Imaginary Numbers Are Real [Part 13: Riemann Surfaces]
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- Опубліковано 2 вер 2016
- Want to learn more or teach this series? Check out the Imaginary Numbers are Real Workbook: www.welchlabs.com/resources.
Supporting Code: github.com/stephencwelch/Imag...
Imaginary numbers are not some wild invention, they are the deep and natural result of extending our number system. Imaginary numbers are all about the discovery of numbers existing not in one dimension along the number line, but in full two dimensional space. Accepting this not only gives us more rich and complete mathematics, but also unlocks a ridiculous amount of very real, very tangible problems in science and engineering.
Part 1: Introduction
Part 2: A Little History
Part 3: Cardan's Problem
Part 4: Bombelli's Solution
Part 5: Numbers are Two Dimensional
Part 6: The Complex Plane
Part 7: Complex Multiplication
Part 8: Math Wizardry
Part 9: Closure
Part 10: Complex Functions
Part 11: Wandering in Four Dimensions
Part 12: Riemann's Solution
Part 13: Riemann Surfaces
Special thanks to the fantastic Charles Jackson Young for helping me get the motion rig done in time for this video. You Rock!
Want to learn more or teach this series? Check out the Imaginary Numbers are Real Workbook: www.welchlabs.com/resources. - Наука та технологія
Wow, this is super well done.
Woah, thanks 3Blue1Brown! I love your channel!
3Blue1Brown You both people doing great job for mankind Sir :)
3Blue1Brown omg , I came here aftr a marathon in your algebra playlist!!
You're both sooooooo amazing!!
Coming from you who makes awesome visualizations too, hats off all the more to imaginary stuff
3Blue1Brown ha I was literally just watching some of your vids!
Episodes 1-10: “Ah this is quite easy and relaxing”
Episodes 11-12: “Um what are we even doing?”
Episode 13: “HOW THE HELL AM I SUPPOSE TO VISUALIZEE 4D SURFACES IN A 3D WORLD ON a 2D SCREEN IN MY 1-D BRAIN?!”
TO 0D ATOMS?!?!?!?!?!?!?!?!?!?!?!?!?!?!?!?!??!?!?!?!!?!??!?!?!?!
@@gabrieldoughty6239 nah my 0D understanding
lol - your not.
0D: Wtf are numbers and lines? Wtf is an angle?
1d brain lol
It took me over thirty years, two math degrees and a decade of teaching math to even know of this shadow's existence, much less the meaningful layers behind it. This was incredibly humbling and illuminating.
I believe Riemann hypothesis can be proven using something similar to a 3d magic cubes like magic squares and could predict them easier but what do you think?
more proof that degrees are worthless lol.
Me too!
I had to pause and cup my hands over my mouth multiple times while watching this video. Thank you for such a beautiful explanation.
They are astounded by the beauty of mathematical elegance
Episode 1: what is a number
Episode 13: *COMPLEX MULTIDIMENSIONAL MATHEMATICS*
This is a multiverse extra-complex-mathematics. As predicted.
@@RobertoArteaga The real jump was after episode 10. Whe you saw the hell introduced by multidimensional funtions.
man i binged the shit out of those videos. what a ride!
@@eduardopablo7123 Aren't Multifunctions called that way because of several outputs for an input?
@@fatitankeris6327 man that was so long ago. But no. Multidimensional funtions implies several inputs. She you reach tre inputs you can no longer representate the output on a grafic.
Watching the series feels like a top detective-documentary about math. Brilliant work, stunning storytelling.
Thanks for watching!
Can someone please get me the equation of that plane (hyperplane actually, projected in 3-D Plane; I guess) in terms of "x,y,u&v".
>>>With same meaning of the notations as stated.
8:18 This is exactly similar to how you CAN return to a place you have been to previously, but CAN NOT return the same _time_ AND _place_ simultaneously.
Yeah!! Nice understanding......👍👍
Very well framed
Does this suggest that even if we're not moving, as long as time is a concept, we always move on 1 specific direction in 4D?
what about same time and different place?
i exists in math, not in science.
@@khytron06 Actually we are all moving at the speed of light in spacetime. This video ua-cam.com/video/au0QJYISe4c/v-deo.html visualizes it perfectly. This is the best channel I know of regarding visualizations of physics- ans math-concepts.
