Imaginary Numbers, Functions of Complex Variables: 3D animations.
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- Опубліковано 7 вер 2024
- Visualization explaining imaginary numbers and functions of complex variables. Includes exponentials (Euler’s Formula) and the sine and cosine of complex numbers.
![The most beautiful equation in math, explained visually [Euler’s Formula]](/img/n.gif)
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What is the difference in color represent?
Pls
What is the name of the beginning music
All the music in this video is from the free UA-cam audio library, and the names of the songs are the following.
Renaissance_Castle
Sicilian_Breeze
Wigs
Allemande
@@EugeneKhutoryansky לא
Good evening sir/ma'am please try to make a video on usage of complex numbers in circuit theory, it would be very helpful for me in understanding the impedance.
Thank you.
You do a great service to mankind with these videos.
+AlphaOmega, thanks. It is nice to be appreciated.
+Physics Videos by Eugene Khutoryansky You're awesome, Eugene. You're revolutionizing Science education to a whole new dimension. Just can't thank you enough. Your channel deserves just as much attention and support as Khan Academy. Keep them coming. All the best!
Down vote.
@@marshacd what's the point of commenting that
uh
I feel like I won the lotto finding this channel. I'm taking complex analysis and it's nice seeing what this looks like. I'm struggling to get my brain wrapped around the principle branch lol.
I am also glad that you found my channel, and glad that you like my videos. Thanks.
I recently created a Patreon account for people who want to help support my channel. The link is on my UA-cam home page. Also, in case, you have not already seen them, I uploaded several other videos recently. As always, for each video that you like, you can help more people find it in their UA-cam search engine by clicking the like button, and writing a comment. Lots more videos are coming very soon. Thanks.
Great, you really
Great, you really empower my imagination and boost the spirit of creativity. Any video about the dot and cross products of vectors?
+Physics Videos by Eugene Khutoryansky I sent you a message about doing a one time donation, as you have set up a monthly donation in your Patron Account. You responded, but I can't access your response anywhere! Will you please respond below this? Thanks, Eugene!
+Donna Valentine, Hi Donna. Thanks for wanting to make a donation on Patreon. I really appreciate that. The way Patreon works is that it only supports ongoing donations, and not one time payments. But, there is the option of canceling the ongoing donations at any time, so if someone wants to only make a one time payment, one possibility is signing up and then canceling after the first month's payment is made. It's up to you if you want to do it this way, but in any case, as I said, I really appreciate your interest supporting my channel, and I am glad that you like my videos. Thanks.
You are DOING GREAT ! PLEASE DO MORE !
I've never imagined this kind of animations were possible, and what's more, they just show the beauty of physics, maths...
Just wow... 10 out of 10. Brilliant job done here.
Thanks for the compliment about my video. I am glad you liked it that much.
@@EugeneKhutoryanskyWhich software do you use to make these beautiful animations ?
@@AK56fire poser 11
I really hope we look back on these videos in 50 years as the start of an educational revolution. Education all the way from K-12 to university is nothing but a cash cow for adults that don't give a damn about their students. Free access, repeatable, archived, and in an ideal world you get paid more the better job you do. Contrast that to even small non-doctoral schools whose admins are pushing teachers to get NSF/DOD grants because they get to suck up half of the award. Somehow they have to do that and teach 4.5 classes a semester. The students aren't the ones winning in that system
+Dave M look up Charlotte Iserbyt's "The Deliberate Dumbing Down of America" i.e. this is intentional not accidental
BEAUTIFUL!! You, sir, are a genius and a prodigy! Keep 'em coming! I can't get enough of these intellectually intoxicating grand opuses.
+SOBIESKI, Thanks for that really great compliment. Lots more videos are on their way.
Eugene Khutoryansky GOOD!!! :)))
I love you. Because of you I studied math and physics and intuitively understand better than I would have ever done from school.
Thanks. I am glad my videos have made a difference and that they are helpful.
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Thanks.
Can you please make a video on C.F and P.I of differential equation ?
What exactly are they ?
