Old English half, halb (Mercian), healf (W. Saxon) "side, part," not necessarily of equal division (original sense preserved in behalf), noun, adjective, and adverb all in Old English, from Proto-Germanic *khalbas "something divided" (cf. Old Saxon halba, Old Norse halfr, Old Frisian, Middle Dutch half, German halb, Gothic halbs "half"), perhaps from PIE (s)kel- "to cut."
Used also in Old English phrases as in modern German, to mean "one half unit less than," e.g. þridda healf "two and a half," literally "half third." The construction in two and a half, etc., is first recorded c.1200. Of time, in half past ten, etc., first attested 1750; in Scottish, the half often is prefixed to the following hour, as in German (e.g. halb elf "ten thirty"). To go off half-cocked "speak or act too hastily" (1833) is in allusion to firearms going off prematurely.
c.1200, halfen "to divide in halves;" see half. Meaning "to reduce by half" is from c.1400. Related: Halved; halving.