Видео

22:23 - What you did is represented negative numbers with arguments of -pi and pi, got two adjacent roots of the equation, rotated them by third of a circle getting a third one and an overlap, and rotated them again, just getting two more overlaps. It makes sense to look at it that way! But to get all the possible outputs for f(x) = x^(1/p) from the very beginning, consider negative values to be of any argument of the form pi + 2pi•k, with the distinct solutions for k є {pi, 3pi, ..., 2pi•(p-1)} and for circular arguments to be from θ є [0, 2pi•p).

Note for myself at 22:02 - this only gives imaginary values because with the inputs of the same moduli we only consider arguments ranging from -pi to pi, so the function output for the negative numbers (we assign the arguments of -pi and pi to them) will be -1/3•pi and 1/3•pi respectively. But there is another argument that gives negative numbers corresponding to a different function output - multiples of 3pi, which will give negative numbers of argument pi. The part of the cube root curve is missing because we only represented negative numbers with two arguments of -pi and pi, whereas they could be anything of the form of pi + 2pi•k, where k is an integer. They are periodic for rational powers, but if you consider irrational powers it'd give a different complex number for every value of k. z=(-1)^√2 gives infinite solutions.

Just to note for myself, 19:03 - The line that splits is the transformation of the real axis via z = x + if(x), where negative numbers split in two because for odd values of pi, which are in their argument, something like e^(1.1pi•i) and e^(-1.1pi•i) will not match. The circular curves of the same colour contain the transformations of all numbers with the same modulus - distance from the origin, starting from -pi to pi. They intersect with the real number transformation curve at three arguments: -pi, 0 and pi; for even powers the numbers of argument -pi and pi always end up at the same place, so the real transformation curve won't split and circle transformation curves will be enclosed.

Hello there. Thank you for creating such an informative and engaging video! Congratulations on its success. I plan to incorporate it into my high school Advanced Topics class, which consists mostly of Grade 11 and 12 students who have completed AP Calculus BC. This year, we dedicate a full semester to Complex Numbers, including visualizing functions like f(z). Your video is perfectly suited for my class, and I will definitely use it in my lessons. Additionally, I plan to use Diffit AI to transcribe the video for further reference. However, I intend to utilize GeoGebra instead of Desmos (which you demonstrated) to help my students visually explore conformal mappings, specifically the mapping of points z on the unit circle to the complex plane using f(z)=z^2. Based on your video, I believe the mapping was z=x+(x^2)*i. I attempted to replicate the graph using the following commands in GeoGebra, but my graph differs from yours. I would greatly appreciate your guidance on how to adjust my approach. I've attached my GeoGebra file setup for your reference: www.geogebra.org/calculator/fmdscxat Thank you in advance for your time and insights! Best regards, Misael Fisico, ASIJ Tokyo Japan mfisico@asij.ac.jp

You've made really amazing video ! Thanks y lot !

Beautifully done

O okay I like that generalization because the focus is on a specific case that doesn’t generalize well It’s a useful and easy case but composite numbers get tricky

This overlaps with the 'Rethinking the Real line' video really well

I had an interesting idea. What would happen if you took Pascal's Triangle and made it 1 face of a square base pyramid, and made the other 3 sides the equivalents of the triangle but for i, -1, and -i? What would happen on the inside?

I suppose it's infinite if it's infinitely nothing. 😂

this makes no sense. why are you talking about oranges?

15:05 German Iron Cross

So why is (the inverse of) temperature equivalent to cyclic imaginary time? Is there an intuitive way to think about that?

Hm but then by this logic there's also no interpretation here of dividing by 1/2, but we have defined this still. Clearly something not making sense in this scenario doesn't imply it can't be mathematically sound

Something you could look into would be Geometric Algebra, seeing as it allows you to divide vectors, would be cool to see if everything still works out the same. These videos have all been extremely interesting, and I can't wait to see what topic you will cover next. Notifications on for sure!

Usually you can't divide by zero just because you can't.) It's not undefined.....it is impossible.) Common...... 20/5=4... Right? Means keep subtracting five from twenty till you get to zero and count the amount of subtractions.... But try to subtract zero.... ................ Man......something is wrong...... I can subtract zero.....infinite number of times. On an abstract level. (Like, subtract number zero from number twenty, get twenty....repeat....get twenty again, repeat....and go on forever.) Though, I can't subtract zero amount of things on a concrete level..... Wtf. (Like, there's no way am capable of subtracting zero amount of sticks, from a set of sticks.) Just when I thought I was gonna super smart here.... So, the conclusion. With real, actual things....I can't divide by zero. But with numbers....I, seemingly, can.

hi :3 UwU

Goto 3d animation, [[x, Re(x^n), Im(x^n)], x = -10 .. 10], n = 0 .. 2 n is the animation variable

The Phiangle!

20:50 Finally, imaginary angles!

I wish I could have seen those last animations on an oscillating 3d graph😭

hi

"Complex Variables" by John W. Dettman (published by Dover) is a great read: the first part covers the geometry/topology of the complex plane from a Mathematician's perspective, and the second part covers application of complex analysis to differential equations and integral transformations, etc. from a Physicist's perspective. For practical reasons, a typical Math Methods for Physics course covers the Cauchy-Riemann Conditions, Conformal Mapping, and applications of the Residue Theorem. I've used Smith Charts for years, but learned from Dettman that the "Smith Chart" is an instance of a Möbius Transformation. The Schaum's Outline on "Complex Variables" is a great companion book for more problems/solutions and content.