I absolutely adore the work that you are doing. It is helping me soooo much thaat I will probably donate my first salary to your channel. These videos, if taught by a professor would cost in order of hundreds of dollar per class but here, it is free. I promise to make a humble donation next month when I get some money. I have already put that on my calendar. Thank you extremely very much for your videos, I hope you find everything that you are looking for from life!
Aakash, I am glad to hear that my videos are helpful, and that you like them that much. I really appreciate your interest in donating, and I appreciate your support. Thanks.
Aakash Pathak nice spirit
That's an awesome idea bro , extremely good, you are such a sweet person
@@kanishkagadiparthi1789 His comment is 5 years old
@@lambda1004 indeed bro 😅
But I have seen it now 😁
who are these 29 sinners who dared to dislike your video??? they mus be punished.
Great job!!!!! btw, later half of the video flew above my head, but i encountered somebody for the first time who could make me understand this topic in this much detail. THANKS!!!!!!!!!!!!!!!!!!!!
1 possibility is they can't understand. 2 possibility is the trick in this video is that 'log' builds the shape of funnel, without this, the video will not exist.
I finally understand why i^2 = -1. Thank you
Though to get there, the creator did gloss over the origin of the rules of multiplying complex numbers, thus this might be an example of circular reasoning.
You would normally start from the definition of i^2 = -1 to show that products of 2 imaginary numbers is the same as multiplying the magnitude and adding the phase angle. 1·i = 90 degrees rotations then pops out from that.
i^2=-1 by definition. It's not an implication.
i is defined simply as the square root of -1. So multiplying it by itself (i²) gives -1, just like squaring the square root of 9 would give you 9.
MoonlapseVertigo 確かに。i^2=-1が、iの定義です。
I reply that -1 equals 1i (i) times 1i (i).
Thanks to you, Dante.
I do not know how many people worked on this, but I am so much grateful for making such videos. They help and impact a lot. Thank you very much!!!!!! I hope your work gets the necessary awards in real life.
Thanks for the compliment about my videos. I made all the animations myself. The narration is done by my friend, Kira Vincent. Thanks.
When a professor cannot be understood , find relevant UA-cam videos you should.
And oh boy... I stuck gold. This video is the best video I've seen on the subject.
Very good for visual learners. Very well made.
This is the most honest and deserved sub I gave anyone.
Thank you
As a note of improvement: I would replace the medieval fair music with jazzy-synths and a mellow hip-hop beats. Or even ambient music.
The medieval music is a little distracting and quite off-putting to be honest.
If you need someone to make you something like that I'd be willing to do it for free (I owe you one for the education after all)
This channel is a life saver! I'm very much a visual type learner and benefit massively from the videos this channel provides, great for Electrical Engineering students - Thanks a lot!
Glad to hear that my videos are useful. Thanks.
these videos make me scared
the music, the animation, the voice, the subject!
and the background
Math is my religion and knowledge is my weapon. I am truly honored to
live long enough to understand this. Bless all of you who feel the same.
The light is spreading slowly but at least its spreading. Now if we
could stop the money pigs from destroying Earth in my lifetime that
would be awesome! Join me and lets change the system from the inside out!
changing the system has relatives meanings....but we can help each other loving and respecting all opinions.
greetings from Brazil!
What I love about your videos Eugene is that instead of just learning a concept, I actually understand it and how it works, it is actually quite interesting once you understand a concept for its entirety. Thank you
+Destin Pearson, thanks. It is nice to know that my videos are appreciated.
6:25 Usuallly, the length of a complex number is called the _magnitude_ of it (as you said), however its angle is called the _argument_ of it. When dealing with signals (as in AC circuits), the magnitude is then usually called as the _amplitude_ of the signal, and the angle is then called the _phase_ of the signal.
This is not a mistake you made ;) I'm just saying this so people can have a better vocabulary.
As always, these videos are extremely awesome!!!
these complex functions' representations are just simple magic!
Perfect. Thank you Eugene for all of your and your friends' efforts.
+M. Yakup Özer, thanks.
what's with the music???? i feel like I'm at a medieval festival or in a video game.
