Etymology
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half- 
in genealogical combinations, "sharing one parent," Middle English, from half.
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semi- 

word-forming element of Latin origin meaning "half," also loosely, "part, partly; partial, almost; imperfect; twice," from Latin semi- "half" (before vowels often sem-, sometimes further reduced to se- before m-), from PIE *semi- "half" (source also of Sanskrit sami "half," Greek hēmi- "half," Old English sam-, Gothic sami- "half").

The Old English cognate, sam-, was used in such compounds as samhal "in poor health, weakly," literally "half-whole;" samsoden "half-cooked" ('half-sodden'), figuratively "stupid" (compare half-baked); samcucu "half-dead," etymologically "half-alive" (see quick (adj.)); and the lingering survivor, sandblind "dim-sighted" (q.v.).

The Latin element was common in formations from Late Latin, as in semi-gravis "half-drunk," semi-hora "half hour," semi-mortuus "half-dead," semi-nudus "half-naked," semi-vir "half-man, hermaphrodite."

The Latin-derived form in English has been active in forming native words since 15c. Semi-bousi "half-drunk" ('semi-boozy'), now obsolete, was among the earliest (c. 1400). As a noun, semi has variously been short for semi-detached house (by 1912), semi-trailer (by 1942), semi-final (by 1942).

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sesqui- 

word-forming element meaning usually "one half more" than the indicated unit, from Latin sesqui-, sesque- "one and a half; one-half more," contraction of *semis-que- "a half in addition," from *semis "a half" (see semi-) + -que "and" (from PIE root *kwe "and, -ever," for which see ubiquity). Latin semi- had a tendency to get partly swallowed in compounds; compare these derivatives listed in de Vaan: selibra "half a libra," semodius "half a modius," sestertius "having the value of two-and-a-half" (as a noun, the name of a silver Roman coin, short for sestertius nummus), contracted from *semistertius; simbella "coin worth half a libella;" sincipitis "a half-head."

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demi- 

word-forming element meaning "half, half-sized, partial," used in English from mid-14c., especially in technical terms from French, from Old French demi "half" (12c.), from Late Latin dimedius, from Latin dimidius "half, one-half," which contains the elements dis- "apart" (see dis-) + medius "in the middle, between; from the middle," as a noun "the middle;" from PIE root *medhyo- "middle." Formerly also demy-, and in early use often written as a separate word.

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hemi- 

word-forming element meaning "half," from Latin hemi- and directly from Greek hēmi- "half," from PIE root *semi-, which is the source of Sanskrit sami, Latin semi- (see semi-), Old High German sami- "half," and Old English sam-, denoting a partial or imperfect condition (see sandblind).

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mixo- 

word-forming element of Greek origin meaning "mixed," from Greek mixo-, from mixis "a mixing, mingling, intercourse," from root of mignynai "to mix, mix up, mingle" (from PIE root *meik- "to mix"). As in mixolydian in reference to a half-Lydian mode in ancient Greek music.

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