Secret Kinks of Elementary Functions
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- Опубліковано 7 чер 2024
- What happens to graphs between degrees of polynomials? How can we draw complex inputs and outputs in 2 dimensions? And what will we see if we try?
00:00 Intro
01:14 The Messy Powers
04:41 About Complex Numbers
07:15 Importing a Function into the Complex Plane
12:19 Overshooting with Euler
17:27 Roots
23:30 Flower Pressing
25:47 Down to and Around Zero
27:06 The Big Bang
29:34 Enjoy!
30:53 Bonus Functions
Correction: 03:26 The first x in the expansion is raised to 1.4. It should be raised to 1.6.
DESMOS GRAPHS:
===============
5th degree polynomial
www.desmos.com/calculator/e9t...
x^p
www.desmos.com/calculator/eob...
x^x
www.desmos.com/calculator/um2...
base Gaussian
www.desmos.com/calculator/sl4...
More on Complex Numbers:
• Counting in Imaginary ...
Music by:
@timkuligfreemusic (intro)
@Lisayamusic (the rest of the video)
Notes:
At 2:46, when we start converting fractional powers, the fractions must be reduced to lowest terms before we use them to assign the degree of root and power under it.
It's a shame this video wasn't released as part of 3blue1brown's SoME series, you'd have blown everyone out of the water!
Thank you for saying that! I guess I needed more practice to iron out the **ahem** kinks 😁
@@imaginaryangleclap…clap…clap
@@hyperduality2838 yeah the number two shows up a lot...almost like it's the commonest nontrivial number.
@@hyperduality2838 ok I guess it's a joke by now. But why can't you replace 'duality' with 'plurality'? Your enlightened BS is based on the most simply ignorant case it can be...why not obsess over trinity?
@@hyperduality2838 sir, this is a Wendy's
When you revealed the "3d" structure of the cube root function by rotating (24:26), wow. I actually gasped.
I'll be honest, I read "Secret Kinks of Elementary" and I was _very_ concerned about how that was gonna end 😅
💀
Never would I expect to find out the exponential function has a sort of focus, but its location does seem to line up with intuition. This was great work!
Thank you! I was also surprised!
didn’t know elementary functions were freaky like that 😳
This is what I was here for
Holy cow, I'm so glad this popped up on my feed. It's not often I learn about an entire new way of visualising and conceptualising the behaviour of simple functions like this that I had no idea about. 3blue1brown hinted at this with the animations of remapping the complex plane, but this takes it so much further and with so many cool insights along the way. I'll have to watch it another one or two times at least to really absorb the details I think.
Really nice work, and I'll share this around. I hope your channel grows because you deserve the audience.
Thank you so much!!
So, uh, thank you, for this.
I haven’t enjoyed a math video this much since I discovered 3blue1brown a few years ago. This was amazing. I really loved that you opened this topic with a seemingly simple question and explored its complexity and beauty, while building a sense of intuition.
Also, the pressed flowers metaphor was everything. So good, I’m obsessed. Chef’s kiss.
Fingers crossed will get to see an explainer on the irrational powers someday.
Thank you so much for sharing those feelings in such detail! It really reaches me and gives me so much joy! I hope I get around to this topic, it's one of my favorites.
Wow, this channel is so underrated
Thank you! 🤗
I completely agree
Amazing
Criminally so
@@enantiodromia maybe, but my comment speaks 100% truth
@@imaginaryanglebtw, congrats, this is now your most viewed video
This channel is one of my secret kinks.
Thank you!
wow I didn’t know elementary functions could be kinky
I just love how the visualized elements jiggle when you mention them. "He's talking about me, yay!" ❤
Gotta show them affection 😉
whow, redefining plotting as `z = x + f(x)i` is such a brilliant idea!
This was awesome, I love seeing the Riemann sphere existing still hidden in this. Maybe you can do a video showing what's going on when you go into the extended complex plane with the Riemann sphere?
Thank you! That might come up at some point if I find a good story to tie it together.
@@hyperduality2838but then why do we exist in three dimensions and not a multiple of two?
