As you can see, when splitting an $~$n$~$\-digit, the fair split occurs at the largest whole number $~$x$~$ such that $~$x \\cdot x \\le n$~$, in which case we both get to record one $~$x$~$\-message\. This is why it is 316, and not 500, that can most naturally be seen as "about 2\.5 decimal digits long:" The number that is 2\.5 decimal digits long is the largest number $~$x$~$ such that, given five digits, both you and I can use those 5 digits to store independent $~$x$~$\-messages\. If you store a 500\-message, that only leaves enough space for me to store a 200\-message\. If you only store a 300\-message, that leaves enough space for me to store a 333\-message\. $~$x\=316$~$ is the point where we can both store an $~$x$~$\-message, because 316 is the largest whole number such that $~$x^2 \\le 100000.$~$

Because 5 digits is equal to 2, 2.5 digits.

(By the way, it would be good if there were a way for me to take notes, on the text that didn't require me to spam the comments section.)