Today is Pi Approximation Day, at least in countries which use DD/MM date format.
No new outstanding pi approximations, just a minor improvement
on one of Ramanujan's:
\(\sqrt[4]{\frac{2143+\left({6+\sqrt{6-\frac{6}{6^6}}}\right)^{-6}}{22}}\)
on one of mine:
\(\frac{\ln\left({\frac{16\times\ln\left({878}\right)}{\ln\left({16\ln\left({878}\right)}\right)}}\right)}{1+\left({\frac{5}{94+\sqrt{2}}}\right)^8}\)
Happy Pi Approximation Day!
(Edited to fix a typo)
Impressive and mind-blowing.
Looks like both of you should receive the Nobel Prize!
(07-22-2023 09:00 PM) Gerson W. Barbosa Wrote: [ -> ] [image]https://live.staticflickr.com/65535/53041849078_abc8fd75a9_z.jpg[/img]
Today is Pi Approximation Day, at least in countries which use DD/MM date format.
No new outstanding pi approximations, just a minor improvement
on one of Ramanujan's:
\(\sqrt[4]{\frac{2143+\left({6+\sqrt{6-\frac{6}{6^6}}}\right)^{-6}}{22}}\)
on one of mine:
\(\frac{\ln\left({\frac{16\times\ln\left({878}\right)}{\ln\left({16\ln\left({878}\right)}\right)}}\right)}{1+\left({\frac{5}{94+\sqrt{2}}}\right)^8}\)
Happy Pi Approximation Day!
(Edited to fix a typo)
There's a recent Mathologer video:
VIDEO
Or you can read about it here:
The Infinite series of Pi and it’s approximation by Madhava
This gives the following approximation:
\(
\begin{align}
\pi &\approx 4 \cdot \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11} + \frac{37}{149 \cdot 6} \right) \\
&\approx 3.14159427448...
\end{align}
\)
(07-23-2023 11:52 PM) Thomas Klemm Wrote: [ -> ] There's a recent Mathologer video:
VIDEO
Or you can read about it here: The Infinite series of Pi and it’s approximation by Madhava
This gives the following approximation:
\(
\begin{align}
\pi &\approx 4 \cdot \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11} + \frac{37}{149 \cdot 6} \right) \\
&\approx 3.14159427448...
\end{align}
\)
Nice video and article, thanks! I had watched the video already, but the article is new to me.
The correction term is equivalent to the continued fraction
\(\frac{1}{4\times6+\frac{1}{6+\frac{4}{4\times6}}}=\frac{37}{894}\)
For maximum efficiency the continued fraction would need to have four more terms, matching the number of terms of the series, as we have seen in
this post from 2014 .
Interestingly the video suggests that Madhava was able to derive the correction continued fraction by comparing a few partial sums of the series with the approximation 355/113, which gives only the first six decimal digits of pi, which is quite a remarkable feat. It was not difficult for me to find it, but I had at hand a wp34s set to double precision (34 significant digits).
This post on another subject might give an idea of how these continued fractions can be found numerically.
Gerson.