*F.A.Q:* *Q. How did you do the 3D animations?* *A.* I programmed them in Python, using the manim library: www.manim.community *Q. How can I play around with these types of graphs myself?* *A.* Any software that can graph 3D parametric curves will do, but you do have to define the real and imaginary parts of the parametric function yourself. Here's an interactive example for x^a using GeoGebra 3D: www.geogebra.org/m/wtqwhswu *Q. Isn't x^2.4 just the 5th-degree root of x^12? (and thus just a negative number for negative x?)* *A.* That is one possible was to interpret it, yes. There are other just-as-valid interpretations. This is what I wanted you to think about with the suggested exercise at 13:50. When the exponent a in x^a can be written as a fraction with an odd base (which is the case for 2.4 = 12/5) then one of the many values of x^a will be a real number, even for negative x. Which one of the multiple values to pick from can be subject to debate, and you can see that in the showcase starting at 00:23 only one of the three graphing tools that we tested (Desmos) chooses this interpretation of always assigning a real value to x^a when possible. *Q. So you are defining x^a by using e^something-else? Isn't that a circular definition?* *A.* No. The function e^x or exp(x) is not understood to be an exponentiation, but rather a function defined by its series expansion (this expansion can be seen at 07:07). *Q. Where do all of these definitions come from anyway?* *A.* That's a great question, hopefully for another video.
My jaw hit the floor when x^a started rotating around the real axis and suddenly I understood exactly why even a curves up and odd a curves down. Fantastic video, thanks for sharing these concepts!
I didn't guess what was going to happen, but when it did happen it made so much sense already knowing how multiplication of complex numbers introduces the idea of rotation in the complex plane. Like, _oh_ I might have been able to predict that if I'd paused, maybe, because it just makes so much sense.
well, an even number of negatives multiplied together make a positive number (curving up), and an odd number of negative numbers multiplied make a negative number (curving down), so it's relatively easy to understand even with very basic math
I did my Engineering degree many years ago. Currently just over half way through a CS degree. One of the things that has struck me now is the consistency. It like: 1. Hey I have Integers. Lets do some basic operations (+,-, x, /). And it works... 2. Lets add Real numbers. Hey, basic operation still work. 3. Lets add some operations (Differentiate, Integrate, Trig, etc.). Hey, it still works with Real and Integers... 4. Add Complex, Matrices, Vectors, .... Lets try the operations. Hey, they still work. It's like a big Matrix. 'Numbers' on the one axis, 'Operation' on the other. As you add to either and grow the matrix (very rapidly), it ALL still works!!! That has blown my mind!!!
@@perrymaskell3508 Real numbers don't always work though :d Real numbers are evaluations of limits. We can infinitely refine some number according to some condition, and then the 'target number' approached by that refinement is called a real number. So yeah, a real number is just a made-up number, that comes right after the "last" refinement, in the endless sequence of refinements. And that's why 0.000000...1 is said to be equal 0. So when you use real numbers and talk about numbers infinitely close or equal to 0 (or infinity), you can often run into paradoxes. And not to even mention p-adic numbers here... Integers, on the other hand, aren't paradoxical in their nature. And neither are fractions (scaled down integers).
@@perrymaskell3508 And complex numbers, if you're curious, are just a tinsy-winsy part of geometric algebra. GA is a great simplification of some aspects of linear algebra. In fact, quaternions (or more accurately, rotors) are a small intuitive part of GA. It's a very underrated field with millions of great applications, I highly recommend it.
@@hOREP245 yes tbh I think a majority of complex topics cannot be visualised nicely. Anything dealing with dimensions above 3 is already out of the picture. But the basics of the problem are usually always motivated from ideas we can understand, and these can usually be visualised nicely.
This is HANDS DOWN the best of the 10 or so #SoME2 videos I've watched so far. 1/ It centers around a "simple" (middle/high school) topic: graphing some of the most basic functions. 2/ It reveals beautiful hidden aspects of the topic that are likely surprising to many people, even those with a higher degree in STEM (usually because the visualization of complex functions is HIGHLY underexplored for the sake of rushing to more "interesting" topics in complex analysis). 3/ It encourages the audience to experiment, think about different approaches, and explore exercises. 4/ It is well produced (great visualizations, great audio quality, great script). For my taste, you could have spent a little more time on the general formula for any x in C* and the more technical parts involving algebraic manipulations and the different branches of the logarithm (see how carefully 3b1b does it ;-) ), but in general I think that shouldn't bar you from being one of the top contenders for the win. Keep up the good work!
Thank you so much for putting "simple" in quotation marks. In some schools, complex numbers are presented in a most obfuscating way and for some students, it's really hard to open their brain to the concept. It takes quite a while to get to the hexadecinions for some 🤔
Idk why but watching the "complete" behavior of the functions in the complex plane always fascines me Thanks for the great visuals and amazing video man :]
This answers a problem my buddy and I ran into in our high school Calc I class (effectively (-x)^a; the best we could do was to determine this was a way to determine “whether a fraction is odd or even”). I’ve not done much higher level math since then, but randomly revisited it about a year ago and realized there was probably something in the imaginary plane that this could reveal, but lacked the familiarity with the tools to take it much further than that. All that is to say: thanks for the video. It answered some long standing questions I’ve had about this type of question.
The difference between dumbing down for the viewer and raising the viewer up to the desired level is such a shockingly pleasant one to experience. Brilliantly satisfying video, even a comforting one in its beauty.
As far as I'm concerned, complex numbers are more about rotations than they are about roots. All i² = -1 is saying is that halfway to a reflection across the origin is a 90° rotation.
@@angeldude101 I would not say that as reflection through the origin is not the same as 180 degree rotation, because you end up with a mirrored shape. However rotation by angle a can be defined as two reflections defined by any two lines going through origin with mutual angle a/2. For example 180 deg rotation is mirroring object using x and y axis (works with planes in 3D)
@@pavelperina7629 It depends on your dimension. A reflection in 1D can be realized as a rotation in 2D, hence why taking the square root of a reflection needed a second dimension to rotate through. In 3D, reflection across the origin does invert handedness because it's a composition of 3 reflections, and 3 is odd. In general, a reflection in N dimensions can be realized as a rotation in N+1 dimensions. While it doesn't invert handedness in N+1 dimensions, it does appear to when projected back to N dimensions. There's actually a really elegant manifestation of this in Geometric Algebra involving the even subalgebra.
To be honest, I really like the dense, discontinuous representation of *x^x, x < 0 rational* . Yes, there are implicit assumptions (e.g. any rational *x* is only considered in lowest terms to avoid ambiguities), but that graph really underlines the connection between rational exponents and *n* 'th roots! And let's be real (pun intended): Functions that are discontinuous everywhere (like _Dirichlet's Function_ ) are fascinating to think about and lead to many interesting problems, like Riemann- vs. Lebesgue-Integration!
