A basic vector addition puzzle turns into an exciting dive into Euler's formula and Geometric Series

[@intuimation.314]

Something I'd like to highlight is that I didn't mention how does the sum transform into an integral as n → ∞? The key idea comes from the concept of a Riemann sum, which approximates the integral of a function by summing its values at discrete points and multiplying by the width of the subintervals. As the number of intervals increases, the width of each subinterval becomes smaller, and the Riemann sum becomes a more accurate approximation of the integral. If you'd like to see the detailed explanation with LaTeX code for this transformation, you can check out the document here: www.overleaf.com/read/fxrswmhfmpws#6671bf Also, if you want a full video explaining Riemann sums and how they transform into integrals, feel free to let me know in the comments, and I can make a dedicated video on that topic!

[@shreejalbhattarai4635]

Here are my observations from this video: 1) This might be your best video so far; 10/10 for the concept. The animations have been a lot better. I've noticed that the explanation time and the time consumed by each frame match precisely. The length of the video is fantastic as well. One thing I'd suggest is revising the script once you finalize everything because minor mistakes can make the soup sour for critiques. In 11.41, there is a small spelling mistake, again, which I admit should be the least of concern, but I've included it for your retrospection. 2) I like how you connect a simple thing, like the vector sum of symmetric discrete vectors to zero, with electric field intensity and the fact that you were able to generalize your conclusion to continuous value systems. Excellent work on that. 3) I know that videos like these take a lot of time to prepare, and putting consistent work into these requires a great deal of work and dedication. I'd suggest putting the same work on your vocals as well. Try to pronounce the words a bit clearer and practice the script beforehand. I'd also suggest you record this audio in parts. Again, in no way should my suggestions be among the main takeaways from the audience's reaction, but I tried to sum it up as a well-wisher of this channel.

[@intuimation.314]

Thank you so much Shreejal for the detailed feedback! I'll take your suggestions to heart and make the necessary improvements.

[@nortonsm]

Great video! Appreciate the hard work in putting these together. One small comment on the proof that I had to “prove” to myself and maybe others might be interested. I worried that there could be some n where the denominator would go to zero in that expression for the geometric expansion ( either the real or imaginary part of the denominator since it’s complex, i.e 1 - exp(2*pi*i/n) ) and therefore would not make the proof general for all n integers > 1. But then, it occurred to me that the denominator represents a vector from a point in the complex unit circle to 1 on the real-axis and other than n=1, the trivial case, this will never be zero in the real or imaginary component. Anyways, thanks for pointing out this proof and letting me have some math fun!

[@intuimation.314]

Thanks for sharing your thoughts and for taking the time to dive deeper into the proof. You're absolutely right-if you picture the vectors on the complex plane, it becomes clear that the denominator will only go to 0 if the imaginary part is 0, which happens only for the 0th vector. No other vector is purely real, so the imaginary part is never 0 elsewhere. Glad you enjoyed exploring this proof!

[@nartoomeon9378]

We can also use rotational symmetry - if you myltiply every vector from given set by *exp(i*2π/n)* , the set does not change and vector sum(Σ) too: *Σ=Σ×exp(i×2π/n)=Σx* This equation works for any *Σ* if the coefficient on the right side = 1, but *exp(i×2π/n)* ≠ 1 ! So, only possible solution is Σ=0. But we need to know that *x* coefficient is not (left/right) zero divisor. If so, we get more than one solution.. which breaks vector sum as function. But complex numbers form a number field, any non-zero complex number is not a zero divisor. The answer was right - *Σ=0* ✅

[@intuimation.314]

The rotational symmetry approach is a really cool way to think about it. I hadn’t thought about the role of zero divisors in the complex field like that-great point! Appreciate you taking the time to share this!

[@nartoomeon9378]

​@@intuimation.314 Im sorry, but my point of view was not so clear and full. Equation Σ=Σx for x≠1 has more than one solution not only if x is zero divizor, but if x preserve Σ under multiplication in another way. Such case is possible in rings, for example: 7*2=2 (mod 12) [Z/12 commutative ring]. But wait, we can use properties of the ring and rewrite our equation in the next way: Σ=Σx => Σ-Σx=0 => Σ(1-x)=0. Now, we can see how to solve it.(zero divizors wins) We don't need "preserving" property, multiple solutions possible Only if Σ or (1-x) is one-side zero divizor , or one of them two-sided... Anyway, my intuition was not so bad. All of these is my self-critique, but I think it useful for you too.

[@pizzarickk333]

I remember learning about this proof when I was first introduced to complex numbers. Nice video.

[@intuimation.314]

Thanks! I love how elegant this proof is.

[@scifi2sci]

Great work! The microphone and voice could be better, but overall really great.

[@woaitsbren]

agreed, enjoyed the visuals a lot, would have prefered clearer microphone quality

[@intuimation.314]

Thank you! It's recorded in Mobile. I'll try to improve the audio quality in future videos.

[@benhernandez]

This is awesome,thank you. Im learning higher level math later in life and this fills my curiosity of why things work like they do and how. It was interested that you included code, this is something i want to try out

[@intuimation.314]

I'm happy you enjoyed the code - feel free to play around with it!

[@jakeaustria5445]

Thank You ❤

[@JohnSmall314]

Very interesting and very well done. I subscribed

[@apoorvvyas52]

Beautiful

[@fernandopauli49]

Beautiful!

[@shankar...........]

software used in this video for teaching >????

[@intuimation.314]

It's not a software actually. To make animations I use Python Library called the MANIM. But yeah I use video editing software to combine my animations and voice

[@SEE2080jp]

How do you make the voice + visuals

[@intuimation.314]

I start by writing a script for the video. Then, I animate the visuals using MANIM. For the voiceover, I record it on my mobile phone. Finally, I use video editing tools to combine the animations and voice seamlessly.

[@SEE2080jp]

Do you animate all scene and voice over

[@intuimation.314]

Yeah👍

[@mapleigue]

Are you using manim?

[@intuimation.314]

Yeah the community version

[@AbhishekJohn-g5y]

👍

[@sakshampokhrel42]

Are you from nepal? If not then where are you from?

[@intuimation.314]

Yes I am, but I live in the United States🇺🇸

[@sakshampokhrel42]

@@intuimation.314 as i though man the education system here is trash even torwards acedimic subjects like mathematics i want to leave as soon as i can so i can persuite my dream of being a mathematician one more question were you born there is the us or born here and later went there.

[@antormosabbir4750]

Souns Bangladeshi 🤔

[@intuimation.314]

😂Really. I am from south Asia though.

[@antormosabbir4750]

Yes brother! We speak in this accent, not like Indian accent

[@braineaterzombie3981]

Audio isn't smooth. If you don't have good setup, you can use ai voice. There are many free tools available. Also work done is way less with them However animations and other things are fantastic

[@intuimation.314]

It's definitely something I'm working on improving. I'll also look into using AI voices.

[@crimsnblade8555]

please no ai voice, I like my maths videos human