The beauty of complex numbers
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- เผยแพร่เมื่อ 21 พ.ย. 2019
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This video covers complex numbers, euler's formula, fractals, and more of the mechanics of complex numbers.
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I wasn't concerned with this terminology in the video but note that technically the 'julia set' is the boundary of what was seen here (the fractal part or in one case the circle), while the internal points (the one's that don't diverge) are part of the 'filled Julia set'.
Also I took the difficulty down on this video compared to recent one's but the follow up will focus on applications and that will be more advanced (mainly about control systems and nyquist stability). The geometric intuition of laplace/fourier will be after that.
Keep up the great work 👍🏼
How is this comment pi days old?
I already saw your video. Oops sorry it was done by someone else th-cam.com/video/Y4ICbYtBGzA/w-d-xo.html
Nice! Looking forward to the Laplace/Fourier video!
Actually found out that if you put pictures of Julia set next to each other, guess what you'll see?
th-cam.com/video/JuyPwwG-Cr8/w-d-xo.html
Fun Fact: Some those weird looking Julia sets were originally calculated by hand
Fun fact: things were calculated by hand before computers were invented 🤡
@@louisrobitaille5810no need to be a dick
@@louisrobitaille5810shut up Louis
This is the first time I've seen either the Julia set or the Mandelbrot set actually explained. Bravo!
You need more Numberphile, they cover both.
Just one thing: keep x and y to scale on your plots!
Yeah, I was a bit confused for a second when the point didn't go around in a circular path
You should get a PhD and become a college professor. You're great at explaining complex (no pun intended) concepts and a lot of students would really benefit from it.
The amazing thing is that Julia did all this on paper and in his head.
I’m watching smart people talk about funny numbers
Protip: The more you watch stuff like this, the more sense it will make. A lot of this sorta stuff was over my head at one point, too. But I found the more I watched and the more stuff I looked up, the more pieces started fitting together and making sense. Math is like any language, the more you immerse yourself in it, the more you'll understand.
There's still a fair bit I don't get, but the point is if you take your time and keep at it, you'll find yourself understanding things you never thought you could. Just remember to enjoy the journey and don't worry about how fast you'll get to the end. =) Cheers.
Oh yeah, have a play with Desmos, too. I can't stress enough how much it's helped me to learn. There's an online version that runs in your browser, so you don't even have to download anything.
@@VoidHalo.. Why is this the same for me too? I wish I had chosen to do this kind of stuff when I was 8.. The more I watch this, you start to connect more ideas from math to other subjects.. that one formula that got you wondering-why is this true? (quadratic formula to completing the square; -b/2a to calculus and derivatives)
It's a special feeling when you know you've learned this before school's even taught it to you :)
2 years ahead on the subject of math (at least where I am)-in most other subjects, I suck. It's definitely the interest in watching these types of videos, and I can guarantee you that.
Heh, I actually dropped out in grade 9, so I'm mostly self taught, along with learning from videos and such. Though now that I'm in my 30s I wish I had stayed in school and actually learned this stuff back then. But I'm just not the sort of person who learns well in a school environment. I need to take my time and learn things by making intuitive connections, and be able to really appreciate what I'm learning. Rote memorization like school does just isn't for me. In school, I just dismissed math as a bunch of arbitrary nonsense because they never explained WHY things work. They just make you memorize the fact that they do work and then move on to the next topic. So didn't really care much about it as a result. And having deadlines and being pushed to learn about something in a given period of time just stresses me out and takes the fun out of it.
Watching stuff like this, or learning about math in the context of engineering, physics or some other practical application has given me a whole new appreciation for it. Especially when I start seeing mathematical connections between topics that seem unrelated. Like how often Euler's number crops up. Or seeing the same formula being used to describe alternating current, waves on a pond and sound waves. And it's so satisfying to see these sorts of connections.
If only TH-cam existed when I was in school.
@@VoidHalo I'm in high school now and intrested in physics i try to understand everything intuitively but it take so much time for me to absorb concepts clearly, any suggestions from your side
sorry for my bad english
The visualization tools you young'uns have access to is pretty bitchen!
We are fortunate to have guys like Zack (and a lot of other geeks) for these presentations. ~ GOB (Grouchy Old Bastard)
Literally just learnt about complex numbers last Wednesday and now it’s popped up in my recommendations
Excellent. I wasn't expecting the references to Mandelbrot sets - like watching a movie with a thrilling ending!
