Complex Numbers Have More Uses Than You Think

[@dylanparker130]

"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]

Probably because it's not true in a mathematics context that complex numbers are just for rotation

[@dylanparker130]

@@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?

[@regarrzo]

@@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?

[@dylanparker130]

@@regarrzo I was referring to systems with equilibria whose stability can be studied through the eigenvalues of an associated Jacobian Matrix.

[@regarrzo]

@@dylanparker130 Ahh, then I understand. Thanks for clearing it up!

[@theonearney205]

I would love to see a video on quaternions

[@DoxxTheMathGeek]

Me too! I love them, but I don't understand the polar-form.

[@soyjakchud]

same because i don’t understand them at all

[@lacryman5541]

Probably a series of videos

[@andremaldonado7410]

Also would like to see a video on quaternions

[@marcomoreno6748]

Strange we'd get this video before quaternions, given how widespread they are in applications.

[@marin3546]

Complex Analysis is such an interesting field, and I think everyone would love to see more on this topic. Great video!

[@happmacdonald]

I concur. Let's analyze this complex subject.

[@whatelseison8970]

I found a really excellent lecture playlist that covers the most important parts. kzbin.info/aero/PLMrJAkhIeNNQBRslPb7I0yTnES981R8Cg

[@goodguyamr6996]

the animations are so clean that I almost forgot I was watching a math video I was so mesmerized 😭

[@General12th]

Hi, Morph. This is a really great video! I also appreciate how you include well-written captions. Not every math channel does that.

[@kered13]

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.

[@MsKelvin99]

wow

[@mervynlarrier9424]

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?

[@bjornfeuerbacher5514]

@@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).

[@mgostIH]

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.

[@happmacdonald]

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. 😁

[@lumipakkanen3510]

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.

[@viliml2763]

@@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.

[@lumipakkanen3510]

@@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.

[@fotnite_]

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.

[@AzureLazuline]

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. ❤

[@krigermark]

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!

[@MatthewKelley-mq4ce]

I hope that paper went well.

[@Mathymagical]

Thanks! Please do the quaternion time derivative.

[@mega_mango]

I just want to say that your videos are one of the most interesting thing in math KZbin.

[@Chloe-ov2xr]

Hand down the best explanation of complex arithmetic I’ve ever seen! Thanks for the video!

[@aliberkozderya3112]

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

[@thermonn]

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.

[@janemcelroy6044]

19:21 I would love to see a video about quaternions from you in the future! I loved this one!

[@DoxxTheMathGeek]

I love complex numbers!

[@deadheat1635]

Same

[@ujjawalk6780]

Have seggsss

[@DoxxTheMathGeek]

@@ujjawalk6780 Should I have sex with complex numbers? I think it's going to take forever because there are so many of them.

[@FunnyAndCleverHandle]

I love undertime slopper!

[@DoxxTheMathGeek]

@@FunnyAndCleverHandle What's that?`Is that a guy on Tiktok?

[@aditya007asva]

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.

[@hanspeter5118]

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

[@lox7182]

Um what about theta = 0?

[@hanspeter5118]

for Θ=0 => y=0, regardless from which side you approach the x-axis so Θ' = y' / x which is the correct result

[@matthewrayner571]

Great video! As a physics student with a passion for maths, this was really interesting and useful to watch.

[@tmarvel4347]

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 👍

[@brandonprescott5525]

Wow! Top notch content. Cannot wait to watch the quaternion video.

[@michalnemecek3575]

6:12 Wikipedia gives an interesting proof of Euler's formula via the Taylor expansions of e^x, sin(x) and cos(x)

[@kevy1yt]

Thanks!

[@stevewhitt9109]

I do look forward to quaternions also. Your unique viewpoint helped me to see more. Thks

[@carteiraricaeducacaoeinter3882]

First time I fully understood this topic. One of the most useful vídeos for me in internet.

[@henryginn7490]

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

[@polishane8837]

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

[@InfiniteRegress]

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. ^_^

[@jonathandavis2731]

Love your videos! First time catching one on release

[@loicgeeraerts]

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.

[@ecologypig]

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.

[@danielcorrea2396]

love how you put the background in a dimmed yellow, so my eyes won't get tired

[@J.B.L2227]

Amazing your channel is so underrated

[@johnstuder847]

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!

[@elijahshadbolt7334]

Could checkout 3blue1brown's video on quaternions

[@dylanparker130]

I loved that step at 12:00 - genius!

