Fundamental Theorem of Algebra

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  • เผยแพร่เมื่อ 29 ธ.ค. 2024

ความคิดเห็น • 79

  • @blackpenredpen
    @blackpenredpen 6 ปีที่แล้ว +34

    Wait... I thought you were uploading Peyam's NY Conjecture today?

    • @drpeyam
      @drpeyam  6 ปีที่แล้ว +16

      Hahahahaha 😂😂😂 It’s too long to fit in one year 😂

    • @TheNachoesuncapo
      @TheNachoesuncapo 6 ปีที่แล้ว +6

      @@drpeyam just imagine how long it's proof would be then

    • @khanster
      @khanster 6 ปีที่แล้ว +1

      Dr. Peyam’s conjecture? Now I’m curious

  • @antimatter2376
    @antimatter2376 6 ปีที่แล้ว +9

    Absolute madlad. Proved the fundamental theorem of algebra.

  • @ricaanne7398
    @ricaanne7398 5 ปีที่แล้ว +8

    "FTA not to be confused with GTA" - loved it from then on

  • @alexanderskladovski
    @alexanderskladovski 6 ปีที่แล้ว +34

    Please prove the fundamental theorem of engineering

    • @WhattheHectogon
      @WhattheHectogon 6 ปีที่แล้ว +40

      pi=e=3?

    • @TheNachoesuncapo
      @TheNachoesuncapo 6 ปีที่แล้ว

      thats easy

    • @samisit0
      @samisit0 6 ปีที่แล้ว +19

      The proof is trivial and left as an exercise for the reader

    • @Sporkabyte
      @Sporkabyte 6 ปีที่แล้ว +7

      Let f:R->R be a continuous function given by the map x -> sin(x). Then the corresponding nth Taylor polynomial is P_n(x)= x -x^3/3! +x^5/5!-...+(-1)^(n+1)x^n/n!
      Suppose x is smol, then cx^n is also smol for real c and n>1, then the sequence of partial sums n -> P_n(x) converges to x. Thus sin(x) = x.
      Bob’s your uncle

  • @aniruddhvasishta8334
    @aniruddhvasishta8334 6 ปีที่แล้ว +25

    Proof of the Riemann Hypothesis:
    This proof is trivial and left as an exercise for the reader.

    • @chronicsnail6675
      @chronicsnail6675 3 ปีที่แล้ว

      "I have found a proof but the margin is to small so cba"

  • @shiina_mahiru_9067
    @shiina_mahiru_9067 6 ปีที่แล้ว +7

    Btw, The FTA can be directly proven with the Cauchy Integral Formula. Instead of proving 1/p(z) is bounded, you would be proving 1/p(0) = 0

    • @PedroTricking
      @PedroTricking 6 ปีที่แล้ว +1

      How do you use the Cauchy Formula to prove there's a c such that 1/p(c)=0?

  • @jwyliecullick8976
    @jwyliecullick8976 4 ปีที่แล้ว

    Best math content on TH-cam by far. Thank you!

  • @felipegiglio2047
    @felipegiglio2047 ปีที่แล้ว +1

    I have a shorter proof for the claim that g is bounded. Suppose not. Then there is a sequence z_1,z_2,z_3,... such that |g(z_k)| goes to infinity. But this means that |p(z_k)| goes to 0. Because big |z| gives big p(|z|), the sequence |z_k| is bounded. So there must be a convergent subsequence that converges to z. As p is continuous, p(z) = 0. Contradiction.

  • @AndDiracisHisProphet
    @AndDiracisHisProphet 6 ปีที่แล้ว +6

    One of my favourite memories of my whole studies was when my Analysis Prof in my first semester proved the FTA with the words that this was the triumph of the Analysis over the Algebra since it is not possible (is this really true?) to prove it using algebraic means. And instead of starting the proof with "BEW:" (for Beweis, proof) he started it with "Triumph:"

    • @jeremicao131
      @jeremicao131 5 ปีที่แล้ว

      Hi do you have any sources on mathematicians stating that this was the "triumph" of algebra over analysis? Because i would need this for an important paper, but i can not seem to find it anywhere
      (i speak german english and french)

    • @rockinroggenrola7277
      @rockinroggenrola7277 2 ปีที่แล้ว

      You can prove the Fundamental Theorem of Algebra using Galois Theory and the Sylow Theorems, but if you prove it that way, you still have to use the Intermediate Value Theorem and the fact that every positive real number has a real square root.

    • @AndDiracisHisProphet
      @AndDiracisHisProphet 2 ปีที่แล้ว

      @@rockinroggenrola7277 Well that is a blast from the past. But you are basically saying "no it can't be proven just by algebraic means"?

    • @rockinroggenrola7277
      @rockinroggenrola7277 2 ปีที่แล้ว

      @@AndDiracisHisProphet Not that I know of unfortunately.