I love how this video started off boring and square and Cartesian and went all trippy and psychedelic and fun all by itself. Julia must have been a really cool person. Such is the power of mathematical logic. :-)
Here's the real question: how do we know the shape of the julia set? Why does it make fractals? I'm really interested in why fractals show up here!
And why on earth it is connected only if C is in the Mandelbrot set!!!
watch video on fractals the equation z^2-1 is the hint
@@Nickesponja, the *_definition_* of the Mandelbrot set is the set of all complex numbers _c_ such that the Julia set of the function z ↦ z² + _c_ is connected. It turns out that an equivalent definition of the Mandelbrot set is the set of all complex numbers _c_ such that the magnitude of the sequence f(n) doesn't tend to infinity, where f(n) = [f(n−1)]² + _c_ and f(0) = 0.
You start by taking each point on the xy plane and performing the calculation as he has shown here. After each iteration you calculate the magnitude of the result x^2+y^2 >= 4 then the result will continue to infinity. If after a number of iterations the result is still less than 4 it is probably in the Julia or Mandelbrot set.
You assign a color to the point by (# of iterations) mod (size of color palette) for points that go to infinity
or black for points that are in the set.
If you want to know more about why 4 is important look up some videos on Feigenbaums constant.
We were just starting on complex analysis. You gave a nice overview of the topic. Thanks
There is another Mandelbrot set. It's based on the ellipse rather than the circle. Eisenstein triples. I tried to update the wiki but they wouldn't let me.
This video is so great! Hope you continue to make such videos. Great work.
Definitely one of your more useful videos. Thank you
Oh my gosh, I have UNDERSTOOD the way how the fractals are built, it's just clicked for me like a thumbler inside my head! Thank you for the video!
Wow. I've read about fractals before but never really understood them as complex numbers. But now I have a better idea, and they're beautiful. Please make more videos about complex numbers!
Best introductory video about conformal mappings and juila sets. You are awesome. Really jealous about you. Majorprep rockzzz
That explanation / visualization was the thing I was missing to have the intuition, very cool, thanks
Amazing... Thanks for making such videos... 🙏
You, along with Vsause and 3blue1brown are my fav channels. You post such amazing videos and explain stuff in such a simple and understandable way! Thank a lot for your hard work!
You are simply....most brilliant...thanks😉
Thank you, man, now I understood why discrete transfer functions' poles with modulus inferior to 1 are stable and don't go to infinity. At college I was never told about "Julia set", it's probably too mathematical a concept for my engineer teachers.
I really love this
Eagerly craving for the next video in the series !!!?
Amazing!
Beautiful video, mate, the complexity of "complex numbers" is so fascinating. I think this subject might be one of the best parts of maths advancements of the 20th century or even just in recent civilization. I'd reckon patterns like this ought to emerge linked with electron (or other fields) descriptions as they're investigated over time. I remember back to learning about electron "orbitals"/"orbital clouds" in my university chemistry days and I remember how big of a difference it was from my previous conception about electrons being these little particles orbiting like planets in the planetary model. Given that EM waves behave like complex numbers and can be modeled as phasors, it would be interesting to know if electron orbitals can exhibit fractal-like patterns somehow. Cool vid
Jackson Carroll Everything is related.
Amazingly beautiful video
Reminds me of the Numberphile video on Mandelbrot set 😊
Grammar says "I am beautiful"
Mathematician "i is beautiful"
Subject verb agreement ----Am i joke to you😁😁🤣🤣🤣😇😇😇
I dont understand why indians crop up everywhere with there stupid english grammar classes. Grow up, kids. Nobody cares for grammar that much. You should go learn your language and get some wisdom.
@@Astro-ms6zo The joke went straight over your head!!
And what's your problem with Indians?
@@Astro-ms6zo why so serious man? You have grown up but what about your thinking man?? And yes I know my language and yours too ,i wrote my comment in English so that you can understand as you don't have understanding of hindi....i think u got my answer.......
@@randomdude9135 😀
@@Astro-ms6zo Can you judge whole Indians just with the help of my above witty response.??
I enjoyed it. Thank You.
Great visuals
Awesome video I was wondering you know about the art of computer programming books series if you could cover some of the heavy math calculus related topics there 😄 I always found that book series alluring however many say the math is heavily advanced however I still look at it from time to time and some videos going thru some of it in presentation I think would be unique and awesome content
Another excellent video! What software do you use to do your complex plane animations?
