You’re not wrong, I just feel like the way the post was written it was a little bit disingenuous of the author not to at least include the series and discuss it

You’re not wrong, I just feel like the way the post was written it was a little bit disingenuous of the author not to at least include the series and discuss it. They make 355/113 seem mysterious, or that it might point to a new series. In fact, it comes from a boring/inevitable series. Maybe it’s my physics training coming through, but asking “why” pi is close to this cute fraction is the wrong question. I would be much more interested in knowing “given some transcendental number, what is the likelihood that there exists some fraction with at most $x digits that approximates it to $y decimal places?” For instance, 577/408 approximates sqrt(2) even more closely than 355/113. Is this usual? Can I come up with a number between 1 and 10 that has no “good” 3 decimal approximation? My very basic understanding of measure theory is that it is actually hard to find numbers that are not well approximated by /some/ integer fraction. After all there are a lot of fractions…