I've got Pi memorised more precisely than 355/113 as well for no good reason, and for engineering applications it doesn't serve any purpose.

But the fraction is weirdly close and that's interesting for other (more academic) reasons, such as explaining why plotting primes in polar coordinates looks like a pattern: https://www.youtube.com/watch?v=EK32jo7i5LQ

i just memorized it as / 1 1 3 3 5 5 / where you start with the denominator and circle back.

So you've got to remember three digits, the fact they are doubled, the place to start, and the place circle back (or deduce any of those).

Or you can remember 1592 as the year Trinity College in Dublin was founded. You didn't know that? You probably won't forget now.

Ha. I don't think I'll remember '1592' directly. Easier to remember the offset of '+100' and combine it with my existing knowledge of who sailed the ocean blue.

Think of it the way we wrote division in grade school, like:

      ___
  113|355

Memorize six digits to get seven back?