This is awesome
I lowkey wish I studied complex analysis in phd studies. No other math is this beautiful
Golden Ratio is beautiful too. :)
Great stuff, 👍
In Electronic and Electrical Engineering, 'we' use 'j' rather than 'i' to represent the square root of -1.
This is simply because 'i' was already (often) used as a variable representing an Electrical Current.
hey dude, EE here myself, check out my comment, if you please. It would be good to hear the perspective of a fellow EE.
I study Electrical Engineering too, but was never told why we used "j" instead of "i" for complex numbers.
This is interestingly not as common in the UK, where common convention is the use a capital I to denote current, rather than lowercase, so there is no such clash.
@@JivanPal I got my BSc. Electronic and Electrical Engineering in Scotland/UK.
'We' would tend to use lowercase, such as 'i' or 'v' when referring to an alternating or signal (non-dc) current or voltage.
Having said that, one can use any characters for variables, as long as what they represent is stated.
'We' had to remember to write 'j = ✔️-1', wrt each set of calculations, in the exams. 😱
(I Graduated in 1994).
@@JivanPal btw: I watched your 6 May 2019 video, and gave a thumbs-up 👍
Good to see you have keen and concise interest in mathematics.
I just wanted to add that a friend doing a Physics BSc, at the same time, used 'i = ✔️-1'
Therefore that/the 'convention' is probably down to the discipline studied and the teachers, etc.
Where did you go to school? My school did a real bad job at explaining the mathmatics to us, we weren't even required to to take single linear algebra class which made understanding all the topics covered in the more advances courses that much harder to understand. Im now starting grad school and am scrambling to have a solid understanding of the mathmatics before classes start in the fall.
Good video!
WOW! Cool Stuff! You certainly bring mathematics to life! 🎉 😊
nice i just searched your channel to see if you had uploaded so,mething
To anybody who wants to play with this stuff themselves, I strongly suggest checking out graphing apps like Desmos or Geogebra. They both have versions that run in your browser, though as far as I know those ones only do 2D graphing. But I know Geogebra has a mobile version that will make 3D graphs. And there is a mobile version of Desmos as well. I can't speak for any desktop versions, but it would be worth looking into that as well.
It's amazing what you can learn simply by playing with various functions and equations, plugging in different numbers, and playing with the variables and seeing what happens. Before I tried Desmos, I didn't even know what a function was. But between playing with graphing apps and watching videos like this, I've learned a ton. They're pretty easy to figure out how to use, and you can make some pretty cool looking things. Even fractals. Happy Learning, guys. =)
Oh wow, this is sooo cool.
I'm curious why the mandelbrot set seems to represent or be related to the cos(t) parametric function.
nice video, super interesting :)
Mind blowing
As a highscooler i came across 3:30 as modulus argument form.
Great ... Now I come to know what is Julia set and Mandelbrot set, although I watched many complex videos but this is the first time to notice that there is a Julia set for each function and there is also a single Mandelbrot set ! I hope I catched it right
I wonder why I didn't hear that before !
hey, great vedio. but,
how did u animate it... with desmos??
could u share the code please?
WOW, it is a very interesting thing, and the fact that I don't understand anything doesn't make these videos less interesting
Fractals = Self similarity = self equivalence = self duality
Duality = equivalence principle = The laws of physics are the same for all observers = Objective democracy.
Subjective is dual to objective
Can you make a video on complex numbers and real life applications
(Will need to watch again. Using Polar-Cartesian coordinates in conjunction with Survey tools tends to set the Mathematical Intuitions in concrete mental habits, so the visual synthesis of the vector graphics "does not compute" automatically. That's why reiteration-research improvements, is Science)
Every student should have a feel for vector fields in perspective?, the navigational approach. The same teaching and learning process as simultaneously putting math-physics/science in the picture by Geometrical Drawing and Perspective, the navigational/functional approach to Polar-Cartesian coordination.
U R great.. 🌹
Majorprep I just wanted to know out of curiosity how are u able to come up with patterns that complex numbers form ....is it there in any book. If it is there so I would love to read that thanks
4:48 he started with 45 complex points and ended up with 18
I know complex numbers are used to understand quantum qbits. Could this be the reason why qbits look so random?
Do a video about the differences between electrical engineering and electronics engineering plz
Amazing
9:40 that’s amazing
Thanks
Are there any 3D plotting tools out there to help visualize things like this and other plotting that requires more than 2 axes?
