What is the graph of x^a when a is not an integer? An unusual look at familiar functions

Рет қаралды 431,292

Күн бұрын

This is my entry into the Second Summer of Math Exposition. I used to wonder about these questions ever since my first year in university, so in a sense this is a rather personal subject for me. I hope that this video reaches someone else who is also wondering about these issues.
Here's the Desmos document for visualizing the n complex roots of any complex number:
www.desmos.com/calculator/1v4...
Videos mentioned in order of apparition:
e to the pi i, a nontraditional take (old version)
• e to the pi i, a nontr...
The 'Everything' Formula - Numberphile
• The 'Everything' Formu...
Winding numbers and domain coloring
• Winding numbers and do...
The 5 ways to visualize complex functions | Essence of complex analysis #3
• The 5 ways to visualiz...
Visualizing Complex-Valued Functions
• Visualizing Complex-Va...
Jeff Tupper's webpage on the graph of x^x
www.peda.com/grafeq/gallery/ro...
A relevant Stack Exchange question and answer that touches on these ideas:
mathematica.stackexchange.com...
Chapters:
00:00 Intro and problem motivation
01:49 The R-to-C graph of x^a
04:37 The R-to-C graph of a^x
06:37 Rotations in complex exponentiation
07:27 The R-to-C graph of x^x
08:17 An alternative version of the graph of x^x
12:37 Recap of the various graphs of x^x
13:49 Homework
14:09 Summary and outro
Music by Vincent Rubinetti
Download the music on Bandcamp:
vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
open.spotify.com/album/1dVyjw...

Пікірлер: 1 000

@a.arredondo Жыл бұрын

*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.

@LunaAlphaKretin Жыл бұрын

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!

@SirRebrl Жыл бұрын

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.

@eriktempelman2097 Жыл бұрын

... not to mention the link to sines and cosines. My jaw was in the basement at that point.

@Kapoosh000 Жыл бұрын

@@SirRebrl I love that about math. I would have never guessed it but when you see it makes so much sense

@G8tr1522 Жыл бұрын

same

@luckylmj Жыл бұрын

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

@DynestiGTI 2 жыл бұрын

Complex topics don't have to be dumbed down, all it takes is a good visualisation and it can be taught to anyone.

@hOREP245 2 жыл бұрын

Some topics cannot be visualised in an easy way, or a way that doesn't obscure everything.

@DynestiGTI 2 жыл бұрын

@@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.

@clobre_ Жыл бұрын

@@DynestiGTI this deals with different values of x, so it's kind of already 4 dimensional

@MeshremMath Жыл бұрын

You can visualize a 4D perspective quite well with a volumetric view.

@harriehausenman8623 Жыл бұрын

Complex problems need to be approached very real. 😆

@PretzelBS Жыл бұрын

I love how math is always self consistent. Literally spins around in the complex plane to get to where it needs to go

@blinded6502 Жыл бұрын

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.

@perrymaskell3508 Жыл бұрын

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!!!

@blinded6502 Жыл бұрын

@@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).

@blinded6502 Жыл бұрын

@@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.

@warmesuppe Жыл бұрын

@@blinded6502 I think kurt gödel will disagree.

@johnredberg 2 жыл бұрын

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!

@harriehausenman8623 Жыл бұрын

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 🤔

@JivanPal Жыл бұрын

@@harriehausenman8623 Sedenions.

@harriehausenman8623 Жыл бұрын

@@JivanPal thx!

@samuelding7854 Жыл бұрын

I agree, this video os amazing at generalizing functions that look like they abruptly cut off at the y-axis

@adrians9367 2 жыл бұрын

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 :]

@harriehausenman8623 Жыл бұрын

absolutely fantastic

@theshuman100 Жыл бұрын

ah nothing like the extradimentional rotating arm

@mihailmilev9909 Жыл бұрын

@@theshuman100 exactly lmao

@drawing-ology 2 жыл бұрын

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

@usm1le Жыл бұрын

You're not dumb. My dumbass got taught this in 11th grade

@Amoeby Жыл бұрын

It's not really a high level math but it's still beautiful nonetheless.

@harriehausenman8623 Жыл бұрын

@@Amoeby C'mon. for some it is. 🤗 For a while. 🤓

@angelmendez-rivera351 Жыл бұрын

@barutaji "will never hear about it" does not equal "advanced."

@impanthering Жыл бұрын

@@angelmendez-rivera351 ok bufdy

@lloydgush 2 жыл бұрын

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.

@angeldude101 Жыл бұрын

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.

@rokisok9618 Жыл бұрын

@@angeldude101 holy I never thought about it like that

@pavelperina7629 Жыл бұрын

@@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)

@angeldude101 Жыл бұрын

@@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.

@TheLangendork Жыл бұрын

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.

@michaelleue7594 Жыл бұрын

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.

@carstenmeyer7786 Жыл бұрын

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!