The real question is Discord serber wen

fun convergence to phi fact: in geometric algebra, you can (kind of) divide vectors (most vectors have multiplicative inverses), and by starting with two vectors and following the Fibonacci recurrence relation with vector addition, these two vectors actually do converge to the golden ratio, i.e. multiplying one by the inverse of the other approaches the golden ratio

"divide the cake into 0 identical pieces" ok I eat the cake, give it a few hours for my digestive system to make infinite cuts and you are left with no cake.

Cool stuff, but something I've come across is partial derivatives and whatnot, and the obvious fact that although mapping from complexes to complexes may appear to require 4D plotting, on the contrary, every x + iy for reals x and y defines a plane, but the conjunction of two planes orthogonal to each other with one of their axes collinear suffices to define an infinite Euclidean 3-space just as the conjunction of two sets A*B forms a plane. Letting C denote the complexes, our Euclidean 3-space formed from C*C would be visualized by mapping all X + iy to f(X+iy), and all x + iY to f(x + iY) where X and Y are held constant as x and y vary over time. Wolfram Alpha just visualizes complex plots by showing the 3D plot of the real and imaginary parts separately since at that point, you have only 3 variables. Alternatively, perhaps it might be shown in 3-space two planes parallel to each other, and the path that each point follows according to the rules of the particular function over the complexes, so for example, f(z) = z would, for some finite rectangular neighborhood of input points, just show a rectangular volume; f(z) = z^2 would probably show for real t, |z|^(t+1) e^(i(t+1) arg(z)) as t increases linearly with time which would show some kind of cylindrical solid when Re(z)^2 + Im(z)^2 is less than or equal to 1 with funny transitional stuff in the neighborhood of 1, but a sort of hyperboloid volume outside that; etc. Sorry, I went on a tangent there (pun not intended). Anyways, is there a definition or analog of the derivative for 3-space for the intuitive idea of the plane tangent to a point on a surface?

0:30 "and along the way, learn how to *extend* the concept of derivatives to complex valued function" That was nice

Your videos are getting better and better. What I like most is your ability to describe familiar concepts (here, the derivative as a ratio of input to output) in unfamiliar ways. It feels like taking the ideas I first learned in high school and slightly tweaking an “abstraction” knob. Fitting, given the beautiful images at the end of this video!

"Even if everything else somehow made sense, we still could not divide vectors." That's where you're wrong kiddo! 😎 It wouldn't make a huge difference though, since the result is effectively a Complex number anyway (assuming the vectors are 2D), which is distinct from a 2D "vector" in this system despite having the same components.

Polar coordinates of a parameterized space are very similar to a Fourier transform.

A truly remarkable video...thank you so much for the education and entertainment!

Nifty AF !

Absolutely beautiful.

I took a complex analysis course a few years ago, so my recollection of it is somewhat limited now. Videos like this really make me stop to think about about questions I don't even really know how to ask, such as questions that involve exploring non-analytical cases with singularities. I love these videos, and they are a massive breath of fresh air to get my mind off my studies and onto something else, if only for a few minutes.

I am just learning the derivative concept, and seeing this with complex numbers is just astounding. Also great explanation, I could understand pretty much everything even though my math level is below this topic, which makes me way more curious. Your content deserves to be at the top

I've just got to say that this is the best explanation of the complex derivative that I have ever seen.

I like your channel, but I really dislike this approach to imaginary numbers, saying that they are just a lateral movement in space. If that's true, we could just as well use 2D vectors. It's the algebra embedded within complex numbers that makes them special, and the deeper intuition is that complex numbers completely classify all translations, rotations, reflections, and scalings of 2D space, just like all real numbers classify translations and scalings of 1D space

Espectacular 💯🙌🏼

I wonder ... well one thing I have wondered about with graphical approaches is: what happens to curves and lines when the axes values are rescaled? It could be either axis on its own or multiple axes on a 1:1 ratio or some other ratios. Would it give further graphical insights into patterns and lead to pattern recognition? Excellent video!

Absolutely fascinating!

1:29 "Group" is a mathematical term reserved for group theory. Please do not use it in this context, unless you are relating complex analysis with group theory.

I tried playing with parametric representation on desmos to visualize space transformation when using complex variables, this really reminded me of that. In a similar vein I wanted to understand and play with raising non-unitary complex numbers (a+bi) to non-unitary complex powers which lead me to finding my favorite number: Gelfond's Constant (-1)^(-i)=e^pi.

2:26 bruhken

ITS beautiful, to See that there ist only one Infinity in complex world, Not a plus and minus Infinity.

in physics, many of these properties occur in quasicrystal diffraction

That was a very high quality video of getting deep with fibonacci's sequence. I understood that there always is more to explore, just by using the basics and converting them one to another.

2:53 two equals signs on the short bar

These animations, together with their 'complex analysis', are 'real' gems of mathematical 'imagination'! (all 'puns' intended)

Wow, subscribed and added to favorites!