I also really liked this visualization - I'd never seen x^x like this, it's very cool. It's more complete in a sense, since you're looking at all multiples k at once: the resulting graph is the *envelope* of all possible complex curves, for every k at once! And picking a higher k just makes the 3D function graph oscillate through this envelope with a higher frequency. I do think that the plot at 11:19 is better than that at 12:34, since there's no real reason to leave out the points in the half-plane with positive real part, and doing it like at 11:19 has a nice symmetry to it. Also it's interesting to note that k, strictly speaking, does not need to be an integer to be able to plot the parametric function. The interpretation may become a bit more difficult, but algebraically and qualitatively, it's totally fine. Some Mathematica code to reproduce, with interactive choice of k: y[x_, k_] := Exp[x (Log[Abs[x]] + I (Arg[x] + 2 k \[Pi]))] Manipulate[ ParametricPlot3D[ {x, Re[y[x, k]], Im[y[x, k]]}, {x, -5, 5}, PlotRange -> {{-5, 5}, {-20, 20}, {-20, 20}}, AxesLabel -> {"x", "Re(y)", "Im(y)"}, AspectRatio -> Full ], {{k, 0}, -5, 10, 1} ]
Actually, you can get a similar "envelope plot" for an entire family of functions: Exp[x (f[x] + I (Arg[x] + 2 k \[Pi]))] where f: R -> C is an arbitrary function. But I think the subfamily of functions f[x]=Log[g[Abs[x]]] for arbitrary g: R+ -> C are most interesting to look at :)
My only problem with it is that it constrains itself to just Real outputs. I'd rather treat the equation as the rotationally symmetric solid 2D surface in 3D space.
When I went for my undergrad, I had to take analysis, which jumped straight from R -> R into C -> C without any visual assistance for how to interpret the latter. I think getting some exposure to R -> C (and following that up with R+ki -> C) would be a really good way to get a handle on why n-dimensional space needs to give way to transformations as the best visualization, as you move towards C -> C. The fact that R -> C can be informative in both frames without one or the other being obviously better makes it a really useful example.
I absolutely LOVED this video. It's no exaggeration to say that the beauty of those curves on in R X C (especially when "animated") nearly moves me to tears. Thank you very much.
I’ve seen complex numbers make usual high school math look very interesting during one of my undergrad courses. This video though provides so many amazing visuals showing me that I should have been more interested lol. Congrats on this amazing video!!
I need to watch this a few times to really get this. Amazing to think just how much complexity and beauty is revealed when you look at simple functions from a complex perspective.
I have only seen around 15 #SoME2 videos so far, but I think I have found my personal winner. Great job! I hope you will continue to make videos about math, I would be definitely interested to watch more.
Excellent video! I am an EE student currently interested in complex analysis, the classes for which my uni doesn't provide, so I am always lurking for more visualizations/intuitions online. I have never considered that complex functions - namely ones involving exponentials - can have different visual interpretations for different multiples of the principal value of the number. Thank you for your magnificent work Armando!
You should make more maths videos like this. You have a great talent for teaching and discovery. PLEASE make more videos like this. You will gain many views and subscribers.
Thanks for an excellent video on this topic with superb graphics. In my view there are two valid perspectives regarding graphs involving real powers of negative numbers: either consider real-valued real powers of real numbers exclusively from a real number perspective, or consider a complex-number definition of exponents. The second approach is more common and the one used in this video, whilst the first approach, though much less common, has some advantages. To illustrate what I mean, consider the graph of x^(1/3)=³√x. In the complex approach, for negative x, you have to decide whether to choose one branch (e.g. the principal branch, where numbers have argument π/3) or all three branches (with arguments π/3, π and -π/3). In the real approach, no such ambiguity arises, for the function ³√x is, by definition, the inverse function of x³, and so we get real values for negative x. In particular, in the real world, we have ³√(-1)=-1, rather than ³√(-1)=cis(π/3) as you would get with the principal value complex approach. The same applies to any odd root of negative numbers, and in particular any odd root of -1. Consequently, from a real perspective, the graph of (-1)ˣ is defined not only for all integers, but for all rational numbers with odd denominators (when in lowest terms). Similarly to Tupper's graph of xˣ for x
Thank you for this comment Michael. I attempted to convey this idea about the two valid perspectives with the x^x example at the end, but alas with limited success since I've seen some confusion about this in the comments. While making the video I considered at one point going back to the a^x example and re-graph it using the second interpretation, in which case the graph would be one which combines the curves (-a)^x and -(-a)^x for the appropriate rational values of x. In the end I decided against it in favor of not making the video too long, and instead I tried to nudge the viewers to figure this out by themselves using Desmos' behavior for x^a as the motivation.
@@a.arredondo Thanks, understood. Desmos is quite exceptional in doing a really good job of showing all real-valued real powers of real numbers, even when this is challenging, as in the case of xˣ for x
Beautiful! It was really satisfying to try and guess what x^a was doing for non-integer a, then immediately see the answer confirmed visually. Very cool.
This is so relaxing and easy to follow along with the visual aid! I highly recommend you try to make a couple more videos like this and see how it goes! Though please do take your time :)
Absolute masterclass in teaching this topic, bravo. I remember thinking about these spirals in RxC back when I was in high school, which is part of what eventually led me to go after a PhD in math. The discontinuous x^x graph is a really cool approach as well. When I get the chance to teach complex analysis, I'll definitely refer to this video when thinking about general exponentiation.
@@devd_rx Haha, I was bored with the standard curriculum and spent a lot of time thinking about complex numbers/exceptions to the rules I was being taught in algebra. It was pretty natural to ask "What about negative bases?" when going over exponentials, or to ask what irrational exponents even meant. It's also a whole lot easier to visualize paths in C as graphs in RxC than to even try to understand graphs in CxC, so as far as I was concerned, I was starting small. Also, I had a great math teacher who encouraged me to ask questions like that. If I had been ignored back then, I might not be a professor now.
I remember coming up with the graph at 8:55 in high school, as probably the (only?) extension of x^x which preserved its multiplicative derivative of x*e. This was before I found out multiplicative calculus was a real thing people before me had done, and they knew how to actually take multiplicative integrals. I think this was in Trigonometry class.
I showed some of my experiments with x^x (and x^x^x^x^...) to my high school math teachers and all they had to say was "hmm, never saw that before" and stopped wanting to talk about it.
I knew there would be complex plane related rotations because I was already familiar with e's role in both representing general exponentiation (particularly useful for writing an exponentiation algorithm in something like assembly code) and rotations in the complex plane, but seeing the 3D graph have the negative side of the function spinning was worth watching the video for on its own. Then the video just kept going and the whole story behind x^x was revealed and now I want to see the whole 3D vase as k approaches infinity...
Amazing explanation about what is the extension of x^a to complex numbers. It remind me about how gamma function is an extension of factorial to negative (non integer) numbers. [good bgm too]
When I had the idea of the most discontinuous function possible (-1)^x back in highschool I tried to graph it on my calculator (a casio graph 35+ iirc) and it did display little dots where x was an integer, and funnily enough when zooming out enough it would kinda look like a sinewave (the engine probably assumed the points where close enough to be jointed by a line). Very interesting video👍
This was a really great video. It's no doubt the latter portion doesn't have the same large scale appeal of the 3blue1brown or Mathologer, especially when we get to x^x, but certainly from my Physics background I'd learnt enough to appreciate just how cool that exploration was after being guided there. I'm sure I can't be the only one wanting this style and depth and it was taught incredibly cleanly with great pacing. Looking at the channel it appears the first video on the subject, but watching it felt like you'd had years of practice like the previously mentioned channels. Very impressed.
THANK YOUUUUUUUUUUUUUUUUUUUUUU I've been pulling my hair out since middle school trying to understand x^x and now 15 years later FINALLY a complete and beautifully explained explanation I can finally die in peace, thank you so much
Great visualization, I really appreciated how you kept the input variable as a real number. A lot of complex visualizations jump straight to C \to C functions, but in lots of application of complex numbers (engineering, physics) it is more useful to map R \to C. The culmination to x^x was the perfect question to ask, and the presentation and pacing were very engaging.
This is a fantastic explanation. It revealed somethings I didn't already know about these functions even though I was familiar with complex numbers (that is, I knew there were valid solutions, but not their relationship).
