Let $a_{(k)}$ be the $k$th smallest number of $n$ real numbers $a_1,a_2,\ldots,a_n$. Is there a formula for $a_{(k)}$?
I know $a_{(1)}$ can be found recursively using $$a_{(1)}=\min\{a_1,a_2,\ldots,a_n\},\\ \min\{a,b\}=\frac{1}{2}\left[ a+b-\lvert a-b\rvert\right],\\ \min\{a_1,a_2,\ldots,a_{n+1}\}=\min\{\min\{a_1,a_2,\ldots,a_n\},a_{n+1}\}.$$ Similarly, $a_{(n)}=\max\{a_1,a_2,\ldots,a_n\}$ can be found with $\max\{a,b\}=\frac{1}{2}\left[ a+b+\lvert a-b\rvert\right]$.