Thanks for this - I enjoyed learning more about complex numbers
Hi i love your channel as well
Can someone please get me the equation of that plane (hyperplane actually, projected in 3-D Plane; I guess) in terms of "x,y,u&v".
>>>With same meaning of the notations as stated.
This. Is. Always will be. My favorite video series I've ever watched.
I'm in university now working for a degree in Mechanical Engineering.
I never enjoyed math.
That has always made working for this degree difficult.
Ever since following this series, math is the most beautiful thing I've ever seen.
Thank you, Welch Labs.
You gave me something I've been desiring my whole life.
Matt Bob no u
Dude, I don't know why, but some of the people still using this pepe meme are the worst commenters on youtube. Btw, hope you're doing great on your degree man
Try "Essence of Linear Algebra", by 3Blue1Brown
good luck mark! I have a professor who teaches analysis II which is basically differential equations. He always says that we have to love math. He gives a particular example "imagine if you are a dr. and you are doing heart surgery and you tell the patient ooh man i hated anatomy, but its ok i can operate you " lol it is the same thing for engineers you have to love math lol
Numberphile, 3Blue1Brown and Welch Labs did the same for me. I started in Mechanical Engineering and now wish I had started in Mathematics and Computer Science. I guess I'll be a self-taught Applied Mathematician because that's where my heart is now.
I rate this series *i out of 10*
uh, thanks?
Its actually an "i" instead of an 1 xD
Its a rate in a whole new dimension
That's very lateral thinking
Why not -10i^2
Fahrul Razi All those in favor, say "i"
Never in my life have I ever had tears in my eyes from watching a video/film/series.
It had to be math.
Thank you so much for this journey, besides that it was very well made and very effective, I think I've truly learnt to see numbers different for the rest of my life.
SAMEE!!!!!!!!!
It’s just beautiful
Teacher: This will be on the test so pay attention *shows part 1*
Test: *part 13*
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@@isaniruch ...
Can someone please get me the equation of that plane (hyperplane actually, projected in 3-D Plane; I guess) in terms of "x,y,u&v".
>>>With same meaning of the notations as stated.
I actually had tears in my eyes at the conclusion of this series. For the nearly two decades I've been studying mathematics, teaching, and using "lateral" numbers...I never truly grasped their meaning until the very end of this video, which ends where the entire series began.
Thank You so much, this was gorgeous and very well made.
Exactly, so many things we 'know' but in fact don't know.
Me too !!!!!
Psychodelics were involved
I had tears in my eyes
@@timpower9742 why are people crying lmao
4D visualization in 3D + Bass in the background? you guys need some oscars man!
On a 2D screen
With a 1D time line :D
Now it's complete
Aman Mahendroo guy.... Imagine all this is done by one guy! (Except captions)
Delivered by 0-dimensional photons to our eyes.
Yeah, remined me Interstellar a little bit :D
watching this series after 6 years of its release, and i'm really impressed of how beautiful this is, one of the most fun and amazing series I've ever watched
Te recomiendo la serie de algo está pasando con phi
Me: watching all parts in 70 minutes
Well that escalated quickly
IKR!
I too did that lol 😂
It's because he left cliffhangers at the end of every video. Clever way to keep attention
Yes
shit, has it already been 70 minutes? i just watched all of it and had no idea that much time passed
Best educational series I saw. Strong plot. Many catchy moments. Tons of history and examples. Astonishing visual effects. I want to rewatch this with every of my friends. Thank you.
Thanks for watching!
Oh My GOD!!! You are amazing! I actually felt emotional at the end of the story "Imaginary Numbers are Real". This video series is beyond this 3d space we live in! :P
All the efforts you put into this videos series is truly apreciated.
Thanks for watching!
After all these years, this series is still the greatest I’ve ever watched. It’s breathtaking. If this was the only video on your channel I would regard you as a math phantom.