Very nicely presented! I hope Grant has recognized your effort. I'll admit I didn't follow every detail, but it's one of those things that I know I'll learn more every time I watch it. Some of those 3-d graphs were as beautiful as they are surprising. Thanks!
I was hoping for more similar videos like this on your channel. I really love the subject, it’s right on the edge of my understanding which feeds both my interest in maths and the astonishment when magic happens and my mind gets blown. Besides, you have an incredibly soothing voice. ♥
Thank you for bringing this to light! I’ve always known this, but you really brought life how the density of these functions comes together once you consider the complex plane and the C^2 world. It’s even more amazing to see them actually visualized instead of just trying to imagine them in the mind.
12:45 Bottom-right is the correct graph of the function mapping R to R. Top-right is the correct graph of the function mapping R to C. Top-left is the correct graph of the relation between R and R. And bottom-left is the same as bottom-right, but with discontinuities removed. For any given application, I believe these are the main concerns to be taken into account when deciding which definition to use.
I never took a complex math class in school. I knew fractional exponents had to be spending some time in the complex plane, but didn't have any intuition for it. This is a great way to visualize them and the different generalized formulas for exponentiation completely blew my mind. Thank you!
When showing the different branches of x^x, it looked as though each of the different branches were tracing out a surface. The densly discontinuous function was effectively just a cross-section of the surface, but discarding the negative solutions for positive inputs (because it is continuous on the positive solutions). In fact, the surface it was tracing out appeared to just be a surface of revolution using any branch of x^x.
This video has really good pacing, just the right amount of detail. I can't imagine trying to explain this to someone without getting sidetracked and rambling.
This was fascinating, good job. Funny enough, a while ago I was thinking about what would happen if we considered every complex number not just single pair (r, a) but infinite set of angles a that matches the same complex number. This was exactly a small glimpse into this world. And many interesting things may await anybody brave enough to explore it more thoroughly.
Great video! Some months ago I was looking for the X^X graph for negative numbers because of reasons (just curiosity), and I found the same 3D graph you showed on the video. I was blown away for the 3D graph with imaginary numbers, and for me is the best way to show this kind of graphs. Additional to X^X, I wanted to look at the X^(1/X) or xth root of X for negative values, but sadly I couldn't find anything on a 3D graph as the first one, so in a sleepless night I put some numbers on a calculator for negative values of X and start plotting those results on a 3D graph, and I was again blown away because of this new graph! It results that, in very small negative values of X (close to zero), the function goes to infinity in an spiral between real and imaginary axis, and when you start increasing the X value (for example, -100), the graph tends to 1 on the Y axis and to -0 on the imaginary axis. Now that I look at your video I can understand why, so thanks for the info! I'm just a nerd who is interested in this kind of nerd content, but not a mathematicians in any way
you're right, I can't think of the last time I heard about a dense discontinuous function that wasn't some insane fractal nonsense. this was a brilliant video and I really hope you continue making videos about content like this!
This is amazing. I was playing with geogebra and trying to find answers for this by myself and it happens I came across with this video... sadly it was a few months later! Congrats on this!
As a programmer who lives firmly in the Real Number plane... Complex mathematics are like reading poetry in another language- it's gibberish to me, but beautifully self-consistent gibberish with a hidden meaning which I can't parse behind it. Lovely video! The visualisations really helped ground me, given my tenuous grasp of what imaginary numbers even are.
8:55 I remember finding this graph on my own in high school, after taking the multiplicative derivative [which I had also rediscovered] of x^x. I got f'(x^x)=x*e, which is wonderful. I thought of using this to extend x^x to the negatives, and realized that would make the value alternate every point. It's nice to know my reasoning made sense; I had thought maybe it was completely stupid. It's my favorite graph of x^x, because (as far as I can tell; I don't know much here) it's the only one that keeps the multiplicative derivative equal to x*e.
I full heartedly hope for a revolution in teaching in the near future. I have had only glimpses of this knowledge on my engineering course, 30 years ago, and now it is being presented in a way a high school, maybe lesser, student could appreciate and even well understand it. And they will, for the newer children are being so so smarter than our generation. That's a wonder.
"These *R* × *C* graphs need more love." - I totally agree! For those who are interested in quantum physics, you will see these a lot when discussing wavefunctions in 1D space: the domain *R* will be the position _x_ (or momentum _p_ if you're working in the frequency domain) in 1D space, and the codomain *C* will be the value of the wavefunction Ψ(x) at that position. Here's a great video on wavefunction collapse / the effect of measurement on the wavefunction that uses such graphs: th-cam.com/video/p7bzE1E5PMY/w-d-xo.html Of course, when considering the 3D space of reality, such graphs would become 5-dimensional *R³* × *C* graphs.
@@JivanPal for a single quantum-mechanical particle. So to describe a single hydrogen atom you're already looking at 24 dimensional graphs (taking the proton as 3 quarks) and that is without added dimensions for spin (which you definitely need fornthe fine structure). Mind boggling how quickly the dimensionality of quantum mechanical systems increase. We're unable (now and in the foreseeable future) to simulate an iron atom, yet we've been smithing iron for millenia.
@@JoQeZzZ To be clear, for a single _type/flavour_ of quantum-mechanical particle, and the wavefunction is only 2 of the 6 dimensions mentioned. So for _x_ types of particle, the total number of dimensions in this graph would be 4+2x. That's 3 dimensions of space, 1 dimension of time, and 2 dimensions for each wavefunction since that's a complex number. Given the 17 particles of the Standard Model, that's 38 dimensions.
@@JivanPal Your math is wrong. It's not 2D for complex numbers. Complex numbers are 1D, 1 complex dimension, but 1 dimension nonetheless. Also, we generally don't take the output dimension into account, 3D vector fields aren't 6 dimensional either, so not only is it not 1D for complex values, but the output space isn't counted as a part od the dimensionality (obviously I thought) Also, I wasn't talking about the full standard model. I was talking specifically about a (1) proton and an (1) electron. Finally, particles interact with eachother, so you can't just give each particle it's own dimensions like it's an independent function.
All of these visualizations are incredibly cool. One thing that might make the “basic” x^n visualization work even better is to add a new grid along the angle that the results of negative values sweep out, so that the perspective is a bit easier to follow. As for x^x visualizations, the coolest one to me is the spiral. I should probably have better criteria than “looks cool” but…
This is amazing! I used to not really care much about complex numbers. But now I am going to start studying them as soon as I get some time. Just incredible.
This is, put simply, amazing. I didn’t quite understand everything, but it was easy to follow, and all around the visuals were beautiful. 100/10, amazing job!
absolutely beautiful visualization of these graphs! I experimented in desmos for quite a while, thinking about how rational indices can also produce real outputs, but the alternate visualization of y=x^x was really pretty.
Thanks for this trip down memory lane, because I had forgotten about learning to understand and derive (and at some point even 'just see') stuff like this. I loved complex graphing and signal analysis in university.
I'm a graduate student in mathematics and I never thought about these nice visualizations. What is your background? More on branches of the logarithm would be a nice idea for another video in my opinion.
I'm a 14 years old kid Who loves maths (currently i'm studying multivariable calcus and group theory) and I love this content I'm glad that I was able to see this video
A quick note on the audio. At certain moments when you speak into the microphone there are these low-frequency bursts which can be quite disturbing when listening (for instance, they come when pronouncing words with the letter 'p' in them). This seems to be caused by the simple fact that you're facing the microphone too directly when speaking into it. The problem seems to go away more or less completely every time you face away from the microphone to speak to the camera, so if you want to improve the quality of this even further, I'd recommend that you look into the best angle from which to speak into the mic as it really makes a huge difference to the end result imo. Overall however, the video was very enjoyable. Congratulations on a job well done!