Kylie Reich RPG game?
the witcher 3
I wish I can download the music. I really like them
HOI4
I loved it actually, haha
This actually gave me a really nice visualisation for de moivre's theorem
You are actually the one guy who doesn't make a lot of woo woo about hard math, but actually helps to really understand it
Euler's formula suddenly makes a lot more sense!
e^(iπ) + 1 = 0
Yes it’s true...
🙆🙆🙆🙆 what a super human brain behind this video..
You must be the education minister of INDIA 🇮🇳
I am a senior in mechanical engineering. Pretty cool to see some of the theory behind what we're doing when we're taking a Laplace transform, which we do all the time when dealing with LTI systems and control systems. The mathematical background behind it was never delved into very deeply, which it really doesn't need to be for our purposes, but it's a little less mysterious now. Great video
+Joe Blow, thanks. I am glad that you liked my video.
Awesome. Cleared all the concepts in Control System Engineering. Lucky to have such video with this great animation.
Glad you liked my video. Thanks.
I've added this to my playlist on Complex Numbers. This is one of the few videos that actually demonstrates the uses of imaginary numbers without it being a lecture. I look forward to watching some more of your videos - maybe they'll end up on more playlists!
The genius who makes the complicated simple that's exactly what are you doing, hats off to you and to your team. Thanks
Thanks.
Impressive! I've always wondered what purpose those colors on complex number graphs had, and now I have my answer. Every difficult concept you intuitively explain makes me go, "Oh, that makes sense!" and helps me retain the information longer. As always, thanks for your massively helpful videos, and I can't wait to watch more of them! 😉
Thanks. I am glad my videos are helpful.
@@EugeneKhutoryansky You're very welcome! :)
This is truly the best science channel on UA-cam.
Thanks for that really great compliment.
This is what I was missing from my understanding of complex numbers. Thank you very much Eugene
+Adam Kahin, glad I was able to help. Thanks.
I follow your work for quite some time, for some reason this video on Imaginary Numbers slipped under my radar. It's the best video on UA-cam right now for people that are starting to study this area of Mathemathics.
Your videos gave me insights that I would never have elsewhere.
Thanks for the compliment, and I am glad to hear that my videos are useful.
Best explanation of complex analysis I've seen! thank you very much!
Thanks for the compliment.
Your videos is & will be seen till mankind seizes to exist.
Thank you a lot for your service 🙏❤️
I love you ❤️❤️❤️
Thanks for that really great compliment. I am glad that you like my videos.
@@EugeneKhutoryansky I'm glad that people like you exist😄❤️
OMG, Thank you so much!, We cover Imaginary numbers in class next week! (I study electrical engeneering)
Your timing is perfect! :D
+Nikola Tesla, you are welcome. Glad I made the video just in time for you.
I’m sure discovering your channel is the best thing happened in my life.
I am glad that you like my videos that much.
Me at the begining: yep man i may be a genious i understand everything
Me when colored graph apperar: º_º
It's really not that difficult. The xy plane is the input, the height (in logarithmic scale) is the real part of the output and the colors are the imaginary part of the output. We use colors because we can't draw in 4D.
@シ yeah something like that. There are many ways to help us visualize complex functions and this is one of them. Another one is a 2D color map with different color brightness for different magnitudes. One of my favorite ones is thinking of them as a mapping. Meaning, they transform one coordinate system to another coordinate system or transforming a shape to another shape. For example a line into a circle.
@@shayanmoosavi9139 if x = 1, F(x) = 1, so output is 1, real part of output is 1, log(1) = ***0***, NOT what is illustrated at 7:20. The graph should intersect the complex plain at every |x| = 1. It looks like the graph has been transposed downward by 1 (log(10)) unit. What am i missing here?
@@mikewaxx yeah I think that's an animation error. The origin should be 1 but instead it's 10. I didn't even notice it before.
@@shayanmoosavi9139 thanx i'm not crazy :-)
The concept of comlex functions is very beautifully and comperehensibly depicted.
Thanks for the compliment.
This is education :)
Education that enlightens :)
I would like to thank from the bottom of my heart for putting these concepts into such greatly simplified illustrations and clean explanations. You have made me feel the worth of many subjects in electrical engineering that I have learnt till now. I will pray that all your videos get the views they deserve. I suggest you start a website too.
+Junaid Memon, thanks. I appreciate that.