12:32-13:49
It could've been a plain lecture but instead it's a movie's finale recapping over critical events leading up to the resolution
The soundtrack fits
i'm really crying at the end 😭😭😭
that was so beautiful
because we can really understand those concepts in a level that we can see the beauty it has? much deeper than men? of course you wanted to keep your privilege of being so pathetical in the past :D
but now all that you have is sweet male tears
Little Bey You're the one crying sweetheart, and I seriously doubt you understand mathematics as deeply as you say considering this video describes basic entry level concepts...Besides, it was a man that first described the beauty of mathematics that lesser people constantly copy in order to appear intelligent....It was also a man that made this video that made you cry so take your box of feminist tampons and fuck off back to the princess tea party...
bitch, go cry in another place and fuck you
such a poor crybaby
go home so your mama could clean your tears :)
why you need so much attention of a woman?
ask some psychologist, maybe they could help with your frustration with them lol
I have never seen a better series of on any subject anywhere else on youtube. This series was just perfectly made.
I recommend this to anyone even if their field is not math related.
Thank you!
Check out 3Blue1Brown! His serie on Analysis and Linear Algebra, as well as all his other videos, are just mind blowing!
Indeed, it was brilliant and - what i find the most surprising - it was very entertaining. It's amazing what a valuable content can be found on youtube underneath all that meaningless, popular bullshit.
Waaaaay late, but this series was certainly one of the best things I've seen on UA-cam. I have no words to express, mate.
Someone said below "I actually had tears in my eyes at the conclusion of this series.". I am happy I was not the only one.
Congratulations!
I had rewatched this series twice but I didn't still get that gut feeling understanding. I ended up writing a small article on it to improve my understanding. The end result is that I finally understood the whole series and I ended up using the article as my math project.
Making a 17-year-old comprehend these things clearly shows how great the videos are!
I usually don't comment on UA-cam. But, I couldn't leave without expressing my gratitude to your effort in this series. I spent almost two minutes trying to find out the right words to describe your work but there are none. Maybe, they reside in 4D space just like complex functions :)
Thanks for watching and for the kind words.
Very well put.
MaShaAllah
This should be compiled into a movie.. and should be released in theaters so that other people will know how much interesting mathematics really is.. 😀
That is exactly what I was thinking!!!!!!!!! I WOULD WATCH THAT
they should put it in netflix, so people would spend their time watching something actually meaningful
I wish I could understand what you do! It may as well be Latin
I actually ended up closing Netflix so I could completely watch this instead...
I wish so too, but while you and I would find this interesting the majority of the "general audience" would be looking elsewhere. There is a reason "reality t.v." was so sucessful.
I've been having much more fun watching this series than watching a Netflix series. Even though I haven't understood what's going on here, it's been an immense pleasure to be a part of this journey.
I've watched this three times over the last few years. The final scene brings me to tears every time. Thank you so much for this beautiful series. Special thanks for the incredible visualizations too. These are, hands-down, the best math visualizations I've ever seen.
This series deserves a field's medal. This is Graduate level Math for toddlers. You guys just exploded the power of imagination. This is education AS REAL AS IT CAN GET !! Thanks a ton !!
This is nowhere near Graduate level
@@thedoublehelix5661 it’s still like 3rd or fourth year dumbed down to the point that I a high school student can understand it… mostly
@@jameson1239 it's not that either. If you wanna learn complex analysis, pick up a good textbook and start going through it
Well done. I feel like i binged through a season of House of Cards.
Except this was actually useful and opened my eyes to lateral planes.
Thank you, good sir.
Awesome, thanks for watching!
This was fantastic!
I never knew imaginary numbers had such a rich history. Could a programming-literate child do this? Is the programming easy?
Me too! And I'm not even math professional! Very good job Welch Labs!
It’s been years and I’ve still never found the end credit song but it’s so beautiful
There exist no words on this planet to express the awe, surprise and beauty of something so fundamental, elegant and yet so complex as complex numbers and a humble expression as the square root of negative one. I LOVE maths and this playlist made me feel like a student at my school desk just wanting to keep learning. I didn't learn complex numbers in my school due to the restrictions of the syllabus and now I know what I missed out on. My jaw was left hanging open and my brain was itching for more upon the end. That 'i' in the end was literally smiling at me. I now feel that I finally understand an iota of advanced mathematics. Thank you Welch Labs, thank you so much for the beautiful explanation of a humble but all powerful concept
This is amazing,no words to say how happy I was watching this whole series.