Awesome video. You don’t often find quality like this from a channel without 400 other videos in the same style. Super inspiring, and looking forward to more!
THANK YOU SO MUCH FOR THIS VIDEO!! I was thinking of this problem a few days ago and i was so happy to see a video on this. However, I have a question. if a is not an integer, couldnt we just convert a into a fraction and get a real value for x^a? For example, 2^3.5 could be written as 2^3+1/2 which is 2^3 * 2^1/2 which is 8*squareroot2
The problem is that, generally speaking, x^(m/n) is ill-defined. It is problematic to say something like x^(1/2) = radical(x), because in general, radical(x^2) is not equal to x, even though radical(x)^2 is equal to x.
Hello Luc. All you have to do is define a parametric curve with explicit expressions for the real and imaginary parts. Take a look at this GeoGebra 3D interactive demo for instance: www.geogebra.org/m/wtqwhswu
we were wondering about this when we were messing around with fractional exponents in Desmos!! we had no idea you could get complex numbers from positive exponentiation, that's so cool
While I appreciate the effort that went into creating the visualizations and presenting a seemingly unifying theory of exponents using complex analysis, there are actually many conceptual problems with this theory, which may give insight as to why mathematicians do not actually use these visuals. The problems already start with the premise of trying to generalize x^β for non-integers β. We know that x^(m + n) = x^m·x^n holds true for all integers m and n, and it holds true for arbitrary objects x, as long as x can be inverted. What is great about this is that this works not only when x is rational, real, or complex number, but it works as long as x is any object for which multiplication (or some other analogous operation) is well-defined. x can be a function, a matrix, an operator, etc. Let M just be any structure where multiplication (or some analogous binary operation) is defined, with the "usual" properties it has when dealing with numbers. If, we think of f[x] as a function from the integers Z to M, where (f[x])(m) = x^m, then we have the identity (f[x])(m + n) = (f[x])(m)·(f[x])(n). This is the defining functional equation for exponential functions of integers. In other words, we can define f[x] alternatively as the unique function f : Z -> M satisfying f(m + n) = f(m)·f(n) AND satisfying f(1) = x. If we want to be reasonable, extending the definition of exponentiation to rational numbers means that we want to find functions f : Q (the rational numbers) -> M satisfying f(r + s) = f(r)·f(s) for rational r, s. If we find such functions, then we can define f[x] such that (f[x])(1) = x, but the problem is that this assumes these conditions are satisfied by only one function. In reality, they may not be satisfied at all, or more than one function satisfies it. If M = C, the complex numbers, then we are always dealing with multiple such functions. The way the video attempts to fix this is by appealing to the complex logarithm, and pointing out that there are actually infinitely many complex logarithms. This is that familiar appeal to "multivalued" functions I see on TH-cam these days all the time. I take issue with this, because, like with virtually every other video on TH-cam I see that talks about multivalued functions, it abuses the concept and represents it rather inaccurately. For instance, no mathematician thinks of "multivalued" functions as having "multiple outputs," nor are these objects treated in a way where you "evaluate them" at a single point to obtain a "set" of points. You will never find a publication in a mathematics journal where someone writes "log(1) = 0 and log(1) = 2·π·i and ...," because that is not actually how these objects are understood. It may be how they were first discovered, but treating them as objects you can use in arithmetic is incorrect. These objects are instead thought of as Riemann surfaces, used to evaluate contour integrals, for example. This is all to say that while there are Riemann surfaces corresponding to the extensions of exponentiation to arbitrary complex numbers, these are not objects you can do computation with, these are not objects you evaluate at a point to obtain multiple values, nor are they objects you do arithmetic with. So, they do not work at all in the way that the video explains them, and this includes things such as the complex logarithm and the nth roots. Also, none of this is applicable if you are trying to define exponentiation with rational exponents (or beyond) for things like matrices and functions. It only works if x is limited to the complex numbers. As such, this is a lot less illuminating than it looks to be on the surface. Of course, I do want to point out that this is not so much an issue with the video, as it is more generally with how exponentiation is taught by sources online in general, and is also more so an issue with how videos on TH-cam completely and universally misrepresent the topic of the so-called "multivalued functions," which are more properly called Riemann surfaces. Leaving these issues aside, the video does do an excellent job at presenting a new spin to a not very popular concept that is hopefully helpful in some ways.
Beautiful graphs, I'm an architect, but I love looking at graphs of functions like this. I have a massive collection of pictures of curves on my laptop.
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*F.A.Q:*
*Q. How did you do the 3D animations?*
*A.* I programmed them in Python, using the manim library: www.manim.community
*Q. How can I play around with these types of graphs myself?*
*A.* Any software that can graph 3D parametric curves will do, but you do have to define the real and imaginary parts of the parametric function yourself. Here's an interactive example for x^a using GeoGebra 3D: www.geogebra.org/m/wtqwhswu
*Q. Isn't x^2.4 just the 5th-degree root of x^12? (and thus just a negative number for negative x?)*
*A.* That is one possible was to interpret it, yes. There are other just-as-valid interpretations. This is what I wanted you to think about with the suggested exercise at 13:50. When the exponent a in x^a can be written as a fraction with an odd base (which is the case for 2.4 = 12/5) then one of the many values of x^a will be a real number, even for negative x. Which one of the multiple values to pick from can be subject to debate, and you can see that in the showcase starting at 00:23 only one of the three graphing tools that we tested (Desmos) chooses this interpretation of always assigning a real value to x^a when possible.
*Q. So you are defining x^a by using e^something-else? Isn't that a circular definition?*
*A.* No. The function e^x or exp(x) is not understood to be an exponentiation, but rather a function defined by its series expansion (this expansion can be seen at 07:07).
*Q. Where do all of these definitions come from anyway?*
*A.* That's a great question, hopefully for another video.
My jaw hit the floor when x^a started rotating around the real axis and suddenly I understood exactly why even a curves up and odd a curves down. Fantastic video, thanks for sharing these concepts!
I didn't guess what was going to happen, but when it did happen it made so much sense already knowing how multiplication of complex numbers introduces the idea of rotation in the complex plane. Like, _oh_ I might have been able to predict that if I'd paused, maybe, because it just makes so much sense.
... not to mention the link to sines and cosines. My jaw was in the basement at that point.
@@SirRebrl I love that about math. I would have never guessed it but when you see it makes so much sense
same
well, an even number of negatives multiplied together make a positive number (curving up), and an odd number of negative numbers multiplied make a negative number (curving down), so it's relatively easy to understand even with very basic math
I love how math is always self consistent. Literally spins around in the complex plane to get to where it needs to go
That's because math is all about describing logic. And the whole gimmick of logic in our universe, is that it's consistent with itself.
I did my Engineering degree many years ago. Currently just over half way through a CS degree. One of the things that has struck me now is the consistency. It like:
1. Hey I have Integers. Lets do some basic operations (+,-, x, /). And it works...
2. Lets add Real numbers. Hey, basic operation still work.
3. Lets add some operations (Differentiate, Integrate, Trig, etc.). Hey, it still works with Real and Integers...
4. Add Complex, Matrices, Vectors, .... Lets try the operations. Hey, they still work.
It's like a big Matrix. 'Numbers' on the one axis, 'Operation' on the other. As you add to either and grow the matrix (very rapidly), it ALL still works!!! That has blown my mind!!!
