Is it possible for a PC to recover a single arrow fired into battle?
The question "Is ammunition recoverable?" is not specific enough to address the case where a player fires a single piece of ammunition.
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The Ammunition property describes how to recover expended ammunition after a battle:
At the end of the battle, you can recover half your expended ammunition by taking a minute to search the battlefield.
If you expended an odd number of arrows, then you divide by 2 and round down to determine the number of usable arrows recovered, because the general rule is to always round down fractions to the nearest integer when an integer is required (which is definitely the case when counting arrows):
Whenever you divide a number in the game, round down if you end up with a fraction, even if the fraction is one-half or greater.
Unfortunately, this means that by strict RAW, if you attempt to recover half of only one expended arrow, you end up with zero arrows recovered.
If you (as the DM) find this unsatisfying and you are not playing in an official context where strict RAW is important, feel free to implement a special case for when only a single arrow was fired. For example, you can roll a d2 and say that the arrow is unrecoverable on a 1. Any change you make will have a minimal impact in practice because firing exactly one arrow in an encounter is very rare. If nothing else, the vast majority of ranged weapon users will have Extra Attack by level 5, which means that if they fire any arrows at all, they will almost always fire at least 2 of them, which means the special case of 1 arrow will rarely, if ever, come into play from that point onward.