    • @AndDiracisHisProphet
      @AndDiracisHisProphet 2 ปีที่แล้ว

      @@rockinroggenrola7277 So it is a Triumph :D

  • @Goku17yen
    @Goku17yen 6 ปีที่แล้ว +1

    Not to be confused with GTA lmao I love your quirky remarks!! :P FTA is definitely one of the most ‘fundamental’ beautiful theorems just so happen to be necessary for math as a whole

  • @bandamkaromi
    @bandamkaromi 6 ปีที่แล้ว

    Thank you, Dr. Peyam.

  • @valedonjos
    @valedonjos 4 ปีที่แล้ว

    Dr Peyam, in the Liousville Th proof I didnt understand the inequality at 18:00 . Does it mean that the integral of the abs value is less than or equal to the abs value of the integral ?

    • @valedonjos
      @valedonjos 4 ปีที่แล้ว

      Ok, I remember that that is true with real functions so I suppose it is still true with complex functions (?)

    • @drpeyam
      @drpeyam  4 ปีที่แล้ว +1

      Yes, triangle inequality

  • @stydras3380
    @stydras3380 6 ปีที่แล้ว +2

    Very nice c: What do you think of it's generalization, the Weierstraβ Product Theorem or its cousin the Factorization Theorem? A video about them would be awesome aswell :)

    • @steffahn
      @steffahn 6 ปีที่แล้ว +4

      NOOO, PLZ! Don’t write a "β" (beta) for a "ß" (German sharp s). The former is a greek letter related to the latin letter B [in fact, "β" is the lowercase beta, and the uppercase beta "Β" looks identical to the latin "B" in most fonts]; the latter is originally a ligature of a long s "ſ" and a final s "s" (i.e. the only form we still use nowadays). Like this: "ß" vs "ſs" - just connect the lines ;-)
      And if you can’t afford the effort of copying the the "ß" character anywhere, do as the Germans would do and just use the fallback variant "ss" (i.e. two ordinary letters "s"; as in "Weierstrass").

    • @stydras3380
      @stydras3380 6 ปีที่แล้ว +3

      @@steffahn In fact I am German. I use a mathematical keyboard, which is not German and doesn't have the "scharfes s"). Therefore I use Beta xD It just looks too similar, hardly anyone ever notices :P
      Still interesting, didn't knew it arose from the long s and s put together :P In retrospect it makes perfect sense... those lazy Germans :P

    • @steffahn
      @steffahn 6 ปีที่แล้ว

      ​@@stydras3380 Oh.. mathematical keyboards are a thing? That’s something I never heard or thought about..

    • @stydras3380
      @stydras3380 6 ปีที่แล้ว +1

      @@steffahn I use one with the Greek alphabet (upper and lower case) and some other mathematical shannanigans like ∫ ∂ ≠ ± ℝ ℚ ∈ ∞ √ and of course indicies ⁰¹²³⁴⁵⁶⁷⁸⁹₀₁₂₃₄₅₆₇₈₉

  • @bmac3933
    @bmac3933 6 ปีที่แล้ว +1

    Can you please explain why the Liouville Thm isn’t contradicted by the existence of the arc tangent?

    • @drpeyam
      @drpeyam  6 ปีที่แล้ว +7

      While arctan(x) is bounded for real x, arctan(z) is unbounded for complex z

  • @DiegoMathemagician
    @DiegoMathemagician 6 ปีที่แล้ว

    I love this chanel, I am learning a lot!

  • @AlexiLaiho227
    @AlexiLaiho227 6 ปีที่แล้ว

    GREAT VIDEO DR PEYAM! happy 2019 🎉🎊

  • @TimsDaBossMC
    @TimsDaBossMC 5 ปีที่แล้ว

    Hello can anyone help me understand this? Are there some specific keywords I should look up and understand before going deeper into the video? Any help is appreciated and have a wonderful day!

  • @andreapaps
    @andreapaps 3 หลายเดือนก่อน

    I am watching Peyam's videos, I feel like I am on about season 10 now and the show is as good as season 1 :D.

  • @shahinjahanlu2199
    @shahinjahanlu2199 4 ปีที่แล้ว

    سلام .با تشکر.میشه لطفا درباره قضیه چینی و دنباله فری توضیح دهید .مرسی

  • @khanster
    @khanster 6 ปีที่แล้ว +5

    Shorts in January? I miss California. Algebro bruh...

  • @valedonjos
    @valedonjos 4 ปีที่แล้ว

    Beautiful proof in less than 15 min

  • @szymon5830
    @szymon5830 6 ปีที่แล้ว

    Dear Dr. Peyam how about The Lebesgue Integral of the Dirichlet function
    ?

    • @drpeyam
      @drpeyam  6 ปีที่แล้ว

      I’ve done a video on that already, check out my video called Integration Sucks

  • @martinepstein9826
    @martinepstein9826 6 ปีที่แล้ว

    I for one don't find it strange that the proof uses analysis rather than algebra. In algebra you extend the rational number system by constructing quotient fields containing the roots of whatever polynomials you want, and with this construction it doesn't make sense to ask if the new numbers you made are complex or not. So I think of it as more of an analysis theorem.