@@perrymaskell3508 Real numbers don't always work though :d
Real numbers are evaluations of limits. We can infinitely refine some number according to some condition, and then the 'target number' approached by that refinement is called a real number. So yeah, a real number is just a made-up number, that comes right after the "last" refinement, in the endless sequence of refinements. And that's why 0.000000...1 is said to be equal 0.
So when you use real numbers and talk about numbers infinitely close or equal to 0 (or infinity), you can often run into paradoxes.
And not to even mention p-adic numbers here...
Integers, on the other hand, aren't paradoxical in their nature. And neither are fractions (scaled down integers).
@@perrymaskell3508 And complex numbers, if you're curious, are just a tinsy-winsy part of geometric algebra. GA is a great simplification of some aspects of linear algebra. In fact, quaternions (or more accurately, rotors) are a small intuitive part of GA. It's a very underrated field with millions of great applications, I highly recommend it.
@@blinded6502 I think kurt gödel will disagree.
Complex topics don't have to be dumbed down, all it takes is a good visualisation and it can be taught to anyone.
Some topics cannot be visualised in an easy way, or a way that doesn't obscure everything.
@@hOREP245 yes tbh I think a majority of complex topics cannot be visualised nicely. Anything dealing with dimensions above 3 is already out of the picture. But the basics of the problem are usually always motivated from ideas we can understand, and these can usually be visualised nicely.
@@DynestiGTI this deals with different values of x, so it's kind of already 4 dimensional
You can visualize a 4D perspective quite well with a volumetric view.
Complex problems need to be approached very real. 😆
This is HANDS DOWN the best of the 10 or so #SoME2 videos I've watched so far. 1/ It centers around a "simple" (middle/high school) topic: graphing some of the most basic functions. 2/ It reveals beautiful hidden aspects of the topic that are likely surprising to many people, even those with a higher degree in STEM (usually because the visualization of complex functions is HIGHLY underexplored for the sake of rushing to more "interesting" topics in complex analysis). 3/ It encourages the audience to experiment, think about different approaches, and explore exercises. 4/ It is well produced (great visualizations, great audio quality, great script).
For my taste, you could have spent a little more time on the general formula for any x in C* and the more technical parts involving algebraic manipulations and the different branches of the logarithm (see how carefully 3b1b does it ;-) ), but in general I think that shouldn't bar you from being one of the top contenders for the win.
Keep up the good work!
Thank you so much for putting "simple" in quotation marks. In some schools, complex numbers are presented in a most obfuscating way and for some students, it's really hard to open their brain to the concept. It takes quite a while to get to the hexadecinions for some 🤔
@@harriehausenman8623 Sedenions.
@@JivanPal thx!
I agree, this video os amazing at generalizing functions that look like they abruptly cut off at the y-axis
Idk why but watching the "complete" behavior of the functions in the complex plane always fascines me
Thanks for the great visuals and amazing video man :]
absolutely fantastic
ah nothing like the extradimentional rotating arm
@@theshuman100 exactly lmao
This answers a problem my buddy and I ran into in our high school Calc I class (effectively (-x)^a; the best we could do was to determine this was a way to determine “whether a fraction is odd or even”).
I’ve not done much higher level math since then, but randomly revisited it about a year ago and realized there was probably something in the imaginary plane that this could reveal, but lacked the familiarity with the tools to take it much further than that.
All that is to say: thanks for the video. It answered some long standing questions I’ve had about this type of question.
Looked like that's some high level mathematics easily explained to a 16 yr dumb like me . Thank you sir for such a interesting session
You're not dumb. My dumbass got taught this in 11th grade
It's not really a high level math but it's still beautiful nonetheless.
@@Amoeby C'mon. for some it is. 🤗
For a while. 🤓
@barutaji "will never hear about it" does not equal "advanced."
@@angelmendez-rivera351 ok bufdy
This man really dropped one of the most chill math videos ever and then went quiet, I would really watch more of these if you make them
The difference between dumbing down for the viewer and raising the viewer up to the desired level is such a shockingly pleasant one to experience. Brilliantly satisfying video, even a comforting one in its beauty.
The fun thing is how complex numbers have this tendency to create rotations.
Essentially when you have an irrational exponent you are taking a root.
As far as I'm concerned, complex numbers are more about rotations than they are about roots. All i² = -1 is saying is that halfway to a reflection across the origin is a 90° rotation.
@@angeldude101 holy I never thought about it like that
@@angeldude101 I would not say that as reflection through the origin is not the same as 180 degree rotation, because you end up with a mirrored shape. However rotation by angle a can be defined as two reflections defined by any two lines going through origin with mutual angle a/2. For example 180 deg rotation is mirroring object using x and y axis (works with planes in 3D)
@@pavelperina7629 It depends on your dimension. A reflection in 1D can be realized as a rotation in 2D, hence why taking the square root of a reflection needed a second dimension to rotate through.
In 3D, reflection across the origin does invert handedness because it's a composition of 3 reflections, and 3 is odd.
In general, a reflection in N dimensions can be realized as a rotation in N+1 dimensions. While it doesn't invert handedness in N+1 dimensions, it does appear to when projected back to N dimensions. There's actually a really elegant manifestation of this in Geometric Algebra involving the even subalgebra.
To be honest, I really like the dense, discontinuous representation of *x^x, x < 0 rational* . Yes, there are implicit assumptions (e.g. any rational *x* is only considered in lowest terms to avoid ambiguities), but that graph really underlines the connection between rational exponents and *n* 'th roots!
And let's be real (pun intended): Functions that are discontinuous everywhere (like _Dirichlet's Function_ ) are fascinating to think about and lead to many interesting problems, like Riemann- vs. Lebesgue-Integration!
Yeeesss 🥳
I also really liked this visualization - I'd never seen x^x like this, it's very cool. It's more complete in a sense, since you're looking at all multiples k at once: the resulting graph is the *envelope* of all possible complex curves, for every k at once! And picking a higher k just makes the 3D function graph oscillate through this envelope with a higher frequency.
I do think that the plot at 11:19 is better than that at 12:34, since there's no real reason to leave out the points in the half-plane with positive real part, and doing it like at 11:19 has a nice symmetry to it. Also it's interesting to note that k, strictly speaking, does not need to be an integer to be able to plot the parametric function. The interpretation may become a bit more difficult, but algebraically and qualitatively, it's totally fine.
Some Mathematica code to reproduce, with interactive choice of k:
y[x_, k_] := Exp[x (Log[Abs[x]] + I (Arg[x] + 2 k \[Pi]))]
Manipulate[
ParametricPlot3D[
{x, Re[y[x, k]], Im[y[x, k]]},
{x, -5, 5},
PlotRange -> {{-5, 5}, {-20, 20}, {-20, 20}},
AxesLabel -> {"x", "Re(y)", "Im(y)"},
AspectRatio -> Full
],
{{k, 0}, -5, 10, 1}
]
Actually, you can get a similar "envelope plot" for an entire family of functions:
Exp[x (f[x] + I (Arg[x] + 2 k \[Pi]))]
where f: R -> C is an arbitrary function. But I think the subfamily of functions f[x]=Log[g[Abs[x]]] for arbitrary g: R+ -> C are most interesting to look at :)
On top of that, it's not just a matter of opinion, if follows from how x^y is defined for real x and y.
My only problem with it is that it constrains itself to just Real outputs. I'd rather treat the equation as the rotationally symmetric solid 2D surface in 3D space.
When I went for my undergrad, I had to take analysis, which jumped straight from R -> R into C -> C without any visual assistance for how to interpret the latter. I think getting some exposure to R -> C (and following that up with R+ki -> C) would be a really good way to get a handle on why n-dimensional space needs to give way to transformations as the best visualization, as you move towards C -> C. The fact that R -> C can be informative in both frames without one or the other being obviously better makes it a really useful example.
