Nested radical approximation of PI and PI day on the forum

[Thomas Klemm]

(08-04-2023 07:05 PM)EdS2 Wrote:  Is this a happy coincidence or is there some mathematical or geometric way to help explain the close fit?

It is listed among others in the The Contest Center's \(\pi\) Competition.

There are others like:

√√√√√√√√√√√√√√√√√√√√√√√√√√√√8 + √√√√√√√√√√√√14 + √√√√√68

It contains 5 digits, but matches pi to 9 decimal places, so it is considered an outstanding approximation.

Or then the root of:

\(
6x^6−4x^5+5x^4+2x^3−2x^2+3x−5083=0
\)

which has just ten digits of coefficients and leads to the fourteen-digit approximation:

3.1415926535898031685143792

I could be wrong, but this looks more like a selection that's mostly pleasing to our eyes.

I also like Ramanujan's

\(
\frac{9}{5}+\sqrt{\frac{9}{5}}=3.1416 \cdots
\)

It is easy to calculate on an HP calculator:

1.8
\(\sqrt{x}\)
LAST x
+