Mathematics is simply beautiful...It's like staring into the face of the universe
Mind-blowing! When it got to the part on plotting things in 3D, I couldn't believe it's about poles and zeros immediately 🤯🤯🤯 Awesome and mind-boggling!
I am glad you liked my video. Thanks.
Remember! imaginary = lateral numbers. Just call them lateral numbers and lets change the math lingo once and for all. Please!
Amen to that!
tonyrosam LOL
don't bring here the knowledge u got from other utube channels !!!
if so then call em non lateral no. instead of real
plzzz
But they are not what Y or Z axes to the X axis. They are something different, for example in Einstein's theory they appear only to represent time, what makes sense (sorta). What do these numbers represent in reality other than just abstract numbers, because if they do not represent something, I fail to see what practical application they may have "in physics and engineering" (quoting the end of the video).
It's not the name what matters, it's not that they are based on i=sqr(-1), it's what they represent in real terms (like negatives can represent debt or opposite pottential or just the left side of an arbitrary determined mapping axis), what are imaginaries useful for? What do they represent for example in Schrödinger's equation?
@@user-vi3pi9rf7w INSTEAD WE HAD AN EXTENSIVES LOL
Lawliet L although yeah it’s mentioned in a video from a different channel, it was actually Gauss who suggested it be called Lateral, as imaginary sounds misleading to unexistent.
Eugene Khutoryansky & Kira Vincont. You are the most fantastic youtubers who make the best clear mathematics from transcendence mathematics. I'm very appreciate you and the things what you do. Thank you.
Thanks for the compliment.
thank you for the video on complex numbers! you have amazing intuitive insights on math and physics!
+ripsirwin1, thanks for the compliment, and I am glad you liked the video.
When I took calculus 2 in high school we were just told i=-1^(1/2) but never why… it just was… vectors were calc 3 and we never had to use i in any of the lesson plans… so seeing the logic behind i=-1^1/2 is an elegant revelation, and complex numbers make so much more sense now. Thank you you do excellent work
Thanks!!!
Hey, just thanks for being on this Earth. You really explain excellently. Respect!
Thanks for the compliment.
As a high school senior in calc and physics c
Imaginary numbers will always be my favorite (and close in my heart) because they spurred my interest in mathematics. Before then, math was just a bunch of random numbers to me.
+mh8010, I am glad that I made a video on your favorite topic. :)
Amazing videos! I'm going through mechanical engineering and find these extremely useful and also enjoy many of them for my own interest.
+Dan Hanson, thanks. I am glad that you have enjoy my videos and that you have found them useful.
A brilliant video with great simulations and simple explanations. I could watch videos like this all day long. Well done!!!
Thanks for the compliment about my video.
Thank you so much for these videos Eugene, they are extremely helpful.
+Lagareth Stanfield (Git), you are welcome, and thanks. I am glad to hear that my videos are helpful.
Brilliant explanation of complex numbers. Looked thru my 100+ quantum videos for this specific one.
So weird, so beautiful. Thanks.
This is great. If someone was able to create a notebook with 100 Q/A's in chronological steps to understand the matter you would have the perfect notebook.
As a highschool student, this was satisfying to watch although I didn't get much of it
I finally understand where the formula:
e^iθ = cos θ + i sin θ
come from! Thank you so much Eugene!
Thanks.
Math is very beautiful indeed!
+Pär Johansson, yes I agree. Hopefully, colorful 3D animations will help people see that.
Thank you very much! This is the cleanest and clearest explanation of this, I've ever seen. The graphics make the ideas easy to follow.
Thanks for the compliments.
tfw you realize you can represent any two dimensional plane with 1 axis and colors
and you wonder what it would look like if: you represent 4 dimensions with 2 axis and colors and "negative" colors,
you represent complex numbers and its operations by a single axis and colors,
you represent quarternions in this way...........
or even if all videos can actually be representented in numbers in this way................
so many possibilities!
I spend many days trying to understand the visualization of complex function and finally I got it from this amazing video, thanks
Glad to hear that my video was useful. Thanks.
Hey Eugene, hello!
I found your channel recently, and I loved it! your videos have nice animations and awsome explications. Can u make a video talking about generators and AC/DC Current?