2:00 This animation is pure awesome!! I will never be looking at x^a graphs the same way again
I absolutely LOVED this video. It's no exaggeration to say that the beauty of those curves on in R X C (especially when "animated") nearly moves me to tears. Thank you very much.
Why did this video not take all the awards, it was so simple, elegant, clear, and powerfully revealing. I was hooked from start to end. 👏👏👏
I’ve seen complex numbers make usual high school math look very interesting during one of my undergrad courses. This video though provides so many amazing visuals showing me that I should have been more interested lol. Congrats on this amazing video!!
I’ve been looking for this video for half a year because the question in the thumbnail alone is enough to get me hooked
very interesting and educational!
I need to watch this a few times to really get this. Amazing to think just how much complexity and beauty is revealed when you look at simple functions from a complex perspective.
I have only seen around 15 #SoME2 videos so far, but I think I have found my personal winner. Great job! I hope you will continue to make videos about math, I would be definitely interested to watch more.
Awesome video. I love how calm, pedagogical and humble you are.
Excellent video! I am an EE student currently interested in complex analysis, the classes for which my uni doesn't provide, so I am always lurking for more visualizations/intuitions online. I have never considered that complex functions - namely ones involving exponentials - can have different visual interpretations for different multiples of the principal value of the number. Thank you for your magnificent work Armando!
Oh it’s this time of the year we’re my whole TH-cam feed is swarmed with math videos, thanks to all that are contributing these videos!
You should make more maths videos like this. You have a great talent for teaching and discovery. PLEASE make more videos like this. You will gain many views and subscribers.
Thanks for an excellent video on this topic with superb graphics.
In my view there are two valid perspectives regarding graphs involving real powers of negative numbers: either consider real-valued real powers of real numbers exclusively from a real number perspective, or consider a complex-number definition of exponents.
The second approach is more common and the one used in this video, whilst the first approach, though much less common, has some advantages.
To illustrate what I mean, consider the graph of x^(1/3)=³√x.
In the complex approach, for negative x, you have to decide whether to choose one branch (e.g. the principal branch, where numbers have argument π/3) or all three branches (with arguments π/3, π and -π/3).
In the real approach, no such ambiguity arises, for the function ³√x is, by definition, the inverse function of x³, and so we get real values for negative x. In particular, in the real world, we have ³√(-1)=-1, rather than ³√(-1)=cis(π/3) as you would get with the principal value complex approach.
The same applies to any odd root of negative numbers, and in particular any odd root of -1.
Consequently, from a real perspective, the graph of (-1)ˣ is defined not only for all integers, but for all rational numbers with odd denominators (when in lowest terms). Similarly to Tupper's graph of xˣ for x
Thank you for this comment Michael.
I attempted to convey this idea about the two valid perspectives with the x^x example at the end, but alas with limited success since I've seen some confusion about this in the comments.
While making the video I considered at one point going back to the a^x example and re-graph it using the second interpretation, in which case the graph would be one which combines the curves (-a)^x and -(-a)^x for the appropriate rational values of x. In the end I decided against it in favor of not making the video too long, and instead I tried to nudge the viewers to figure this out by themselves using Desmos' behavior for x^a as the motivation.
@@a.arredondo Thanks, understood. Desmos is quite exceptional in doing a really good job of showing all real-valued real powers of real numbers, even when this is challenging, as in the case of xˣ for x
The best visual of "rotating through another dimension" I have ever seen. You did a really good job. 👍❤
Beautiful! It was really satisfying to try and guess what x^a was doing for non-integer a, then immediately see the answer confirmed visually. Very cool.
This is so relaxing and easy to follow along with the visual aid! I highly recommend you try to make a couple more videos like this and see how it goes! Though please do take your time :)
Yes! With this production quality, I would love to see even some more basic topics approached by this creator. 🤗
I love it! I had not seen this definition of x^a in the complex plane. Beautiful.
Absolute masterclass in teaching this topic, bravo. I remember thinking about these spirals in RxC back when I was in high school, which is part of what eventually led me to go after a PhD in math. The discontinuous x^x graph is a really cool approach as well. When I get the chance to teach complex analysis, I'll definitely refer to this video when thinking about general exponentiation.
How did u know about how these graphs are in high school, i just barely got to know about RxC ...
@@devd_rx Haha, I was bored with the standard curriculum and spent a lot of time thinking about complex numbers/exceptions to the rules I was being taught in algebra. It was pretty natural to ask "What about negative bases?" when going over exponentials, or to ask what irrational exponents even meant. It's also a whole lot easier to visualize paths in C as graphs in RxC than to even try to understand graphs in CxC, so as far as I was concerned, I was starting small. Also, I had a great math teacher who encouraged me to ask questions like that. If I had been ignored back then, I might not be a professor now.
I remember coming up with the graph at 8:55 in high school, as probably the (only?) extension of x^x which preserved its multiplicative derivative of x*e.
This was before I found out multiplicative calculus was a real thing people before me had done, and they knew how to actually take multiplicative integrals.
I think this was in Trigonometry class.
I showed some of my experiments with x^x (and x^x^x^x^...) to my high school math teachers and all they had to say was "hmm, never saw that before" and stopped wanting to talk about it.
@@FadkinsDiet Can you give an example?
I knew there would be complex plane related rotations because I was already familiar with e's role in both representing general exponentiation (particularly useful for writing an exponentiation algorithm in something like assembly code) and rotations in the complex plane, but seeing the 3D graph have the negative side of the function spinning was worth watching the video for on its own. Then the video just kept going and the whole story behind x^x was revealed and now I want to see the whole 3D vase as k approaches infinity...
Same! That was be worth a video.
Amazing explanation about what is the extension of x^a to complex numbers.
It remind me about how gamma function is an extension of factorial to negative (non integer) numbers.
[good bgm too]
The animations are beautiful, and you tell the story in a very accessible and engaging manner. I look forward to future videos from you!
I keep coming back to this remarkable video. Watched it at least four times, shared it many more... it's a true masterpiece.
Grant is a genius to help make the community bigger!! These awesome content in such a short period of time is really impactful.
New sub to you!!!
Круто, такой общий случай графиков функций вижу впервые. Спасибо за визуализацию в xyz-осях с облётом.
So fascinating. I can't believe this is the only math vid on this channel. We need more.
This was brilliant. I love getting mew insights to familiar subjects in math, especially in regards to complex numbers.
Marvelous and ingeniously done.
The transition to complex numbers beautifully shown it made my day.
Thanks for pointing out this topic.
When I had the idea of the most discontinuous function possible (-1)^x back in highschool I tried to graph it on my calculator (a casio graph 35+ iirc) and it did display little dots where x was an integer, and funnily enough when zooming out enough it would kinda look like a sinewave (the engine probably assumed the points where close enough to be jointed by a line).
Very interesting video👍
This is definitely one of the most beautiful math videos over the Internet. Thank you mr Armando!
This was a really great video. It's no doubt the latter portion doesn't have the same large scale appeal of the 3blue1brown or Mathologer, especially when we get to x^x, but certainly from my Physics background I'd learnt enough to appreciate just how cool that exploration was after being guided there. I'm sure I can't be the only one wanting this style and depth and it was taught incredibly cleanly with great pacing. Looking at the channel it appears the first video on the subject, but watching it felt like you'd had years of practice like the previously mentioned channels. Very impressed.