Thank you,Welch lab.
Thank you for the kind words.
I need season 2 of this
yes season 2 please
season 2: 0/0
Learn complex analysis! You might have to learn a little more math first though.
only season 2i available
This is fantastic. As a math teacher I’ve only known surface level concepts of complex algebra but this is what I’ve been looking for. The historical background is amazing and leads well into the graduate level branches. This has given me a great idea of what I might study for grad school. Thank you and I’m sure this will help my students
Congrats for finishing this journey. Here's some views/likes statistics as of August 2020:
1. 5,330,727 / 111K
2. 1,557,435 / 29K
3. 1,020,456 / 22K
4. 877,545 / 18K
5. 842,658 / 20K
6. 706,350 / 16K
7. 621,010 / 14K
8. 521,989 / 12K
9. 486,186 / 15K
10. 440,908 / 14K
11. 587,091 / 13K
12. 348,950 / 8,6K
13. 738,382 / 31K
That... was.. the... best... series... of... episodes... of... imaginary (ops... lateral)... numbers... ever.
You misspelled "oops" as "ops".
I agree
no way on earth anyone can explain complex number better than this. thumbs up, hats off, thank you.
I just watched all parts in a row :D
You've done a great job in explaining a simple-to-calculate-but-hard-to-visualise complex functions. Many confusions are cleared. I've watched episodes 11, 12 and 13 over and over again. 10 out of 10. Thank you!
👏🏻👏🏻 awesome Work ❤️ bringing over something that is nearly incrompehensible on first sight is probably one of the most fundamental services you can bring to people - thank you so much - this is the first time that I really got a good understanding about the nature of our numbers 😍 visualizing with all your „paper and scissors“ work helped tremendously to make this abstract topic grabbable 😊👍🏻
I binge watched all the 13 videos, amazing.Hats off to the creator.Every math class should show this to the students.
Thank you!
That's the best storytelling about complex numbers I've ever seen so far!
A special thanks for episode number 11, 12 and 13 that are really awesome!
Thank you.
Can't wait for the.....following!!?
Working on it! Thanks for watching!
Great! subscribing right now for notifications 😜!
This is the best group of math videos I've ever seen.
I have several compliments: the editing is great, the explanation is intuitive, the visuals are insanely good, the music completes it perfectly.
Great video(s)!
The first and only time I've shed tears from the beauty of maths
Did I just binge watch a math series? Yes... yes I did.
You couldn't *imagine* how interesting it was!
Yepp
I was just wondering if I had done the same thing....and then I find your comment!
While I'm meant to be working on an assignment? Yes... yes I did.
Ditto! :-D
What an amazing experience to binge watch this series, can't even imagine the people who had to wait after all those cliffhangers!! Math is really beautiful, and that storytelling was on point, I got the chills in the end of the video. That's an awesome piece of work, keep up with it. Love from Brazil.
Thanks for watching!
"… can't even *imagine* the people…"
@@SreenikethanI no
Wow. This series was of such a high quality. I can't believe what you've done here and I'm so thankful for it. This was beautiful and absolutely inspires me to carry on my studies of complex analysis and is just another beautiful facet from which to appreciate math
One of the most beautiful educational series on youtube! Seriously underrated - although it is a quite niche topic, this deserves more recognition
It took me 2 days (with work and stuff) to finish this series because I just got so into it, i took detours to learn more and more about math and science (as well as do sample practice problems as you went over certain topics) to deepen my understanding.
College really just felt like regurgitation of methods with no context, storytelling, or passion; just some tools to memorize and redo and pass. But now you've single-handedly rekindled my old and BURNED OUT passion for mathematics (which I tutor regularly) and made it my favorite subject again. It's 4 AM and I don't even think I can sleep. I just want to keep learning more and more
I really wish college math classes could be longer than a semester. So much knowledge is missed out on. Every week is just a cram to get all the homework done without being able to enjoy the intricacies. I get that college is expensive and nobody wants to spend an extra 2 years on a bachelor's, but I would love it. My favorite math class ever was AP calculus in high school. It was everyday for a whole year, totaling 180 hours. My teacher really went out of his way to make sure we understood WHY everything worked. If people didn't completely grasp something he would just spend a whole day re-explaining it. Now at a large uni with 150 students in each class, it's just not feasible to get so deep into it.
