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
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.
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:"
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)
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.
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
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 :)
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").
@@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 I use one with the Greek alphabet (upper and lower case) and some other mathematical shannanigans like ∫ ∂ ≠ ± ℝ ℚ ∈ ∞ √ and of course indicies ⁰¹²³⁴⁵⁶⁷⁸⁹₀₁₂₃₄₅₆₇₈₉
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.
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
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!
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 ?
Today I'm pass post graduate in pure mathematics.feeling blessed and thanks for all of my teacher,parents & RUclips friend teacher's.thanks for everone.😇😇😇
@@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
@@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 ❤️
Wait... I thought you were uploading Peyam's NY Conjecture today?
Hahahahaha 😂😂😂 It’s too long to fit in one year 😂
@@drpeyam just imagine how long it's proof would be then
Dr. Peyam’s conjecture? Now I’m curious
Absolute madlad. Proved the fundamental theorem of algebra.
"FTA not to be confused with GTA" - loved it from then on
Please prove the fundamental theorem of engineering
pi=e=3?
thats easy
The proof is trivial and left as an exercise for the reader
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
Proof of the Riemann Hypothesis:
This proof is trivial and left as an exercise for the reader.
"I have found a proof but the margin is to small so cba"
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
How do you use the Cauchy Formula to prove there's a c such that 1/p(c)=0?
Best math content on RUclips by far. Thank you!
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.
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:"
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)
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.
@@rockinroggenrola7277 Well that is a blast from the past. But you are basically saying "no it can't be proven just by algebraic means"?
@@AndDiracisHisProphet Not that I know of unfortunately.
@@rockinroggenrola7277 So it is a Triumph :D
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
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.
Thank you, Dr. Peyam.
Beautiful proof in less than 15 min
Shorts in January? I miss California. Algebro bruh...
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 :)
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").
@@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
@@stydras3380 Oh.. mathematical keyboards are a thing? That’s something I never heard or thought about..
@@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
I love this chanel, I am learning a lot!
GREAT VIDEO DR PEYAM! happy 2019 🎉🎊
Can you please explain why the Liouville Thm isn’t contradicted by the existence of the arc tangent?
While arctan(x) is bounded for real x, arctan(z) is unbounded for complex z
سلام .با تشکر.میشه لطفا درباره قضیه چینی و دنباله فری توضیح دهید .مرسی
not right now babe new peyam just dropped 5 years ago and I'm getting it recommended today
Hahahaha 😅
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
Thanks Dr Peyam's. Not easy complex functions. I've looked on wikipedia.
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.
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
I think I got the proof from Brown and Churchill’s Complex Variables and Applications Book
@@drpeyam thank you so much!!!!
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!
Thank you very much!
Dear Dr. Peyam how about The Lebesgue Integral of the Dirichlet function
?
I’ve done a video on that already, check out my video called Integration Sucks
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 ?
Ok, I remember that that is true with real functions so I suppose it is still true with complex functions (?)
Yes, triangle inequality
Really simple! Really nice : )
Great work👍
Today I'm pass post graduate in pure mathematics.feeling blessed and thanks for all of my teacher,parents & RUclips friend teacher's.thanks for everone.😇😇😇
d'Alembert-Gauss for life
Isn't f(x) = sin(x) a counterexample to Liouville’s Theorem?
It’s bounded in R but not bounded in C!
I always go to your videos expecting that bunny where is it
Oreo is happy at her rabbit hotel 🏨
She’s filled with love ❤️
@@drpeyam and with "love" you mean "chocolate", right?
Love it
SHOW!!!!
Didn't the theorem state that every polynomial with complex coefficients has N roots in C?
Sure, so?
@@drpeyam In the video you said it guarantees ONE root, so I got curious if I misremmembered the theorem :p
At least one root! I agree it’s n roots with multiplicities, but there’s at least one root without multiplicity
@@drpeyam Ooooh I see, thx for the clarification!
@@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
At least algebra wasn't inequalized by the great Grönwall. That would have been a disaster.
🎨⛲️💎🌈🫧🙏✨🪷🦋🛸🧩
is your city differentiable and bounded,lol
This is THE proof of the FTA!
The rest are just wannabe proofs 😂
I agree lol
@@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 ❤️
Neat!