Why imaginary numbers are needed to understand the radius of convergence 

stupid people: ? genius people: { }

Please make a video on what is mechatronics and its future

MATH QUESTION Numerical / algebra qué. ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-soN5NmkaXeM.html One time see

im confused, for stability sigma shud be negative decaying , but region of convergence says it shud be positive . whats happening?

I gotta ask, why does it stop any other values outside of the radius like any value not +/-i? For example the Dirichlet series has an abscissca of convergence instead of a radius where it converges for all values past the singularities, that makes sense. Why is the Taylor series however, a radius?

wHaT dO yOu MeAn?¿?¿? iMaGiNaRy NuMbErS aRe InCrEdIbLy UsEfUl FoR tHiNgS LiKe SiGnAl PrOcEsSiNg!¡!¡! (Btw love your channel and the math community in general, you guys are so tight-knit)

i and j are your best friends in college....

@@phillipgrunkin8050 :)

@@AndrewDotsonvideos Nice to see you on one of Zachs videos.

Imagine not being able to Wick rotate.

@ ikr, could never be me

+1

@ why too dry or dense and boring?

@@leif1075 i don't get the subject of your sentence: If the subject is "reading" then "i don't have an interior monologue, so it sucks for me", if the subject is me then "Yes"

This book is great. Im actually studying it right now

I'm dying to learn more! What I really want to know is what property does a function need to have in the neighbourhood of a point (in addition to being infinitely differentiable) to make it possible for its values to be approximated by a Taylor series?

@@MrAlRats It depends on where you're doing analysis. The nicest set is of course the complex numbers, since there are a lot of conditions that are equivalent to analiticity. For example, if a function is holomorphic at a point, that is enough to ensure the existence of a Taylor expansion (around that point). Obviously this is not true for functions over the reals. As a matter of fact, there aren't any nice characterizations of analytic functions over the reals that I'm aware of. You can also look at analytic functions over the quaternions. Unfortunately, analiticity is a very restrictive condition in this case. If I recall correctly, not even linear functions over the quaternions are "quaternion" differentiable. Some are, but not all of them. In a sense, the reals are too small to see the whole picture and the quaternions are too big to be well-behaved. The sweet spot is the complex numbers.

Complex analysis is one of the most beautiful areas of mathematics.

You probably do 😃 I could swear age doesn't matter here. A minimum age would probably be 5 or 6, by the time you get the hang of talking basically lol Otherwise 40 or 12, you can understand even the "hardest" maths. My personal opinion though.

Yimo Awanardo That may just make me an idiot, but thank you. 🙂

@@anonymousdude7982 dont worry, your not an idiot, I have no idea what kind of crack that dude is smoking. Until you have a fundamental understanding of basic calculus, which requires advanced algebra and trig, you (rightfully) should have no idea what a taylor series is. Just wait and your time will come

Well, you probably already know imaginary numbers, you will soon learn what derivative is and then you will learn Taylor series. Taylor series are just polynomials that approximate functions. They can approximate functions as close as you want them to (by having more and more terms in the polynomial), as long as the function that you want to approximate is analytical. The way you set up the polynomial is that you make sure that derivatives for some input of that polynomial match the derivatives for that same input of the function that you want to approximate. For instance, you set up the first derivative at x = 0 of your polynomial to be equal to the first derivative at x = 0 of the function that you want to approximate. Then add another term in the polynomial such that the second derivative at x = 0 is the same as second derivative at x=0 for the approximated function. And so on. I don't know what math curriculum is where you live, but it is possible that you will learn about derivatives next year. Then you can go back to this video, read this comment again and understand what this video is about.

Fake it till you make it

1D disks/balls exist and they have a radius and a surface. In fact, the intersection of a circle and a plane is exactly a 0-sphere, which is an object in 1D with a center and a radius, but only actually contains 2 points.

he is getting close to his mission of [redacted]

Not a pentagram tho

It's now 667K subscribers.

MATH QUESTION Numerical / algebra qué. ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-soN5NmkaXeM.html One time see

technically it still hasn't, and won't ever be calculated in full

Demir Sezer I thought so, but the urge to point out a slight error overruled

Henry Ginn I don't think its an error, its purposely technically true

-COOKIEZILA - correct, I phrased it badly

Henry Ginn technically at 10^-34 you already calculated pi for all real world applications seeing as Physics breaks down at that point and we don’t know if what happens after that.

All numbers are imaginary

@@user-cs1qv9cm9r And all numbers are real. Even the imaginary ones

Yeah this doesn't actually dive into the 'why' but that's because it is much more difficult to explain that, you have to dive further into complex analysis which is way beyond a video like this.

check out the video on it by 3Blue1Brown

@@mrmoinn Yes bro I have seen it and it was awesome too, but I am suggesting it to him too, because I think he can elaborate it on more of the practical side, with all its abstractness that Maths has to offer.

same

Probably not. Complex numbers complete algebra and perfectly describe polynomials and calculus. You start losing algebraic properties from quaternions and it becomes a mess.

With regard to arithmetic closure, the complex numbers should be all that's needed as far as I'm aware. That said, there are alternatives that are mainly useful for different geometries, so the quaternions are best for representing rotations in 3D space, while the split-complex numbers are great for working with hyperbolic geometry.

For this one you can find the software used in the description :)

@@zachstar thanks a billion

In the description :)

@@zachstar oh ty love ur videos :) Still remember when it was used to be MajorPrep

The software I use is in the description :)

@@zachstar Thank you😃😊

In the description!

Shimmy Shai 😂 yes they did

Your edit is correct, I was just plotting the magnitude since that's all that was needed to show the singularities. I couldn't done phase with color but the program I was using doesn't seem to allow me to do that (I can only change color based on the z value)

@@zachstar Thanks for the reply and the video!

i agre - i am making this comment to see how many comments are added by the time i reload this

Now it's at 120 x 10 views

@@princelumpypackmule1101 Now it's at 120^10 views...or at least it should be