@&rnie To the revolution comrade!
i vfinnesed in like an hour and a half
Why am I crying? I know nothing about math, but I love this series. How can math evoke so much emotion???
guess the music also helped
The beauty of math is that it conveys a simple truth. It's based on a set of very few principles that we can all agree on (axioms), and from there on, impressive discoveries can be made by simple logic. Nevertheless, it requires both creativity and perseverance to make the correct logical leaps to discover the truly impressive and beautiful relations - such as, for example, e^( i * pi ) + 1 = 0 - a relation between the most commonly used constants and identities in mathematics (0, 1, e, i, pi), which also happens to involve the basic operations addition, multiplication and exponentiation once each.
to me math is not beauty, but it's greatness because it permanently extends the reach of our thought. It's touching because of how great it is and what math has accomplished in engineering and science.
It's mostly the BGM
It shows the power that a well crafted script and narrative has especially when applied to a carefully chosen subject. We are curious story-tellers, all of us. You satisfy those two things, and you satisfy totally.
This is Nirvana! With enthralling background score, I felt like I am watching "A Beautiful Mind" again! Well done.
This is one of the most elegant and interesting explanations of this topic that I have ever seen. I'm currently taking complex analysis in university and am having a difficult time understanding what the math actually means. Its one thing to just extend the math we are familiar with into two dimensions (essentially how it is taught) but I don't feel that I was able to fully grasp exactly what it means until I watched this series. Incredible job!
Mate, I came to UA-cam to watch a chemistry video for my test tomorrow... Ended up here learning about how imaginary numbers are represented in 4D and then casted down to 3D which I then watched simulated on a 2D screen. All I wanted was to know more about redox reactions and now I'm questioning existence, thanks -.-
I just finished watching the whole series for the second time, a breathtaking journey to the most real imagination...Well done.
I'm here in my second time too, simply amazing
I have watched it atleast 5 times already, each time with a little more understanding of underlying concepts of mathematics that i havw been learning in school
This is truly amazing
My second time too, one of the best series on youtube
@@underfilho Complexly amazing.
Me too
Man you are great, I wish you are posting more video series I love them, they are elegant, well visualized and very well-thought, the music the physical objects you create of these concepts and pen and paper with the music have the best psychological relaxing and tangling effect, It is a miracle that someone can learn such a topic in such a way thanks a lot for it!
MIND BLOWN! just watched the whole series, very well explained, put together, structured. I think another thing that's blowing my mind is how one unassuming and seemingly simple problem led down a deep rabbit hole that led us all the way to the craziness of Riemann. Wild stuff.
I have a final exam tomorrow (not mathematics), but here I am watching all of your 13 videos about imaginary numbers. These videos were great, I finally understood what these "lateral" numbers I've been using all along.
I never imagined (lateraled?) I would escape from studying for my exam, by studying something else. You managed to grab my attention throughout the whole 13 videos. You deserve a cookie (and a nobel prize) #subscribed
Ahmad Aldulaie I agree
the power of lateralisation
This series was probably my most beautiful discovery of 2018. The final parts in specific where all you had tought us came into a shadow of a 4D plane were really emotional for me :')
Thank you for all the effort in making a short and understandable explanation of imaginary numbers. You have earned yourself a happy sub
Woohoo!
This is a superb series presented by a brilliant teacher. I learnt so much and have come to appreciate the beauty of mathematicss. This series has been a revelation for me. Thank you very much for your great contribution. Keep safe and well.
Bro, that was one of the best series of my life! Well explained, "simple language", comparissions to the real world..... Truly amazing!! Thank you!!
For the first time in my life, I got the feeling I grasped what imaginary numbers are about. Thanks for creating this!
Watching this ~1 hour series will save you hundreds of hours trying to study and comprehend Complex Analysis. That is how mathematics are ought to be taught. In fact every teacher in every school everywhere should already start using these videos. Thank you for creating this masterpiece and sharing it with us for free. It really is the most amazing educational project I have ever seen!
Wow, thanks for watching and for the kind words.
