Is there a way to directly find the $n ^{\rm th}$ digit of the fractional part of an irrational number?
For example, how can I find the $1000^{\rm th}$ digit of $\{\pi\}$?
2026-03-28 22:27:05.1774736825
$n ^{\rm th}$ digit of an irrational number
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The family of algorithms called spigot algorithms are designed to do just that: generate digits within the the $b$-ary expansion of irrational constants (for some integer $b>2$).
For your example of $\pi$, the Bailey-Borwein-Plouffe formula gives rise to such a spigot algorithm specifically for $\pi$.