Svijet Matematike

[Bluemoon]

Obožavam ovog lika.



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[nicky]

https://risingentropy.com/ramanujans-infinite-radicals/

The Problem



Solve the following equation:
Background
This problem was initially posed by the most brilliant mathematician of the past century, an eccentric prodigy named Srinivasa Ramanujan. He submitted it to the Journal of Indian Mathematical Society without a solution, challenging his fellow mathematicians to solve it. Eventually, when nobody could do it, he gave in and revealed the answer along with a simple proof that probably put them all to shame.
Solution
Ok, so seriously, try this for yourself before reading on. Most people haven’t ever done anything with infinite nested square roots, so this is a good opportunity to be creative and come up with weird ways of solving the problem. Play around with it, give random things names, and see what you can do!
You’ll notice (as I did when I first tried working it out) that solving nested radicals can be really really hard. Things quickly get really ugly. Luckily, the answer is beautifully simple. It’s just 3!
Let’s see why. We start with a very simple identity:

Taking the square root, we get

If we just plug in a few integer values, we see the following:

What happens if we plug in the value of 4 from the second line to the 4 in the first? We get

Hm, already looking familiar! We’re not done yet – we now plug in the value of 5 from the third line to this new equation, obtaining:

And once more:

And etc off to infinity.

And that’s the proof!


Generalizing
Now, Ramanujan actually proved a much more general result than this. I’ll prove a less general result than his, but still more general than what we just saw.
The generalization comes from noticing that our initial identity didn’t have to involve adding 1 in (x+1)2. We could have added any number to x, squared it, and then gotten a brand new general theorem about a whole class of continued root problems.
Thus we start with the identity



or



Now we perform substitutions exactly as we did above:



In the end, what we get is a solution for a whole class of initially very difficult seeming problems:


And if we plug in m = 1 and n = 2, we get our initial problem!

[Solid]

@nicky
Dobar post.


Calculus Visualized - by Dennis F Davis

[nicky]

Ako nijesi gledao, preporuka: https://www.imdb.com/title/tt0787524/
moze i https://www.imdb.com/title/tt0138704/ samo je lud skroz

[Solid]

Ovaj Pi sam baš davno neđe gledao, i koliko se sjećam bio je dobar. A ovaj film o Ramanudžanu nisam gledao, mada je meni lično on nekako kao matematičar nije puno interesantan jer je imao neke kontraverzne objave kao npr. ona da je 1 + 2 + 3 + ⋯ + ∞ = -1/12 itd., ali se mora priznati da je imao masu kreativan i jedinstven pristup matematici.

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[nicky]

Dobro si zapamtio
Interesantan mu je zivot, a iz glave je pisao gotove funkcije za redove, sto je potpuno neshvatljivo. Poslije kad je dosao u Englesku, morao je da uci napredniju matematiku kako bi dokazao to sto je pisao. Meni je bas interesantan film bio.
Ova suma ima veze sa Rimanovom zeta funkcijom (koja se dosta pominje u kvantnoj fizici), davno sam lomio glavu oko toga jer je zbilja van svake pameti. Ali u tom kontekstu je okej i moze se dokazati relativno lako.

[Bluemoon]

Najbolji momci u forumskoj zajednici ove filmove moram dodati i pogledati hlava ti nicky.

Ovo izgleda brutalno dobro.

[nicky]

Ako ti se svidi zahvali. Kad pomenuh Rimana, iskoci mi nedavno:

Bernhard Riemann died in 1866 at the age of 39. Here is a list of things named after him.