I come back here often. This is such a joy to watch. Functions of a complex variable are notoriously hard to visualize, and this is masterfully done.
Still one of the best math series on UA-cam. I come back and watch the entire thing about every year and I always understand it more than I did the last time I watched it. Words can't do justice to describe how much I appreciate this series.
This is one of the most beautiful mathematical realizations Ive ever seen! I would have never imagined that the standard Cartesian graphing techniques I learned in grade 9 had such a completely hidden world just under the surface!
Thank you!
This is so fantastic! Thank you so much for making this
+Henry Smith Hey Henry, I noticed we concur the same maths channels so I went through your subscriptions in case I missed any of them. Maybe you've checked it already but I thought you'd like one called hackerdashery. Cheers
Wow, I've never seen such a well put explanation of the P vs NP problem. Thanks for the suggestion, I subscribed.
you're welcome!
Hi, do you have any other math channels to reccomend? The ones I'm already subscribed are numberphile, mathologer, 3blue1brown, and njwildberger, in case you don't know them yet. I'd also suggest the Banach-Tarski video from vsauce.
Hi! 3blue1brown is my personal favourite, but if you like mathologer and njwildberger you may as well like Mathoma (or even XylyXylyX). More divulgative/entertaining, there's a couple of videos about possible curvatures of the universe on Zogg from Betelgeuse, explained by an alien. Much more arty (you may know this one) is ViHart, who plays around with fractals and other cool mathematical concepts. You can find them on my subscriptions as well
This whole series on Imaginary ('Lateral' precisely) numbers is amazing. Explained 'complex' thing very lucidly. A big Thanks !
This was just amazing. I came here for a school project and this series was just... wow. Thank you so much for all the effort you put into this and the workbook, this is the most fun and interesting math I have ever done.
I just watched this whole series and learned more than I did in the past 2 years of math class.
Triamcinolone Acetonide I want to like. But you have 69 likes
True :D man true
Classes are really more about memorizing then actually understanding the subject,
Simply Magnificent. Guys I want you to know that I rarely post any comments on videos, but this serie deserve one. It was wonderful, yet mind blowing, because we barely know the capacity of "lateral numbers (I'll stick to that name) and those numbers could provide a huge advantages, both cultural and scientific, to the human race. If the Number Universe was big enough, now has double it's size. I want to congratulate you for the explanation, it was clear and easy, yet clever and logical. Thank you very much
Thanks for watching!
Wow. One of the best series i have ever watched. It was amazing how you broke it down. Truly imagenary ;) Thank you for the tremendous effort.
By the way, I am an Electrical Engineer that has dealt with these number throughout my degree and have used the techniques you have explained but it finally clicked after watching it. I have graduated 5 years ago but only truly understood it not.
Amazing series. You sir have helped me understand what has always eluded me in university. The way you present the concepts and ideas in such a simple and clear way should be the method all schools teach math. One of the reasons people are afraid of math is because they don't have good explanations like this one.
Thank you so much!
I just watched the whole thing lmao, fucking quality mate!!
Mind Blown. I remember working the formulas, but never understanding what the point was.
Sébastien Tuyizere same here. Although I must admit I didn't quite put my head around Riemann surfaces, but still I came out learning a lot. And it was impossible to stop watching.
Sébastien Tuyizere same hahahaha
this is by far one of the best mathematical series on the internet, your diligent work is much appreciated sir, thank you for this eyeopening outstanding series!
Wow, this was a mind-blowing playlist exposing the real intricacies behind such a powerful concept in math. Thank you so much.
I landed at the first video but binged the whole series.
This short series was absolutely amazing! I never really understood the whole circle and other axis stuff, because my teachers just weren't explaining it well enough. The visualisations really helped to understand the whole picture, although I didn't really get some parts of it like z, u, v, w. In conclusion, very good thorough explanation and journey to understanding imaginary (or lateral) and complex numbers.
"Thanks for watching"
*holding back tears*
No Sir *snif* Thank you
Riemann was a genius. He had pure imagination. I admire him. I cant imagine how advanced the science/mathematics today would be had he not died too early :(
A lot like Ramanujan, another brilliant mathematician who died too early...
@Noor Wachid Better question is what if we had 100% focus for extended periods of time?