@@imaginaryangle yes the Riemann Sphere was the only thing missing from this video. Adding it would've been perfect! An entirely geometric interpretation of all complex numbers, while also making your "complex infinity" (really just an unsigned infinity like 0) make intuitive sense. Better yet, graphing on the actual Riemann Sphere would show what physically happens when your graphs like 1/x shoot off to infinity. Hint: 1/x is just a Mobius transformation.
That was just wonderful....very enlightening, you should be quite proud of this work!
Thank you so much! A lot of love went into it!
@@imaginaryangle I'm curious to know if you have found any kind of formula for the location of those "foci" for functions other than degree 0, 1, or 2 or the simple higher degree ones like x^n with no other terms
@@WhattheHectogon No, just the simple cases you've listed here (you can see it in the Desmos graphs given in the description). I also didn't come up with an elegant approach to look for it. Do you have an idea?
yeah, this is awesome
@@imaginaryangle It seems like they happen where the derivative of x + f(x)i is 0, since the lines 'bunch up' around the kinks. So they would be the x where f'(x) = i, plugged back into x + f(x)i. Don't know about a general explicit formula though, and it doesn't explain why they would be like foci.
Great video by the way!
this is literally the best day of my life
Marvelous and gorgeous! Please produce more like this. Truly enlightening and edifying. It would be fantastic to see more of the 3D renderings, though. All becomes clear when you add more dimensions.
Keep up the good work. 👌👍👏
Thank you! As for 3D, my animation skills are not keeping up with everything I'd want to show. But I'm learning!
Wow! I studied complex analysis many years ago, and while I understood it well enough to get good marks in assignments and exams, I always felt that I didn't really understand it. It's like a jigsaw puzzle where I have all of the pieces and I know which pieces connect to which other pieces, but I can't see the whole picture. This gives me a new way to visualize and think about analytic functions and see the whole picture.
I find, that I can't understand anything in mathematics unless I can find a way to visualize it.
Thank you!
@@hyperduality2838 I do agree. I came to the same conclusion myself.
"Being and non-being create each other.
Difficult and easy support each other.
Long and short define each other.
High and low depend on each other.
Before and after follow each other."
-- Lao Tzu
Visual representation of curves on complex plane, done beautifully and explained clearly in details.
If this was submitted in #some, it will be easily at the very top.
agreed, it would be a clear prize winner I think
Thank you so much! It makes me happy you think that!
This is the best video I have watched in a long while, teaching me something elementary yet revealing about math. Looking forward to other videos!
This is excellent! Thank you for making it.
My pleasure! 😊
This is the most intuitive way anyone has tried to explain to me the connections that each power graph has to any other power graph. Using complex number space and compressing it to fit on two and three axes really does show a lot of what's hidden on the real number line. And it was done in a way that retained the shape and certain key features of each power graph. Bravo!!!
Wowwww amazing visualisations! I love seeing functions' level curves and I don't think I've seen this used as a way of visualising complex functions before - I feel like I understand these functions better now, thank you so much for making this!!
Fantastic. One of the best ways to visualise, theorise and conceptualise so many different parts of geometry, rays, graphs, and ellipses I have ever encountered.
Truly amazing video, both in terms of explanation and aesthetics! Providing the desmos links is a nice treat as well.
This is brilliant! Very brilliant! This channel belongs in the same league as Mathologer and 3Blue1Brown.
This has echoes in three other videos:
Mathologer - Times Tables, Mandelbrot and the Heart of Mathematics - where multiple foci of cardioids appear
Welch Labs - Imaginary Numbers Are Real [Part 13: Riemann Surfaces]
3Blue1Brown - Taylor series | Chapter 11, Essence of calculus
I wish imaginary numbers had more real name (like orthogonal numbers or some such - this idea was suggested by Riemann I think) so that they will not get a short shrift and thus allow development of more intuition about them. I know "imaginary" is just a word but sometime sociologically it has an effect of apathy.
I think you are helping us develop that intuition which of course should be followed by mathematical rigor. But creativity starts with intuition.
Thank you so much, this means a lot! And you're right, my focus is more on assisting the building of intuition. It's awesome that there's a whole ecosystem of math educators and each of us can dive into our own approach without fear that something important won't be covered.