THANK YOUUUUUUUUUUUUUUUUUUUUUU
I've been pulling my hair out since middle school trying to understand x^x and now 15 years later FINALLY a complete and beautifully explained explanation
I can finally die in peace, thank you so much
Great visualization, I really appreciated how you kept the input variable as a real number. A lot of complex visualizations jump straight to C \to C functions, but in lots of application of complex numbers (engineering, physics) it is more useful to map R \to C. The culmination to x^x was the perfect question to ask, and the presentation and pacing were very engaging.
Excellent to show and explain the full catalog of visualizations. Very satisfying
Another new channel with such a great video! Amazing work dude, keep growing. Hope to see more from you soon. Good luck with the journey on TH-cam.
u don't know how long I was looking for the graph of x^x on the complex plane, thank you so much !!
Really interesting, especially the x^x part! Nicely visualized too.
This is a fantastic explanation. It revealed somethings I didn't already know about these functions even though I was familiar with complex numbers (that is, I knew there were valid solutions, but not their relationship).
Very nicely presented! I hope Grant has recognized your effort. I'll admit I didn't follow every detail, but it's one of those things that I know I'll learn more every time I watch it. Some of those 3-d graphs were as beautiful as they are surprising. Thanks!
You sir have a gift. Keep doing what you’re doing, you’re brilliant
I was hoping for more similar videos like this on your channel. I really love the subject, it’s right on the edge of my understanding which feeds both my interest in maths and the astonishment when magic happens and my mind gets blown. Besides, you have an incredibly soothing voice. ♥
Thank you for bringing this to light! I’ve always known this, but you really brought life how the density of these functions comes together once you consider the complex plane and the C^2 world. It’s even more amazing to see them actually visualized instead of just trying to imagine them in the mind.
12:45 Bottom-right is the correct graph of the function mapping R to R. Top-right is the correct graph of the function mapping R to C. Top-left is the correct graph of the relation between R and R. And bottom-left is the same as bottom-right, but with discontinuities removed. For any given application, I believe these are the main concerns to be taken into account when deciding which definition to use.
"Bottom-right is the correct graph of the function mapping R to R". No it isn't. When working in R, x^x is defined for x
I never took a complex math class in school. I knew fractional exponents had to be spending some time in the complex plane, but didn't have any intuition for it. This is a great way to visualize them and the different generalized formulas for exponentiation completely blew my mind. Thank you!
The fact that exponentiation can be multi valued like a square root blew my mind, despite having been studying math for the past 5 years
I was not expecting this. Such a simple question with such an interesting answer. Love it!
When showing the different branches of x^x, it looked as though each of the different branches were tracing out a surface. The densly discontinuous function was effectively just a cross-section of the surface, but discarding the negative solutions for positive inputs (because it is continuous on the positive solutions).
In fact, the surface it was tracing out appeared to just be a surface of revolution using any branch of x^x.
This video has really good pacing, just the right amount of detail. I can't imagine trying to explain this to someone without getting sidetracked and rambling.
Please make more of this kind of videos. It was awesome!
Wow. Its amazing how an additional dimension, makes the jump from even to odd a values continous. I never thought about it but it makes so much sense.
This was fascinating, good job.
Funny enough, a while ago I was thinking about what would happen if we considered every complex number not just single pair (r, a) but infinite set of angles a that matches the same complex number. This was exactly a small glimpse into this world. And many interesting things may await anybody brave enough to explore it more thoroughly.
Great video!
Some months ago I was looking for the X^X graph for negative numbers because of reasons (just curiosity), and I found the same 3D graph you showed on the video. I was blown away for the 3D graph with imaginary numbers, and for me is the best way to show this kind of graphs.
Additional to X^X, I wanted to look at the X^(1/X) or xth root of X for negative values, but sadly I couldn't find anything on a 3D graph as the first one, so in a sleepless night I put some numbers on a calculator for negative values of X and start plotting those results on a 3D graph, and I was again blown away because of this new graph!
It results that, in very small negative values of X (close to zero), the function goes to infinity in an spiral between real and imaginary axis, and when you start increasing the X value (for example, -100), the graph tends to 1 on the Y axis and to -0 on the imaginary axis.
Now that I look at your video I can understand why, so thanks for the info! I'm just a nerd who is interested in this kind of nerd content, but not a mathematicians in any way
you're right, I can't think of the last time I heard about a dense discontinuous function that wasn't some insane fractal nonsense. this was a brilliant video and I really hope you continue making videos about content like this!
This is so cool! The graph crossing the real at odd and even integers while spinning around in the complex third dimension is amazing
This is amazing. I was playing with geogebra and trying to find answers for this by myself and it happens I came across with this video... sadly it was a few months later! Congrats on this!
As a programmer who lives firmly in the Real Number plane... Complex mathematics are like reading poetry in another language- it's gibberish to me, but beautifully self-consistent gibberish with a hidden meaning which I can't parse behind it. Lovely video! The visualisations really helped ground me, given my tenuous grasp of what imaginary numbers even are.
8:55
I remember finding this graph on my own in high school, after taking the multiplicative derivative [which I had also rediscovered] of x^x. I got f'(x^x)=x*e, which is wonderful. I thought of using this to extend x^x to the negatives, and realized that would make the value alternate every point.
It's nice to know my reasoning made sense; I had thought maybe it was completely stupid.
It's my favorite graph of x^x, because (as far as I can tell; I don't know much here) it's the only one that keeps the multiplicative derivative equal to x*e.
I full heartedly hope for a revolution in teaching in the near future. I have had only glimpses of this knowledge on my engineering course, 30 years ago, and now it is being presented in a way a high school, maybe lesser, student could appreciate and even well understand it. And they will, for the newer children are being so so smarter than our generation. That's a wonder.
This animation was AMAZING. As a TH-camr myself, I understand how much effort must have been put into this. Liked and subscribed :)
My mind is blown!! I've always wondered what's going on with those graphs!! Amazing!!!
"These *R* × *C* graphs need more love." - I totally agree! For those who are interested in quantum physics, you will see these a lot when discussing wavefunctions in 1D space: the domain *R* will be the position _x_ (or momentum _p_ if you're working in the frequency domain) in 1D space, and the codomain *C* will be the value of the wavefunction Ψ(x) at that position.
Here's a great video on wavefunction collapse / the effect of measurement on the wavefunction that uses such graphs: th-cam.com/video/p7bzE1E5PMY/w-d-xo.html
Of course, when considering the 3D space of reality, such graphs would become 5-dimensional *R³* × *C* graphs.
Add time to that and you get yet another axis.
@ And that 6D space is everything that is, was, and ever will be!
@@JivanPal for a single quantum-mechanical particle. So to describe a single hydrogen atom you're already looking at 24 dimensional graphs (taking the proton as 3 quarks) and that is without added dimensions for spin (which you definitely need fornthe fine structure).
Mind boggling how quickly the dimensionality of quantum mechanical systems increase. We're unable (now and in the foreseeable future) to simulate an iron atom, yet we've been smithing iron for millenia.
@@JoQeZzZ To be clear, for a single _type/flavour_ of quantum-mechanical particle, and the wavefunction is only 2 of the 6 dimensions mentioned. So for _x_ types of particle, the total number of dimensions in this graph would be 4+2x. That's 3 dimensions of space, 1 dimension of time, and 2 dimensions for each wavefunction since that's a complex number. Given the 17 particles of the Standard Model, that's 38 dimensions.
@@JivanPal Your math is wrong. It's not 2D for complex numbers. Complex numbers are 1D, 1 complex dimension, but 1 dimension nonetheless. Also, we generally don't take the output dimension into account, 3D vector fields aren't 6 dimensional either, so not only is it not 1D for complex values, but the output space isn't counted as a part od the dimensionality (obviously I thought)
Also, I wasn't talking about the full standard model. I was talking specifically about a (1) proton and an (1) electron.