This is pure art! What great service you have been done! Thank you so much!
Thanks.
How tf can we touch real numbers?
Think about it like the area of a square or the number of apples someone has. You can’t have negative area and you can’t have negative apples.
@@prestonakantimatter?
Then ask yourself how you can touch words, or your brain
@@prestonak I'm sure the bank can tell you what negative apples looks like.
With these animation you're saving life of students and serving humanity 👍❤️
Thanks for the compliment.
We can see and touch real numbers??
I am not so sure I would make the claim that you can see, touch or use any other sense to divine what a number really is or why it is so useful to us. Numbers are constructs of our mind and it is their relationships between one another that can tell us a lot about the world. Numbers on their own have no substance. Even in the Platonic sense, numbers are somewhere out beyond the realm of what we would consider our own reality. I think we can definitely say that numbers have a certain kind of reality to them, but I really think that is the best we can say about them.
I obviously understood your meaning from the opening monologue, but I wanted to say that I would argue the point based on the sheer fact that our minds are the only things we can depend on to give a us a sense of the world around us. This ability is flawed, as countless scientists have shown, and therefore the best we can say about the reality of numbers is that they are useful and we can develop a language based on these "things"..
+Mark G, even if it turns out that positive real numbers are not really real, this actually would not change the point that imaginary numbers are "just as real" as positive real numbers. That is, they would still be lumped together into the same category.
Eugene, when i first started to seeing your videos i was on high school. Now i'm studying electrical engineering! Thanks for helping me discover my interest for the way that nature works!!!
so "i" is half a flip in direction of a vector. Then what is 1/3 of a flip?
I guess it would be adding the vector by a vector in the complex plane. But what does x times y mean on a xy plane?
A one third flip in magnitude-angle form would be 1 angle(pi/3)
In A + Bi form this would be 1*cos(pi/3) + 1*sin(pi/3)*I
In the x-y plane, multiplying a vector by another vector can mean either taking the dot product or the cross product, both of which are different than multiplication in the imaginary plane.
This video was very intuitive! I am glad I will now have a greater understanding of the plane of a function with complex outputs in my head when dealing with these numbers.
The sucker punch: 7:05
Yeah, I pretty much lost it there
nvshd why did I skip all those days of class. on the other hand felt food about my self till 7:05.
*good
I’m lost too...
🤣🤣 I am feeling so lucky that i asked my doubt on google to find this kind of a channel is a blessing
I am glad you like my videos. Thanks.
I just watched a 14 min video on Caramel Dynamics
These videos are just awesome.....they helped me visualise these concept so beautifully. A huge thanks to you Mr.Khutoryansky.
Thanks for the compliment and I am glad that my videos are helpful.
i am currently too uneducated/dumb to understand this video past the half way.
you were setting yourself up so perfectly to depict Euler's Identity. this video was good as always, but I wish you had ended with the most beautiful equation ever discovered. you did a great job preparing the prepared the viewer to be introduced to it.
I really wanted to see mandelbrot in this video - those functions wasn't that cool
you are right. Mandelbrot and conplex numbers shouldn't go apart
Mandelbrot and alike are color maps of the complex plane. Functions like Y = cos X or Y = e^X are 4D beasts in C^2 or R^4 space. It's a matter of viewing them in 3D or 2D. There are cool ways however to do so, making one realise and visualise e.g. the complex "identity" of circle and hyperbola, or, of goniometric and hyperbolic functions. These can be seen on my webpage
home.scarlet.be/wugi/qbComplex.html
few years back, i always wondered why do complex numbers numbers even show up while doing physics, etc.
now that i realise, it had all do with the way of expressing exponentials in terms of trigonometric functions: e^{i\theta}=\cos(\theta}+i\sin{\theta}
truly beautiful i must say
I hate how this video states that the real numbers are the only numbers you can "touch and see", you cannot touch or see any numbers. Numbers are purely an abstraction and no number is more "real" than another.
They mean that you can touch and see something like 5 apples, but not 0 apples, -5 apples, or 5i apples. The quantity, not the number itself.