@harriehausenman8623 Жыл бұрын

Yeeesss 🥳

@gooblepls3985 Жыл бұрын

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} ]

@gooblepls3985 Жыл бұрын

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 :)

@FedericoStra Жыл бұрын

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.

@angeldude101 Жыл бұрын

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.

@simlee6177 Жыл бұрын

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.

@johnchessant3012 Жыл бұрын

2:00 This animation is pure awesome!! I will never be looking at x^a graphs the same way again

@Hextator Жыл бұрын

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...

@Tabu11211 Жыл бұрын

Same! That was be worth a video.

@hkayakh 7 ай бұрын

I’ve been looking for this video for half a year because the question in the thumbnail alone is enough to get me hooked

@WillyFlowerz 11 күн бұрын

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

@IqbalHamid Жыл бұрын

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.

@edswalkthroughs 2 жыл бұрын

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!!

@kharybdis Жыл бұрын

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.

@tomasarredondo6524 2 жыл бұрын

very interesting and educational!

@andysleeper1283 2 жыл бұрын

The animations are beautiful, and you tell the story in a very accessible and engaging manner. I look forward to future videos from you!

@SzanyiAtti Жыл бұрын

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.

@aquamanGR Жыл бұрын

I love it! I had not seen this definition of x^a in the complex plane. Beautiful.

@brunesi Жыл бұрын

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.

@mattkerle81 Жыл бұрын

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.

@Itangalo Жыл бұрын

Awesome video. I love how calm, pedagogical and humble you are.

@primenumberbuster404 2 жыл бұрын

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!!!

@draco18s Жыл бұрын

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).

@jasperiscool Жыл бұрын

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. ♥

@CreativeDimension 2 жыл бұрын

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 KZfaq.

@eriktempelman2097 Жыл бұрын

I keep coming back to this remarkable video. Watched it at least four times, shared it many more... it's a true masterpiece.

@monoamiga Жыл бұрын

This is definitely one of the most beautiful math videos over the Internet. Thank you mr Armando!

@theodote_ 2 жыл бұрын

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!

@silviavalentine3812 2 жыл бұрын

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 :)

@harriehausenman8623 Жыл бұрын

Yes! With this production quality, I would love to see even some more basic topics approached by this creator. 🤗

@jamesdurtka2709 Жыл бұрын

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.

@ClementinesmWTF Жыл бұрын

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.

@chennebicken372 Жыл бұрын

The best visual of "rotating through another dimension" I have ever seen. You did a really good job. 👍❤

@elliotb1137 Жыл бұрын

This was brilliant. I love getting mew insights to familiar subjects in math, especially in regards to complex numbers.

@quintusthegnome7429 Жыл бұрын

I have been wondering about this since forever! Thanks so much for the detailed and concise video. Awesome work!

@gregorymorse8423 Жыл бұрын

Excellent to show and explain the full catalog of visualizations. Very satisfying

@deadall127 Жыл бұрын

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👍

@kenhaley4 Жыл бұрын

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!

@leofoxpro2841 2 жыл бұрын

u don't know how long I was looking for the graph of x^x on the complex plane, thank you so much !!

@newperspective5918 Жыл бұрын

I was not expecting this. Such a simple question with such an interesting answer. Love it!

@topekian Жыл бұрын

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.

@EAdano77 Жыл бұрын

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 Жыл бұрын

How did u know about how these graphs are in high school, i just barely got to know about RxC ...

@EAdano77 Жыл бұрын

@@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.

@DavidSartor0 Жыл бұрын

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.

@FadkinsDiet Жыл бұрын

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.

@DavidSartor0 Жыл бұрын

@@FadkinsDiet Can you give an example?

@BartvandenDonk Жыл бұрын

Marvelous and ingeniously done. The transition to complex numbers beautifully shown it made my day. Thanks for pointing out this topic.

@whyami5355 Жыл бұрын

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.

@brollejunior 2 жыл бұрын

Really interesting, especially the x^x part! Nicely visualized too.

@belgaer4943 Жыл бұрын

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

@Mo-dg2iu Жыл бұрын

I really like the calming vibe in the video, the music goes perfectly well with your voice and style of explaining. Keep it up!

@livinlicious Жыл бұрын

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.

@olivialambert4124 Жыл бұрын

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.

@lexinwonderland5741 2 жыл бұрын

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!

@benjamingraemegorman7657 Жыл бұрын

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!

@zhoutai182 10 ай бұрын

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

@alekseyprokopev7409 Жыл бұрын

Please make more of this kind of videos. It was awesome!

@kevdrag Жыл бұрын

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!

@fmcountto123 Жыл бұрын

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!

@tekkorb778 Жыл бұрын

This is so cool! The graph crossing the real at odd and even integers while spinning around in the complex third dimension is amazing

@kasuha 2 жыл бұрын

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.

