an amazing and mysterious approximation for pi! 

3:25 Engineers

(sinx)*(cosx) is just (sin2x)/2, the Taylor Series is easy to find 5:30

20:49

yesterday I saw a cheese going for 3.14. I bought it.

This is kinda cool but seems pretty useless when you can get simple rational approximations that are closer. For example, 312689/99532 already gives you a better approximation than the one at the end (and if you were calculating it you still have the issue of needing to know sqrt(3) accurately, which I guess is much easier than pi but still).

Just a small detail, In the Newton final fórmula should be 3sqrt(2) in the denominator. Great video, as always

x= pi/4 for n=1 has relative error of about 2.27x10^-3, and x= pi/6 (still 1) gives 12*(2sqrt(2)-1)/7, which has relative error of about 4.31x10^-4. The thumbnail (n=3, x= pi/6) has relative error of 4.13x10^-11 + O(10^-13).

The RU-vid thumbnail is (currently) wrong saying "80405" instead of "80504"

The guy's not named Snellins, but Snellius. It is the latinised version of the name Snel (meaning quick).

Yesterday I was reading Dan Pedoe’s Geometry in which he proved the equivalence of Snell’s Law and Fermat’s Principle of Least Time using Ptolemy’s Theorem. I thought "This is something Michael might turn into a video." Today I turned on RU-vid and you mentioned Snell’s Law. Though relatively simple Pedoe’s proof is still interesting, at least to me. As is Snell's Window.

it would be interesting to see this for padé approximations of the basis functions. but that would be even more hard to follow on a black board. 🙂

I believe it's "Snellius", not "Snellins" 😅

Taylor's expansion for the sinx should be x^(2n+1) not just x^n

I was wondering if you are currently thinking about that Hank Hill sum problem or whatever it was called. It looked to me like there was an underlying theme to the recent videos on the channel and was curious if you were occupied with that intriguing sum. It looks preposterous, that its convergence is unknown, but the more I thought about it the more I saw, and understood that it was reasonable that it is a difficult problem. It still seems weird that someone hasn't shown that the reciprocal cube term totally dominates the reciprocal squares of sines at a given magnitude of n, even though that latter term does become arbitrarily large. But, I guess that's the interest of the problem, we need a proof of that if it's true.

nice but I do not like the sqrt(3) in the final answer. it is an irrational. would be nicer if it were just rationals in the approximation. You still have to calculate sqrt(3)...

Do you have any content on hypergeometric series?

So,does this system of equations has a solution always? I know that we can use the rank theorem and say that rank of the matrix with and without the (1000...) cell are the same which implies that the solution exists,but it is not much obvious i wish...

The pi approximation of Ramanujan would be interesting. But it is probably out of reach(?)

There must be a way to invert the matrix of coefficients in general, right?!

Makes sense that the error at the end (10^-10) would be as good as twice the number of digits in the approximation (5 digits)

Wow this is interesting❤. How about (355/113)-Tan(355/113), where do you think it comes from?

I'm wondering if there's any method to pop out 355/113 ~ pi, or if someone just noticed it was only off by less than one part in a million?

Calculus was still in its infancy at the time of Newton. Would he have had access to Taylor expansions like that? Or this presentation in modern terms, when Newton would have thought of it very differently?

Sin(x)*cos(x) = 0.5*sin(2x).. It seems not to match the product of the two aporoximation... Why?

Isn’t pi just the relationship between coordinate systems? Doesn’t the fact that it’s not resolvable prove that the systems are not extensions of each other? So exploring approaches to pi gives insights into the shortcomings of mathematical processes. Or am I missing something?

Is this how those projects for calculating 1 million digits of pi work?

But don't forget that for most real life applications 22/7 is an *acceptable* approximation.

9:17 that's not right. That fraction gives me an answer of 3.940603027...