Also dont forget Galois. He kickstarted the foundation of a whole new field of mathematics (pun intended, lol) before he died at age 20
@Noor Wachid would people eat more or less bugs ? I don't know...
Hello, man from the past! I'm sorry to disappoint you, but we nowdays have only way more flat Earthers and anti-science activists, rather than geniuses like Riemann...
Bro, it's the best video I've found on all of youtube. I really congratulate you on your great passion for mathematics.
Excellent!! :D
Everything I've mindlessly memorized and calculated in high school to just get good grades had such a mind-blowing deep side all along. I've continuously played back this video multiple times to get the idea, and boy it is amazing. Thank you so much for such a great series of videos, and this will be really helpful for me.
This is a masterpiece. When I asked myself at the beginning which one I should watch, this series of a movie, I opted this series and I am dumbfounded. This is purely amazing.
Thank you for watching and for the kind words.
Top class education! Thank you Sir
Thank you!
This could do well in classrooms!
For free, shoud I add
This has been amazing. For the first time in my almost 50 years on this planet I can grasp what complex numbers are, and how they are as real as negative numbers. It's videos like yours that make me feel truly in awe of the deep beauty and elegance of mathematics.
This series is absolutely amazing. Can't say I understood what was going on, but it certainly made me try to see the bigger picture and realize that there is way more to maths and "geometry" than what we actually see
This series should have billions of views. It's the best thing I've seen on UA-cam so far. And I've seen a lot.
hey, i don't know how to thank you. I love to study beyond whats i'm having in school. I was having some problems with complex numbers and I hit into your videos and know I learned even more. Hope youre making more of this, because you got a new subscriber :)
Woohoo! Thanks for watching!
I'm studying electrical and software engineering. I'm 2 years at college and i have never seen an explanation like that! Thank you for making me understand what complex numbers are and how they work! It's a great pleasure for me finding this channel! Congrats!
Take love, bro. Such an wow presentation. I just clicked the video of the part 1. And couldn’t but keep watching till the end. You just increased my thirst for learning more mathematics.
I just wanted to let you know that your series has given me such a fascination with mathematics that I have switched my majors from CS to Computational Mathematics.
Woah, really?!
Incredible. BEST explanation of any mathematical concept ever. This is teaching. You should be proud.
one of the best mathematical series i have ever seen so far. therefore have many thanks for your incredible visual intuition about a really hard topic, which you have also made accessible to a mathematical unexperienced layman.
Wow......I wish I saw this 4 years ago, this is nothing but absolute brilliance. It is amazing, the 4d projections make me wish I was living in a 4D world but Math got there first... Thanks for this series, finished it at a go today.👏👏👏👏👏👏👏👏👍..Wished our algebra teachers could show us this side of Math...
I am still in High school and I still found this follow able. You guys deserve an award. I have always loved math but this... This is beautiful. How knew we could graph in 4 dimensions on a 3 dimensional plane.
Next serie: Laplace transform, please.
That would be an interesting to visualize
I learned the very basics of imaginary numbers in high school in learning to solve quadratic equations, but years later this series gives me a visualization for what the complex solutions look like, which I haven't fully gotten from some other great videos about the subject. It's a bit mind-blowing with the four dimensions and I don't fully grasp it yet, but it gives my brain something to grab onto. It was cool that you even made a sort of physical model of the Riemann surface before showing computer-generated versions.
Haven't studied Maths for more than a decade. Still love to watch informative videos, and this series is the best. Couldn't stop without finishing, and what imagination these pioneers had. Respect to them, and Welch Labs.
It was like watching a movie come to an end. I mean, these type of things aren't meant to be free, you're doing a great service Stephen. Thank you so much for inspiring us and introducing this whole new world of complex numbers. I will never be able to look at it the way I used to before. You've given me a new dimension (pun-intended) of interpreting and solving complex numbers. Once again, thanks.
I'm so sad it's over. Didn't think math could almost make me cry.
An amazon series, a great perspective!! Kudos to the efforts and how you presented it all along, watching the series was an enriching experience.
Thanks!!
This was fantastically well made. I made a leap whilst watching these last two parts and connected it to the atan2 function, I now have a new found visual understanding on how that function works