@ManjotSingh-sf2ri 2 жыл бұрын

amazing, you deserve more subscribers keep coming up with this awesome content

@ThomasKundera Жыл бұрын

Really love the calmness and effectiveness of your presentation going to the point using the proper math words for it and the beautiful illustration revealing part of the magic of math under our eyes You deserve more subscribers and you just have one more 😀

@bobjohnthemonkeyman Жыл бұрын

So well explained and visualised! I can't believe this is your only maths video! In any case subscribed and happy to see more!!

@DavidSartor0 Жыл бұрын

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.

@PowerhouseCell Жыл бұрын

This animation was AMAZING. As a KZfaqr myself, I understand how much effort must have been put into this. Liked and subscribed :)

@Andrewkosche Жыл бұрын

This answered a questions I’ve been wondering in my head for years now. I really appreciate this video!

@barsanyibela4027 Жыл бұрын

This is an excellent explanation, putting this whole thing into context. Thank you.

@ShapelessMonstrosity Жыл бұрын

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.

@minimath5882 2 жыл бұрын

What a great video! I loved the x^x part!

@filippomariachiappini1257 2 жыл бұрын

The quality of the animations is great! I also saw that weird graph and tried to explain it to myself, and this video helped me nicely.

@quiquito87 Жыл бұрын

This was absolutely awesome! It absolutely blew my mind! And it was so beautifully explained.

@RonWolfHowl Жыл бұрын

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.

@FedericoStra Жыл бұрын

"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

@PaulFisher 2 жыл бұрын

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…

@meteorinc4259 Жыл бұрын

What an amazing explanation of a “complex” idea. Crisp animations and slides with great music. I’m loving your style (that wink was great lol). Please keep making amazing content. Thanks Armando!

@dranrebjamescucio4881 Жыл бұрын

mind blowing content used with good visualisation and animations. I understood very quickly! Thank you for the video!

@whisper3856 Жыл бұрын

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!

@JivanPal Жыл бұрын

"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: kzfaq.info/get/bejne/pp2Srahiqpu0fY0.html Of course, when considering the 3D space of reality, such graphs would become 5-dimensional *R³* × *C* graphs.

Жыл бұрын

Add time to that and you get yet another axis.

@JivanPal Жыл бұрын

@ And that 6D space is everything that is, was, and ever will be!

@JoQeZzZ Жыл бұрын

@@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.

@JivanPal Жыл бұрын

@@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.

@JoQeZzZ Жыл бұрын

@@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.

@eleanorf.7892 Жыл бұрын

This is an awesome video! I’m taking intro to complex analysis this coming year and I’ve been trying to do some learning in advance to wrap my head around complex numbers, and this video made things click in such an interesting way! Thanks for making this!

@timothyrosenvall1496 Жыл бұрын

Phenomenal job! This is the best explanation of these graphs I’ve ever seen! Wonderful work!

@skyetharsis7136 Жыл бұрын

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.

@a.arredondo Жыл бұрын

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

@shiraishichan3944 Жыл бұрын

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

@angelmendez-rivera351 Жыл бұрын

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.

@Benjax_95 Жыл бұрын

Great video! Just found it in my recommendations and I'm absolutely amazed by your analysis and your work. This was very interesting and educational. Lately I've been very interested in the complex numbers world and this just fascinated me by how smooth you described all of this information :) Great work! 😁

@phantoMpikachu Жыл бұрын

this was a fascinating concept to try to grasp and was explained beautifully! also you have a very nice soothing voice (: thanks for this video!!

@PW-qi1gi 2 жыл бұрын

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.

@a.arredondo 2 жыл бұрын

Thank you. I majored in pure math and have a PhD in applied math.

@dannyleung2796 Жыл бұрын

What kind of functions will produce graphs that goes into the fourth or even higher dimensions?

@alanramsayuk Жыл бұрын

This is absolutely gorgeous, in ideas and visuals, and your delivery manner is excellent (I learnt only to a limited level of maths formally, but working in CG had prompted interest in expanding that) I've subscribed in hope that you'll produce more of this sort of thing

@lewismassie 2 жыл бұрын

I saw this video in my feed and thought "oh yeah I'll see what Desmos says about it", saw the weird glitchiness, then watched the video. Very cool stuff

@timpani112 2 жыл бұрын

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!

@user-lo8fb5qf9u Жыл бұрын

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

@ianwood9780 Жыл бұрын

great video, i have forgotten so much. I like your gentle path from Real to complex to expanded definitions of exponents.

@sergiokorochinsky49 Жыл бұрын

This is one of the best some2 videos I've seen so far.

@angelmendez-rivera351 Жыл бұрын

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 KZfaq these days all the time. I take issue with this, because, like with virtually every other video on KZfaq 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 KZfaq 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.

@angelmendez-rivera351 Жыл бұрын

@Jack Salzman Do I care?

@hvok99 Жыл бұрын

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. 👏👏👏

@johanngerell Жыл бұрын

Glad to have found this channel - super high production quality!

@ErikHuizinga Жыл бұрын

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.