  • @jejo6559
    @jejo6559 5 ปีที่แล้ว

    Could you please note the sources, because you explained it outstandingly good, and I would like to use this proof for an important essay (dont know what this essay is called in english)
    I would really appreciate it
    i really mean really

    • @drpeyam
      @drpeyam  5 ปีที่แล้ว

      I think I got the proof from Brown and Churchill’s Complex Variables and Applications Book

    • @jeremicao131
      @jeremicao131 5 ปีที่แล้ว

      @@drpeyam thank you so much!!!!

  • @PedroTricking
    @PedroTricking 6 ปีที่แล้ว

    Really simple! Really nice : )

  • @aniruddhvasishta8334
    @aniruddhvasishta8334 6 ปีที่แล้ว

    Isn't f(x) = sin(x) a counterexample to Liouville’s Theorem?

    • @drpeyam
      @drpeyam  6 ปีที่แล้ว +2

      It’s bounded in R but not bounded in C!

  • @faizanurrehman6220
    @faizanurrehman6220 6 ปีที่แล้ว

    Great work👍

    • @faizanurrehman6220
      @faizanurrehman6220 6 ปีที่แล้ว

      Today I'm pass post graduate in pure mathematics.feeling blessed and thanks for all of my teacher,parents & TH-cam friend teacher's.thanks for everone.😇😇😇

  • @duckymomo7935
    @duckymomo7935 6 ปีที่แล้ว

    i prefer Galois theory route (especially since that was the most natural way to prove it) but it can be proven in many other ways

  • @mpcc2022
    @mpcc2022 6 ปีที่แล้ว

    Thank you very much!

  • @eterty8335
    @eterty8335 8 หลายเดือนก่อน

    not right now babe new peyam just dropped 5 years ago and I'm getting it recommended today

    • @drpeyam
      @drpeyam  8 หลายเดือนก่อน

      Hahahaha 😅

  • @dgrandlapinblanc
    @dgrandlapinblanc 6 ปีที่แล้ว

    Thanks Dr Peyam's. Not easy complex functions. I've looked on wikipedia.

  • @SmileyHuN
    @SmileyHuN 6 ปีที่แล้ว +1

    d'Alembert-Gauss for life

  • @MessedUpSystem
    @MessedUpSystem 2 ปีที่แล้ว

    Didn't the theorem state that every polynomial with complex coefficients has N roots in C?

    • @drpeyam
      @drpeyam  2 ปีที่แล้ว

      Sure, so?

    • @MessedUpSystem
      @MessedUpSystem 2 ปีที่แล้ว

      @@drpeyam In the video you said it guarantees ONE root, so I got curious if I misremmembered the theorem :p

    • @drpeyam
      @drpeyam  2 ปีที่แล้ว +1

      At least one root! I agree it’s n roots with multiplicities, but there’s at least one root without multiplicity

    • @MessedUpSystem
      @MessedUpSystem 2 ปีที่แล้ว

      @@drpeyam Ooooh I see, thx for the clarification!

    • @felipegiglio2047
      @felipegiglio2047 ปีที่แล้ว +1

      ​@@MessedUpSystemthe proof that there are at most N is trivial. you just factor out one root (it exists by this proof) then use induction until youve got N roots and a constant polynomial

  • @sedeanimugamez5418
    @sedeanimugamez5418 6 ปีที่แล้ว +2

    I always go to your videos expecting that bunny where is it

    • @drpeyam
      @drpeyam  6 ปีที่แล้ว

      Oreo is happy at her rabbit hotel 🏨

    • @drpeyam
      @drpeyam  6 ปีที่แล้ว

      She’s filled with love ❤️

    • @AndDiracisHisProphet
      @AndDiracisHisProphet 6 ปีที่แล้ว

      @@drpeyam and with "love" you mean "chocolate", right?

  • @chrissharman8650
    @chrissharman8650 6 ปีที่แล้ว

    Love it

  • @Edsonrsmtm
    @Edsonrsmtm 4 ปีที่แล้ว

    SHOW!!!!

  • @PuzzleQodec
    @PuzzleQodec 6 ปีที่แล้ว

    At least algebra wasn't inequalized by the great Grönwall. That would have been a disaster.

  • @soniahazy4880
    @soniahazy4880 ปีที่แล้ว

    🎨⛲️💎🌈🫧🙏✨🪷🦋🛸🧩

  • @culturecreating
    @culturecreating ปีที่แล้ว

    is your city differentiable and bounded,lol

  • @factsheet4930
    @factsheet4930 4 ปีที่แล้ว +1

    This is THE proof of the FTA!
    The rest are just wannabe proofs 😂

    • @drpeyam
      @drpeyam  4 ปีที่แล้ว +1

      I agree lol

    • @factsheet4930
      @factsheet4930 4 ปีที่แล้ว

      @@drpeyam I really love how your response time to comments remain constant, even as your subscription count goes to infinity and you become more famous ❤️

  • @subnow4862
    @subnow4862 3 ปีที่แล้ว

    Neat!