And I might be biased (I definitely am), but I like the word "imaginary" 😉
@@imaginaryangle I guess "imaginary" frees one to be more creative :) So in that sense I agree.
I've been playing around with polar and parametric graphing and functional analysis in both 2d and 3d desmos for the past few weeks on end, but never once conncted the dots to how the imaginary plane plays a roll in all of it. Seeing this was absolutely mind blowing, as it really just connected so much stuff I've already learned together in ways that are simply incredible to think about. Seeing all the relationships layed out in this manner was just absolutely mind blowing, and it really taught me a lot about how many different areas of math that I'm currently interested in were dreampt up and developed further.
And truly, the foci are OP
"Complex Variables" by John W. Dettman (published by Dover) is a great read: the first part covers the geometry/topology of the complex plane from a Mathematician's perspective, and the second part covers application of complex analysis to differential equations and integral transformations, etc. from a Physicist's perspective. For practical reasons, a typical Math Methods for Physics course covers the Cauchy-Riemann Conditions, Conformal Mapping, and applications of the Residue Theorem. I've used Smith Charts for years, but learned from Dettman that the "Smith Chart" is an instance of a Möbius Transformation.
The Schaum's Outline on "Complex Variables" is a great companion book for more problems/solutions and content.
This was absolutely brilliant the build up to the “flower pressing” was jaw dropping. Immediately liked and subscribed! Keep up the great work!!!
Thank you! Will do!
Thank you, this is great! I get these glimmers of how beautiful complex analysis is, but it's too much to comprehend at once, I only get pieces. So thanks for helping me assemble a little bit more. I was dubious about the x.+ i f(x) trick at first but it's actually a pretty interesting tool for visualization. Thanks again!
Amazing video! Mind blowing stuff - love your initial explanations of imaginary numbers too, somehow it feels intuitive - we gain freedom (of rotation in this case) when negation comes into the picture. Thanks!
Wow... just wow. This video is incredible!
This exemplifies what I love about maths. There are so many deeper layers of understanding.
AMAZING, BRILLIANT, INCREDIBLE, that's just FANTASTIC. Best mathematics video I've seen for complex numbers. Amazing beautiful video THANK YOU.
Wonderful and intuitive visualization! Thank you!
It's so amazing to see new high quality math channels popping up! Keep up the great work!
Thanks! Will do!
Surprisingly low amount of subs for this channel, the vid is really insighful and clearly a lot of effort is put into it! Thanks!
This was AMAZING! Thank you for sharing! I’ll be watching this on the big screen next time!!
This video is simply amazing! Since i introduced myself to complex numbers through some youtube videos and wikipedia articles, i always wondered what were these rainbow-looking images, that on some resources were shown as "graphs". For a high schooler, that did not really learn anything complicated about calculus, (not even mentioning complex "world") this was rather distressing to read the information in overcomplicated and scientific way that is shown in almost all articles and pages. This video just united anything that i knew about essense of graphs and complex numbers and i am absolutely love it!
Dear creator, you really deserve more views and i wish you it! IM IN FOR YOUR NEXT VIDEOS!
Amazing video. I have been wondering about just this (but on a very rudimentary level). Fascinated to learn about a deeper structure here and very good visualizations. Feels there is much more to know here... Thanks for god job with this video.
I have no idea what any of this means, but I like the smooth, soothing and synergistic voice of the narrator.
Comment for the algorithm, this is pretty great and I love how you made the visuals work even under constraints
This is absolutely alien intelligence
So so good man, excellent work. This is one of the best videos I've seen that translates math almost completely into art. You need to take these ideas and perspective shifts and put em up as AR assets because I would totally PAY to get to interact with knowledge like this.
There's some that teach you foil, and there's some that incrementally bump up the understanding of everything you know anything about.
This is totally core human curriculum
Im seriously looking forward to more deep dives from you
Very cool, and amazing visuals! I learned some things. I think I'll have to watch at least one more time to understand it better.
Reaching the end and watching everything come together was really cool to see
Beautiful video. Nice presentation and ideas.
As a non-mathy person, this was fascinating and your way of explaining worked very well.
Complex plane is just so wonderful. Thank you for making this!