[@bigpopakap]

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

[@DavidGrossman-js2xu]

I finally understand this video!! Dope

[@ihsanOWVALCS]

I like this video Makes me excited to learn more about it in my next semester

[@johanngerell]

Thanks for making the background audio stand back a little and not dominate your voiceover

[@SatnamMalhi-p7y]

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.

[@zemoxian]

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.

[@fourierfoyer365]

This video could not have been more timely for me, thank you Morphocular :D

[@evandrofilipe1526]

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

[@LeoDaLionEdits]

Love these videos. So easy to understand and very informative. Can't wait for more to come

[@person1082]

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)

[@madeline_alice]

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!

[@Howtheheckarehandleswit]

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!

[@tasnimul0096]

best video on complex number for understanding its practical use! best

[@bilel114]

Great video as always. Also, was the "angle" at 1:30 an intended pun?

[@mtate405]

Genius. Thank you. I find a great value in your videos

[@Tamir_Karniely]

Amazing. Simply amazing and elegant presentation of this mathematical field. Keep on the excellent work!

[@agrajyadav2951]

took a couple minutes, but i got it, and its absolutely elegant af

[@agrajyadav2951]

and intuitive

[@swordofstrife1174]

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

[@mauriciocarazzodec.209]

loved it dude! keep it up greetings from brazil

[@146fallon]

very inspiring video. Thank you for the masterpiece.

[@rique3012]

You’re as intelligent as you’re kind to us, it’s pleasure to be part of the journey of this channel

[@AriKath]

This is so beautiful , thank you so much! I am so grateful

[@Nusret15220]

Amazing work, I don't know what to say. I really, really appreciate it.

[@iamthebest2662]

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 .

[@arulprakash5420]

Excellent video on this topic, this also explains how rotation matrix works in computer graphics Thank you.

[@parthvarasani495]

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.

[@bjornfeuerbacher5514]

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.

[@MTGreat202]

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

[@xujingzhe82]

Thank you very much for reaching!

[@gijsb4708]

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.

[@sandipmaurya7371]

Loved and Subscribed from India

[@Craig31415]

Great video! The awesome visualizations helped me understand complex numbers a lot more 😃

[@pre-universitygeometricalg5862]

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.

[@EW-mb1ih]

very nice video, hope to see some explanation about quaternions

[@vinbo2232]

Thank you. Hope to see your quaternion video.

[@moodangelatx6580]

Please continue. Great things from small.

[@CreativeDimension]

The new thumbnail is much better

[@SynaTek240]

Wowwowwow, this is really good stuff. I'm in teh first year of my bachelor's studies so I was about to close the video cause it started from stuff I already knew, but man am I glad I just skipped to 10 minutes cause that trick is so cool. I can't believe that I hadn't seen this before.

[@shivamshukla1347]

Great Explanation 😄

[@Redingold]

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).

[@NaN_000]

Nice

[@whatelseison8970]

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.

[@tnk.2033]

please don't stop making these videos

[@plopgoot5458]

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

[@zafran156]

Your videos are sooooooooooo USEFUL! I know you Will say thank you

[@anvayjain4100]

I'm was in awe the whole time 😭

[@plekkchand]

Beautiful, lucid. Similar to another math explaner in format, but without the affectation and twee.

[@dionisiocarmoneto]

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!

[@tubebrocoli]

I'll love it if you ever make a video like this on quaternions!

[@federicomuciaccia9191]

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 :)

[@kilianklaiber6367]

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.

[@mixjzp4357]

Awesome video, neat explanation

[@elliotwilliams7523]

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.

[@gravysnake78]

I think I found one of my new favorite math fields

[@dyld921]

An easier way to derive the identity in 13:35 is to directly compute f' using the product rule and divide through by f

[@ominollo]

Beautiful video 🙂

[@Greenwood394]

16:50 actually blew my mind

[@ΠαναγιώτηςΓιόφτσος]

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?

[@marcospataro4223]

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

[@helene8854]

21:03 wait wait wait, are you also the guy that runs the KZbin channel Serpentine Integral? Those videos helped me so much when I was learning double integration

[@WAMTAT]

Please do a video on Quaternions

[@alejrandom6592]

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'.

[@jaafars.mahdawi6911]

Very well done. Keep up the spirit.

[@DataCraftsman]

17:38 - "You don't even have to be that clever to come up with this formula." - yes, yes you do XD

[@rarebeeph1783]

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.

[@Eniac42]

Can't wait for a video on Quaternions