Finally, particles interact with eachother, so you can't just give each particle it's own dimensions like it's an independent function.
I love watching complex numbers do their funky rotations
All of these visualizations are incredibly cool. One thing that might make the “basic” x^n visualization work even better is to add a new grid along the angle that the results of negative values sweep out, so that the perspective is a bit easier to follow.
As for x^x visualizations, the coolest one to me is the spiral. I should probably have better criteria than “looks cool” but…
This is amazing! I used to not really care much about complex numbers. But now I am going to start studying them as soon as I get some time. Just incredible.
This is, put simply, amazing.
I didn’t quite understand everything, but it was easy to follow, and all around the visuals were beautiful.
100/10, amazing job!
absolutely beautiful visualization of these graphs! I experimented in desmos for quite a while, thinking about how rational indices can also produce real outputs, but the alternate visualization of y=x^x was really pretty.
amazing, you deserve more subscribers
keep coming up with this awesome content
For someone who barely understood this, I have to say the video is really good and I enjoyed it, keep up the good work
What a great video! I loved the x^x part!
Thanks for this trip down memory lane, because I had forgotten about learning to understand and derive (and at some point even 'just see') stuff like this. I loved complex graphing and signal analysis in university.
I'm a graduate student in mathematics and I never thought about these nice visualizations. What is your background?
More on branches of the logarithm would be a nice idea for another video in my opinion.
Thank you. I majored in pure math and have a PhD in applied math.
I'm a 14 years old kid Who loves maths (currently i'm studying multivariable calcus and group theory) and I love this content
I'm glad that I was able to see this video
Multivariable at 14 years is very impressive. Keep up the good work!
A quick note on the audio. At certain moments when you speak into the microphone there are these low-frequency bursts which can be quite disturbing when listening (for instance, they come when pronouncing words with the letter 'p' in them). This seems to be caused by the simple fact that you're facing the microphone too directly when speaking into it. The problem seems to go away more or less completely every time you face away from the microphone to speak to the camera, so if you want to improve the quality of this even further, I'd recommend that you look into the best angle from which to speak into the mic as it really makes a huge difference to the end result imo.
Overall however, the video was very enjoyable. Congratulations on a job well done!
Awesome video. You don’t often find quality like this from a channel without 400 other videos in the same style. Super inspiring, and looking forward to more!
THANK YOU SO MUCH FOR THIS VIDEO!! I was thinking of this problem a few days ago and i was so happy to see a video on this.
However, I have a question. if a is not an integer, couldnt we just convert a into a fraction and get a real value for x^a? For example, 2^3.5 could be written as 2^3+1/2 which is 2^3 * 2^1/2 which is 8*squareroot2
The problem is that, generally speaking, x^(m/n) is ill-defined. It is problematic to say something like x^(1/2) = radical(x), because in general, radical(x^2) is not equal to x, even though radical(x)^2 is equal to x.
I love the combination of math, logic and passion.
What kind of functions will produce graphs that goes into the fourth or even higher dimensions?
Your visualization of how this rotated through the complex plane was absolutely mind blowing. Thank you for making this!
If I would want to explore this concept more for myself, how would I set up a 3d visuslization like that? Cus it's really really good.
Hello Luc. All you have to do is define a parametric curve with explicit expressions for the real and imaginary parts. Take a look at this GeoGebra 3D interactive demo for instance: www.geogebra.org/m/wtqwhswu
we were wondering about this when we were messing around with fractional exponents in Desmos!! we had no idea you could get complex numbers from positive exponentiation, that's so cool
英語得意じゃないけど、2次元の座標では表示できないグラフがどう動くのか視覚的に分かってとても興奮しちまったなー
ちな1000コメだったわ笑
Oh that is beautiful! Thank you. I had completely forgotten this was a thing and never looked at it like this.
While I appreciate the effort that went into creating the visualizations and presenting a seemingly unifying theory of exponents using complex analysis, there are actually many conceptual problems with this theory, which may give insight as to why mathematicians do not actually use these visuals.
The problems already start with the premise of trying to generalize x^β for non-integers β. We know that x^(m + n) = x^m·x^n holds true for all integers m and n, and it holds true for arbitrary objects x, as long as x can be inverted. What is great about this is that this works not only when x is rational, real, or complex number, but it works as long as x is any object for which multiplication (or some other analogous operation) is well-defined. x can be a function, a matrix, an operator, etc. Let M just be any structure where multiplication (or some analogous binary operation) is defined, with the "usual" properties it has when dealing with numbers. If, we think of f[x] as a function from the integers Z to M, where (f[x])(m) = x^m, then we have the identity (f[x])(m + n) = (f[x])(m)·(f[x])(n). This is the defining functional equation for exponential functions of integers. In other words, we can define f[x] alternatively as the unique function f : Z -> M satisfying f(m + n) = f(m)·f(n) AND satisfying f(1) = x. If we want to be reasonable, extending the definition of exponentiation to rational numbers means that we want to find functions f : Q (the rational numbers) -> M satisfying f(r + s) = f(r)·f(s) for rational r, s. If we find such functions, then we can define f[x] such that (f[x])(1) = x, but the problem is that this assumes these conditions are satisfied by only one function. In reality, they may not be satisfied at all, or more than one function satisfies it. If M = C, the complex numbers, then we are always dealing with multiple such functions.
The way the video attempts to fix this is by appealing to the complex logarithm, and pointing out that there are actually infinitely many complex logarithms. This is that familiar appeal to "multivalued" functions I see on TH-cam these days all the time. I take issue with this, because, like with virtually every other video on TH-cam I see that talks about multivalued functions, it abuses the concept and represents it rather inaccurately. For instance, no mathematician thinks of "multivalued" functions as having "multiple outputs," nor are these objects treated in a way where you "evaluate them" at a single point to obtain a "set" of points. You will never find a publication in a mathematics journal where someone writes "log(1) = 0 and log(1) = 2·π·i and ...," because that is not actually how these objects are understood. It may be how they were first discovered, but treating them as objects you can use in arithmetic is incorrect. These objects are instead thought of as Riemann surfaces, used to evaluate contour integrals, for example. This is all to say that while there are Riemann surfaces corresponding to the extensions of exponentiation to arbitrary complex numbers, these are not objects you can do computation with, these are not objects you evaluate at a point to obtain multiple values, nor are they objects you do arithmetic with. So, they do not work at all in the way that the video explains them, and this includes things such as the complex logarithm and the nth roots. Also, none of this is applicable if you are trying to define exponentiation with rational exponents (or beyond) for things like matrices and functions. It only works if x is limited to the complex numbers. As such, this is a lot less illuminating than it looks to be on the surface.
Of course, I do want to point out that this is not so much an issue with the video, as it is more generally with how exponentiation is taught by sources online in general, and is also more so an issue with how videos on TH-cam completely and universally misrepresent the topic of the so-called "multivalued functions," which are more properly called Riemann surfaces. Leaving these issues aside, the video does do an excellent job at presenting a new spin to a not very popular concept that is hopefully helpful in some ways.
@Jack Salzman Do I care?
I love complex analysis stuff, it is so beautiful and elegant.
bro I swear you have to receive the gold medal of this year's SOME
Can't believe this is your only maths video. This is super good!!
Beautiful graphs, I'm an architect, but I love looking at graphs of functions like this. I have a massive collection of pictures of curves on my laptop.
Good luck for SOME2 absolutely brilliant!
Edit: Ok seriously Grant Sanderson, this video deserves at least a mention on your channel...