Complex Numbers Have More Uses Than You Think

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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/morpho-matters/mor...

Опубликовано:

28 июн 2024

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Комментарии : 449

@dylanparker130 Год назад

"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 Год назад

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

@dylanparker130 Год назад

@@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 Год назад

@@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 Год назад

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

@regarrzo Год назад

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

@theonearney205 Год назад

I would love to see a video on quaternions

@DoxxTheMathGeek Год назад

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

@ikilledaman Год назад

same because i don’t understand them at all

@lacryman5541 Год назад

Probably a series of videos

@andremaldonado7410 Год назад

Also would like to see a video on quaternions

@marcomoreno6748 Год назад

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

@goodguyamr6996 Год назад

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

@kered13 Год назад

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 9 месяцев назад

wow

@mervynlarrier9424 5 месяцев назад

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?

@mgostIH Год назад

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 Год назад

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 Год назад

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 Год назад

@@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 Год назад

@@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.

@marin3546 Год назад

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

@happmacdonald Год назад

I concur. Let's analyze this complex subject.

@whatelseison8970 Год назад

I found a really excellent lecture playlist that covers the most important parts. ru-vid.com/group/PLMrJAkhIeNNQBRslPb7I0yTnES981R8Cg

@General12th Год назад

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

@fotnite_ Год назад

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 Год назад

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 Год назад

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!

@Chloe-ov2xr Год назад

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

@charlieb6210 Год назад

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!

@thermon6945 Год назад

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.

@LeoDaLionEdits Год назад

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

@mega_mango Год назад

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

@jonathandavis2731 Год назад

Love your videos! First time catching one on release

@InfiniteRegress Год назад

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. ^_^

@fourierfoyer365 Год назад

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

@tamirkarniely6913 Год назад

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

@Howtheheckarehandleswit Год назад

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!

@matthewrayner571 Год назад

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

@aliberkozderya3112 Год назад

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

@DoxxTheMathGeek Год назад

I love complex numbers!

@deadheat1635 Год назад

Same

@ujjawalk6780 Год назад

Have seggsss

@DoxxTheMathGeek Год назад

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

@FunnyAndCleverHandle Год назад

I love undertime slopper!

@DoxxTheMathGeek Год назад

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

@janemcelroy6044 Год назад

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

@aditya007asva Год назад

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.

@stevewhitt9109 Год назад

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

@plopgoot5458 Год назад

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

@Craig31415 Год назад

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

@danielcorrea2396 Год назад

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

@evandrofilipe1526 Год назад

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

@ecologypig Год назад

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.

@carteiraricaeducacaoeinter3882 7 месяцев назад

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

@johnstuder847 Год назад

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 Год назад

Could checkout 3blue1brown's video on quaternions

@iamthebest2662 Год назад

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 .

@AriKath Год назад

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

@henryginn7490 Год назад

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

@bigpopakap 8 месяцев назад

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

@mtate405 Год назад

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

@brandonprescott5525 11 месяцев назад

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

@polishane8837 11 месяцев назад

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

@mauriciocarazzodec.209 Год назад

loved it dude! keep it up greetings from brazil

@xujingzhe82 Год назад

Thank you very much for reaching!

@mixjzp4357 Год назад

Awesome video, neat explanation

@johanngerell Год назад

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

@parthvarasani495 Год назад

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.

@J.B.L2227 Год назад

Amazing your channel is so underrated

@hanspeter5118 Год назад

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 Год назад

Um what about theta = 0?

@hanspeter5118 Год назад

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

@jaafars.mahdawi6911 Год назад

Very well done. Keep up the spirit.

@HenriqueCosta-fg1pk Год назад

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

@Mathymagical Год назад

Thanks! Please do the quaternion time derivative.

@146fallon9 Год назад

very inspiring video. Thank you for the masterpiece.

@tedsheridan8725 Год назад

Another great video!

@tmarvel4347 Год назад

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 👍

@dylanparker130 Год назад

I loved that step at 12:00 - genius!

@loicgeeraerts 7 дней назад

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.

@arulprakash5420 Год назад

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

@agargamer6759 Год назад

Great video!

@Nusret15220 Год назад

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

@orresearch007 Год назад

this is good work, keep going!

@nathank7569 Год назад

Excellent stuff.

@tubebrocoli Год назад

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

@EW-mb1ih Год назад

very nice video, hope to see some explanation about quaternions

@sanswag Год назад

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

@tasnimul0096 Год назад

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

@ominollo 7 месяцев назад

Beautiful video 🙂

@person1082 Год назад

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)

@zemoxian Год назад

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.

@vinbo2232 Год назад

Thank you. Hope to see your quaternion video.

@tnk.2033 Год назад

please don't stop making these videos

@user-zn2zb7ri6n Год назад

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.

@anvayjain4100 2 месяца назад

I'm was in awe the whole time 😭

@zafran156 Год назад

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

@nouamanmoukassi81 Год назад

love your videos!

@MarcinSzyniszewski Год назад

Great video! :D

@MTGreat202 Год назад

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

@bilel114 Год назад

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

@1495978707 Год назад

Please make a video (series?) on calculus of variations. This is a wide open hole that hasn’t really been covered yet on RU-vid to my knowledge

@DavidGrossman-js2xu Месяц назад

I finally understand this video!! Dope

@dionisiocarmoneto Год назад

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!

@marcelopau2325 Год назад

Amazing, continue this exelent channel

@78Mathius Год назад

Love your videos.

@sandipmaurya7371 Год назад

Loved and Subscribed from India

@swordofstrife1174 Год назад

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

@moodangelatx6580 4 месяца назад

Please continue. Great things from small.

@Eniac42 Год назад

Can't wait for a video on Quaternions

@ceromat2022 10 месяцев назад

Nice to watch

@elliotwilliams7523 Год назад

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.

@youtubeuser7111 7 месяцев назад

Thank you sir 👍

@tanmoypandey Год назад

Wonderful video. Complex number is also handy for AC circuit analysis using phasor. Could you please make a video explaining how complex number is relevant in AC circuit analysis using phasor? Thank you

@Sokhyrr Год назад

You are amazing, thank you

@SynaTek240 Год назад

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.

@rasmusnormannlarsen1972 Год назад

I personally like how you can make Acos, Atan from complex numbers and log, since a number x = r * exp(it) if r is 1, then log(x)/i=t the angle of x. cos(t)=x is the length in horizontal direction of a circle of radius 1, so it is the angle of x+i*sqrt(1-x^ 2) or Acos(x) = log(x+i*sqrt(1-x^ 2))/i. In the same way Atan is the angle from something of horisontal direction 1 and height x. However 1+i*x does not have length 1, but 1-i*x have opposite angle but same length, so Atan(x) = log((1+i*x)/(1-i*x))/(2i), since the ratio have length 1, and when you divide, the angle gets a minus, so you get twice the angle.

@WAMTAT Год назад

Please do a video on Quaternions

@manishp1336 9 месяцев назад

Thanks!

@user-vf5di9nz4s Год назад

graet explanation thanks

@plekkchand Год назад

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

@ArifulIslam-qw6lf Год назад

good stuff homie

@constantinknab 4 месяца назад

this is sick!!

@pre-universitygeometricalg5862 Год назад

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.