This is good stuff. Should put this in submission for Jjjackfilms.
Mind. Blown. Division by zero equals complex infinity, a circle of imaginary radius zero with a direction undetermined. Fantastic video, especially if I ever decide to do mushrooms.
This is an excellent representation of how graphs transform into another function in the complex plane and how would they behave. I might have to look deeper into this to understand what really is going on!! (Congrats in advance when reaching 5k subs btw)
Thank you!
Woooow this was amazing! I learned a lot, and suspect I will rewatch many times
This is excellent. The visuals are really beautiful and everything is presented in a way that flows smoothly. Seeing cardioids (or cardioid-like curves) pop up in the quadratic case was interesting; I'm going to spend some time this weekend understanding why they're there.
Thanks for the great video!
somewhere there's a really cool video digging into the Mandelbrot set and related functions that also points to some fascinating connections to where the cardioid shape in that comes from, as well as some other shapes you get when you generate Mandelbrot-like sets with exponents other than 2. These are somehow clearly related to what's going on in this video but I can't quite articulate it.
Thanks, I'll have to find it. I've found that a circle of radius R is mapped to the cardioid:
r+2R^{2} \sin\left(\theta
ight) = R
But at this point my understanding is algebraic only.
(Edit: Forgot to mention that it's technically mapped to that cardioid translated up by R^{2})
This is the best video on complex numbers since welsh labs blew my mind w his series on them. This is just what ive been waiting for. Godspeed!
Thank you so much! Those videos were amazing!
YOU NEED COMPLEX INFINITY MORE VIEWS WOW
That was such a beautiful video, thank you!
Thank you too!
Absolutely sublime work.
So... I was washing my dishes looking at UA-cam videos because, you know, hardly anyone just enjoys washing dishes.
And then I started looking at your video and actually stopped washing because my brain needs more processing power for your explanation. I stood there, hands wet, for the whole video. Great video sir. I liked it very much.
Beautifully presented and highly engaging, and was I able to follow it.
I love how this title attracted two very different yet equally interesting people, some might fall in the venn diagram intersecting both groups.. Like me who finds maths interesting, and would also like to know about how secretly kinky elementary functions are..
Thank you for scratching a brain itch that has been itching since primary school! Very satisfying!
My only note would be on the animations: in a lot of the animations the mid-point is where the magic happens, but it is also when the animation is fastest.
So I'd suggest either inverting the speed, such that it is slowest around the half-way point, or just leaving it linear. Though linear animations always look a bit stiff.
But the whole point, in my opinion, of this video is showing what happens at the weird transitions, so its a bit of a tease, that exactly that part is sped up.
It was exactly the feeling I had in school! An itch that just wouldn't go away.
Actually one of my favorite videos on UA-cam
Extremely value content, this remains me that everything in nature are dicted by mathematical laws, simply amazing
Beautiful, thank you. 🙏
Wow! That was amazing journey! Thank you!
You deserve millions of subscribers. Keep up the good work!
Wow, thank you so much!
Amazing video. it was very fun & intriguing to watch. (might have to rewatch a couple times to digest it tho lol)
I got like 40% of what was bein said (I barely know enough mafs to get thru highschool).
The visual representations were top-tier. I don't think i would've understood anything without them.
This is great! For what it’s worth, monodromy is a great way to formalise some of the things you mentioned.
Yup, I'm now a sub after having bared witness to all these new secret kinks of elementary functions!
😮 I thought I had seen beauty in maths but it seems I haven't seen anything. Thank you so much for this video. My mind is blown and my curiosity is overflowing. ❤
The form of the function you actually “graphed”
f(x)=x+i•g(x)
seemed kinda arbitrary at first but the you blew my minds at the foci it had when you include the the complex values with the same magnitude.
I’m amazed but am still trying to interpret what I’m seeing here
If one of these days I ever study complex numbers, I'll rewatch this video and let it blow my mind for more than just the pretty visualizations.
Maybe check out my video about them:
ua-cam.com/video/nlqOQ0vJF0Q/v-deo.html
what a beautiful video! the "spirals" from the negative exponents reminded me a lot of the graphs of trig functions in polar coordinates ( r(theta)=cos(a*theta) ).