Why do I understand you so much better than 3 blue 1 brown? I can’t figure out why.
Excellent. but when will I learn all these and connect it with physics or chemistry or any science . Will it not take ages. Plz tell how you learn all these .. How much time did you take
please uses an orthonormed system, especially when you are dealing with angles
Okay maybe I'm just being stupid but which part of the video did I not do this?
@@zachstar At 3:00 for instance, 1 unit on the x axis is larger than 1 unit on the y axis.
And sorry if my English is bad, I am french
4:30 is way worse
excelent
I'm looking at the Mandelbrot Set at 11:10. At -.75+0i and -1.25+0i, does the set break into otherwise disconnected regions that are tangent at those points, or is there really an open "interval" where -.75+yi is in the set for -.05 (or so) < y < =.05?
Love your videos. You should change your channel's name to MathMajor instead of MajorPrep at this point lol
How did u make black bg in desmos??
i just watched smt amazing. wooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo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How do you animate your desmos presentations so well? I need tips!
a lot of sliders! That's really what made the animations look like they did. Also that combined with parametric equations makes for some cool graphics. The next video (I think) looks much cooler and it's basically all parametric equations to represent complex outputs. Here's an example of a file from this video though, www.desmos.com/calculator/ytdemxfw8p
@@zachstar That was exactly my question, so thanks for being willing to answer. Great video...both for the quality and math!
Omg, best things I’ve learnt in 5 years
I feel sad for you then.
Re: Connectivity in the Julia Set - Is the Mandelbrot set a kind of derivative of the Julia Set?
The Mandelbrot set is all of the Julia sets at the point 0+0i, hence calculating the Mandelbrot set starts with z = 0.
Did you use the manim package for animation? (The package 3b1b uses)
No, this was all desmos
Can you please send the link to that website for Julia set!?
just put it in the description!
I love the black background gosh
¿Qué sofware usas para hacer las animaciones?
With the Julia Set, is the border (that surrounds the black area) infinite in length? What is the volume of the black area in the Julia Set? Is it possible for the volume of the black area of the Julia Set, though at first examination appears finite, in reality infinite? If not, is there a fractal that has an infinite border length that contains infinite volume?
I don't have a direct answer for this, but I recall a video describing a sequence of sets that end up with the same feature as the Julia Set. The perimeter has no upper bound, while the area does. That suggests to me that, as your wording suggests you believe, the border of the Julia Set is infinite and the area is finite.
"and now we get to my favourite part in this video... an AD starts xD
dude!
what software application do you use to edit and create your videos
This one was all desmos! Totally free.
Yooo the Mandelbrot set be looking thicc
Wait, does the Julia set consist just of the boundary, or include all the pts that go to 0? Wikipedia seems to disagree
He explains this in a comment near the top.
no, there is more mandelbrot sets, for example with f(z)=z^n+c
"now let's get to my favorite part of the video"
Ad for Swiffer mop and Oral-B
I’m currently eating rice, alone in my bathroom, half naked, on the verge of madness.
MP means maths and physics to me
Not understood after Julia set☹️
What does the notation 2(1)(2)i mean? I thought brackets means that you evaluate what's in them before anything else, but in the his context it doesn't seem correct as 1 is well... 1. Sorry if this is a dumb question, mathematics is more of a spectator sport for me.
It just means multiply
One of my first shaders on Shadertoy was a Julia set renderer: www.shadertoy.com/view/wsdcWf
Pretty neat.
What is the meaning of X+iY Numbers of mangoes actually.
For a more hands-on explanation of the mandelbrot set: th-cam.com/video/6IWXkV82oyY/w-d-xo.html
@2:25 lol
What mathematics class do I need to take to understand this at an undergrad level?
An undergrad? As in you are in college? You should have already been exposed to complex numbers, algebra, and sin/cos functions.
high school math will do
Linear algebra and calculus.
everything is connected
Hi , can you explain more about job outsourcing, H1-B crises and Automation. Impacted industries in next few years
WOAH
Complex numbers : how to add oranges and bananas ? You need fruitagorian theorem !
has anyone researched maths where they used the square root of physical constants?
en.wikipedia.org/wiki/List_of_physical_constants
No steady state AC circuits? :(
I was going to include that in this or the following video but since I've already done one on that I wanted to mix it up. Next will be control's focused.
@@zachstar Don't forget complex impedance, for those ham radio nuts like me.
How do complex numbers