'Purification' in the Bible

Then Paul took the [four] men, and the next day he purified himself along with them [by submitting to the ritual]. He went into the temple to give notice of [the time] when the days of purification [ending each vow] would be fulfilled and the usual offering could be presented on behalf of each one.

They found me in the temple presenting these offerings, after I had undergone [the rites of] purification, without any crowd or uproar. But there were some Jews from [the west coast province of] Asia [Minor],