That's no accident 😉
Oh my god, this is absolutely beautiful
This is the channel that I was looking for, great content!
Welcome aboard!
Beautiful. Well Done.
Such an amazing journey. Thank you so much for your hard work.
Much appreciated, I'm happy you enjoyed it!
Very awesome video, I love the color scheme and great explanations😊
Great visualisations!
Hi @imaginaryangle - if you're ever looking for music for your videos, let me know!
Thank you! I subscribed to your channel, I will keep it in mind 💙🎼
Nice video. I remember similar paterns in videos about fractals Z^n+C, where n changes its values.
Very good video. Much interesting the graphics.
Thank you. This was amazing insight 👏
Amazing video! It’s really amazing how you visualise the mathematics and how you explain it! :o
Thank you!
Absolutely fascinating!
Thank you!
That was an amazingly
well done video
man i know math could be beautiful, but i never expected this! Fantastic work! I really truly believe you are on the same level as 3b1b
Wow, thank you!
Absolutely Beautiful
So nice. So neat. Subscribed.
Thanks and welcome!
Wow this is f*cking amazing -- easily the top math video of the last several months - -instant sub
Thank you and welcome!
honestly one of the coolest videos I’ve ever sene
I tried playing with parametric representation on desmos to visualize space transformation when using complex variables, this really reminded me of that. In a similar vein I wanted to understand and play with raising non-unitary complex numbers (a+bi) to non-unitary complex powers which lead me to finding my favorite number: Gelfond's Constant (-1)^(-i)=e^pi.
This is a pretty demonstration! Not sure what it demonstrates, but very nice regardless!
Great video!
Awesome. Just amazing good stuff.
Captivating. I gorged on number theory and numberphile videos etc.. until I got burned out on the talking head, crude construction paper or blackboard approach. As much as I liked the presenters, I needed a snappier pace. This was the fastest 30 minute math vid I've seen in a while.
great video man. thank you for your work
Thank you!
also notable with the negative powers is folding the rings inside out. it's something i explored a bit as i tried to come up with some way to get the complex conjugate (with the purpose of flipping the phase shift of a filter) a while back (never got anything that works, and moved on to other things before trying to make an approximation), i think because it also changed the phase of the complex number, or it in combination with something else got me close. it was like most of a year ago so my memory is a bit hazy, but it was nice to see something about complex plotting again, very cool topic.
Incredible!
Amazing, thanks!
17:03 so pretty!I left a longer comment but I got shy 😆 Basically, you are a great teacher, and you wield the visual tools effectively. The alt girl on top of the map example got me lol.
Thank you!
Math truly is the heart of beauty. Wonderful presentation all around! A couple ideas on the visualization, for the 3d portion, perhaps some blurring and dual-window cross-eye stereoscoping would be edifying... And pretty please can you make it so only the intersections of the rings are visible with little dots? My intuition says it would look especially cool at higher resolutions and perhaps highlight more complex "channels" through the structure, much like how the line traveled by the focal point jumped out to the eye.
Yes, 3D needs more effects to make spatial comprehension easier, but I'm not that good with tools that let me do that yet. And seeing just the self-intersections of the cycles is an interesting idea. Thanks!
In electrical engineering, we use the omega (w) symbol as an imaginary number, it’s equal to 2πf which is actually angular frequency. When I saw your channel name I immediately thought of this.
"Imaginary" is a stupid name (and was originally _derogatory)._ A better name is hinted by your use of them, as well as by what their graphs tend to do: "spinny numbers." (Or more seriously "spherical/circular/rotational numbers." Complex numbers honestly have more to do with 2D rotations than they do with roots of polynomials.
3D rotations have 2 more orthogonal axes of rotation compared to 2D, so 3D imaginary numbers have 2 extra components compared to 2D ones. Usually these components are called "j" and "k", though in physics they're sometimes called "σ₂σ₃" and "σ₃σ₁".
@@angeldude101 From my understanding, the whole letter thing got out of whack when physics substituted "j" for "i", because of pre-existing uses of "i". Of course adding dimensions, all shifted by one letter, would have made for a mess...