Riemann bilinear relations
Riemann conditions
Riemann form
Riemann function
Riemann–Hurwitz formula
Riemann matrix
Riemann operator
Riemann singularity theorem
Riemann surface
Compact Riemann surface
The tangential Cauchy–Riemann complex
Zariski–Riemann space
Cauchy–Riemann equations
Riemann integral
Generalized Riemann integral
Riemann multiple integral
Riemann invariant
Riemann mapping theorem
Measurable Riemann mapping theorem
Riemann problem
Riemann solver
Riemann sphere
Riemann–Hilbert correspondence
Riemann–Hilbert problem
Riemann–Lebesgue lemma
Riemann–Liouville integral
Riemann–Roch theorem
Arithmetic Riemann–Roch theorem
Riemann–Roch theorem for smooth manifolds
Grothendieck–Hirzebruch–Riemann–Roch theorem
Hirzebruch–Riemann–Roch theorem
Riemann–Stieltjes integral
Riemann series theorem
Riemann sum
Riemann–von Mangoldt formula
Riemann hypothesis
Generalized Riemann hypothesis
Grand Riemann hypothesis
Riemann hypothesis for curves over finite fields
Riemann theta function
Riemann Xi function
Riemann zeta function
Riemann–Siegel formula
Riemann–Siegel theta function
Free Riemann gas
Riemann invariant
Riemann–Cartan geometry
Riemann–Silberstein vector
Riemann-Lebovitz formulation
Riemann curvature tensor
Riemann tensor
Riemannian graph
Riemannian group
Riemannian holonomy
Riemannian manifold also called Riemannian space
Riemannian metric tensor
Riemannian Penrose inequality
Riemannian polyhedron
Riemannian singular value decomposition
Riemannian submanifold
Riemannian submersion
Riemannian volume form
Riemannian wavefield extrapolation
Sub-Riemannian manifold
Riemannian symmetric space
Riemann's differential equation
Riemann's existence theorem
Riemann's explicit formula
Riemann's minimal surface
Riemann's theorem on removable singularities

[Bluemoon]

Ozbiljno dobro no ovo je bilo haj popularno mislim dobar je film ali sad kad si ovo napisao i predstavio tačno mislim da je amaterski naspram toga:



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[nicky]

Odlican. Volim i taj film. A jedan od milijih mi je Good Will Hunting, sjecam se na tabli su bili grafovi i pitao sam se sto je to kad sam prvi put gledao.
Evo ga

[Bluemoon]

Jeste ali tu imaš jak emotivni zahvat (gledala) kao u prethodnom filmu čak još jači a meni je to ozbiljan zajeb da gledam filmove na tu tematiku (drugačije) ako me razumiješ.



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[nicky]

Gledali smo ga jednom godisnje godinama.. filmovi s Robin Williamsom su me samljeli Dead poets society, good will hunting, the fisher king i patch adams.
Prastajte na off

[Bluemoon]

Ođe kad si off đe niko ne zalazi ne brini. Nije ovo tako popularna tema na forum.Podforum nauke ti je u paučinu debelu.

Enivej.

I mene su samljeli nisi jedini.Ali,dobro je to.
Ko razumije.

[nicky]

A to da haha.

Malo sale

[Solid]

@nicky
Dobar post.


Calculus Visualized - by Dennis F Davis

Odgledah ovo do kraja, kakvo remek djelo. Kako lik neće napraviti Calculus 2 i 3.


Čovjek koji stoji iza čuvenog matematičkog kanala 3Blue1Brown je razvio Python package koji se zove Manim da bi koristio animacije koje pravi preko njega u svojim videima. Taj package je kasnije preuzela matemetičko/programerska zajednica i dodatno ga unaprijedila, pojednostavila i napisala dokumentaciju za njega. I sada mnogi matematički i drugi mentori koriste taj package da prave videe i pružaju ovako animiran sadržaj koji je mnogo bolji nego što je to bio slučaj ranije kada su na videima mentori objašnjavali koristeći tablu ili papir i hemijsku.

[nicky]

Super. To bi moglo animirati osnovce i srednjoskolce da zavole malo matematiku. Ali dok ne prodje kilometre olovkom ili hemijskom zalud sva animacija. Jesi li probao taj paket?

[Solid]

Super. To bi moglo animirati osnovce i srednjoskolce da zavole malo matematiku. Ali dok ne prodje kilometre olovkom ili hemijskom zalud sva animacija. Jesi li probao taj paket?

Jesam, pravio sam neke animacije, baš je dobar i ima podršku i za prilaz grafova, za strukture podataka, za 3D, razne transformacije i za razne druge stvari, ali zahtijeva da se čovjek dosta bakće oko toga i da poznaje programiranje.

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[Bluemoon]

[Solid]

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[nicky]

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