Jo Reven indeed you are right. thanks for educating him.
you touch the apples not the numbers, yes the statement was just plainly wrong every number is equally "real" and equally "imaginary"
Well I can't give you 5i apples can I? But I can give you 2 apples, or you can look at a bench and see 0 apples. Negative numbers are real in the case of bank balances or trade accounts as well as vectors in Physics. If you have $0 and you withdraw $100 then you have $-100 owing.
However in the more physical world, negative is simply a direction that is 180 degrees from your original path (which is considered positive). Even transcendental numbers are real in nature and their decimal points never terminate such as e or pi.
Even irrational numbers can exist for example, the golden ratio observed in nature or if you have a square tile that has an area of 2cm^2 then each side would be sqrt(2) long.
But with imaginary or complex numbers where Im(z) ≠ 0 it is impossible to quantify this in the real world. Like I said no one can give you $i, this is impossible! But I can give you $5 (this makes i imaginary and 5 a real number). That's why real numbers are any numbers that are integers, fractions, irrational numbers or transcendental numbers (such as pi). Basically any complex number where Im(z) = 0.
Yes you can give me 5i apples.
If 5 is the quantity and 5i is the quality, 5i apples means 5 times better than 1i apples.
If you give me 5+5i apples, then that means that you gave me 5 apples of 5 quality (as opposed to 1 quality)
5i is just the number 5 existing on a new axes at right angles with the one that 5 lives on.
You can give 5i any meaning you like...
In less than 15 minutes, you explained what my advanced math teachers couldn't back in the day.
We can touch positive integers? Let me see someone touch 5.
You can touch five apples. You can touch zero apples. Try to touch negative five apples.
in that case you are touching the apples not the number itslef
Mubashir Soomro
Now you're touching the subject of semantics, my friend. But not physically touching them. How is that possible? :)
exactly my point haha. I know what he meant when he said we can touch positive integers , I was just being a bit sarcastic.
and I got to learn something new, namely semantics. I didn't know such a subject existed as well.
There are very few channels as smart & knowledgeable channels as yours!
Thanks for the compliment.
I see you are at a lack of music.
Try symphony no 40 Mozart
Wow. Just wow. Everything just makes sense now. Thank you for existing
Thanks. I am glad my video was helpful.
Wtf dares to touch numbers? Outlaw!
Thank you guys for explaining the real science behind every famous thing we learn theoretically only.
In electrical engineering, these concepts are used quite alot, however, we do not use "i" as the imaginary number operator because the letter I (i) is used to designate current. Therefore, EE's use "j" to denote an imaginary number. (x + jy) is rectangular form, X
Perhaps brilliant mathematicians just "get it". But mere mortals like me need to first SEE it in order to get it. Thanks for helping me see it! Brilliant job of imparting knowledge!
Thanks for the compliment and I am glad my video was helpful.
THIS. This is the best way to describe complex functions.
The only thing I would change is using a natural log instead of log base 10 so that e^z has a 45 degree slope.
Thanks.
Amazing graphical video provides nice insight for students in engineering taking control systems/automatic control as poles and zeros are applied in real systems.
I wish we'd had such wonderfully explanatory graphics when I first encountered complex numbers many years ago.
+RayEngineer, at least this will now be available to all the new students we are presently learning this. And thanks for the compliment.
Wow. Today's students are most fortunate to have such wondrous resources.
what an excellent explanation with powerful animation.
Thanks for the compliment.
As a physics freshman, this is a blessing, thank you.
+DeltaAccel, you are welcome and thanks.
sir u Chanel is one of the most beautiful i have never seen before..its all like not only learning the concept but moreover like experiencing the concepts
+Yr Kishore, thanks for the compliment. I am glad that you like my channel and my videos.
You are amazing..Everytime I am fascinated by my new concepts that I learn from these videos
This is strangely relaxing
I like the animation really nice! if someone can learn it what a blessing for his/her students
A great man can make another........Thank You So Much....
Your Hard Working
Thanks.
I really love your elegant graphic conceptualizations Eugene! Your idea of color for phase and height for log of the amplitude of a function is one that I've long believed to be an excellent representation but i've never seen it used before.
+David Caywood, thanks. I am glad you liked my video.