xkcd: Coordinate Precision but pi (π)?

I tried looking for some answer but found mostly

  • People reciting pi
  • People teaching how to memorize pi
  • How to calculate pi using different formula
  • How many digits NASA uses

Update question to be more specific

In case someone see this later, what is the most advanced object you can build or perform its task, with different length of pi?

0, 3 => you can’t make a full circle

1, 3.1 => very wobbly circle

2, 3.14 => perfect hole on a beach

3, 3.142 => ??

4, 3.1416 => ??

5, 3.14159 => ??

Old question below

In practice, the majority of people will never require any extra digit past 3.14. Some engineering may go to 3.1416. And unless you are doing space stuff 3.14159 is probably more than sufficient.

But at which point do a situation require extra digit?
From 3 to 3.1 to 3.14 and so on.

My non-existing rubber duck told me I can just plug these into a graphing calculator. facepalm

y=(2πx−(2·3.14x))

y=abs(2πx−(2·3.142x))

y=abs(2πx−(2·3.1416x))

y=(2πx−(2·3.14159x))

Got adequate answer from @dual_sport_dork and @howrar
Any extra example of big object and its minimum pi approximation still welcome.

    • Kogasa
      3·
      1 year ago

      There are infinitely many. Any sequence of rational numbers converging to pi contains infinitely many. 22/7 and 355/113 are just particularly good ones for their small denominators. You can find such good approximations (“the best rational approximations of a given size”) by truncating the continued fraction representation of pi:

      pi = [3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,2,…]
pi = 3  + 1/(7 + 1/(15 + 1/(1 + ... )))

      These approximations yield:

      pi ~ 3
pi ~ 3 + 1/7 = 22/7
pi ~ 3 + 1/(7 + 1/15) = 333/106
pi ~ 3 + 1/(7 + 1/(15 + 1)) = 355/113
pi ~ 3 + 1/(7 + 1/(15 + 1/(1 + 1/292))) = 97591/31065

      and so on. Note that by pure coincidence one of the first few terms is quite large (292), so the difference between the corresponding partial expansions is small (1 + 1/292 is close to 1). That’s why 355/113 is an unusually good approximation for such a small denominator.