you permute the dimensions, doesn't change for even permutations and flips its sign for odd ones (for n-m=2, that means that transposing a matrix it can give is the same as negating it)
Amazing work, I don't know what to say. I really, really appreciate it.
Please continue. Great things from small.
please don't stop making these videos
very inspiring video. Thank you for the masterpiece.
Thank you very much for reaching!
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 amazed me as someone who's only lightly dabbled in calculus.
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
An easier way to derive the identity in 13:35 is to directly compute f' using the product rule and divide through by f
Can't wait for a video on Quaternions
16:50 actually blew my mind
Please do a video on Quaternions
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)
I'll love it if you ever make a video like this on quaternions!
Your videos are sooooooooooo USEFUL! I know you Will say thank you
Beautiful, lucid. Similar to another math explaner in format, but without the affectation and twee.