Etymology
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hagio- 
before vowels hagi-, word-forming element meaning "of a saint, saintly, holy," from Greek hagios "sacred, devoted to the gods" (of things), "holy, pure" (of persons), in Ecclesiastical Greek, "a saint," which is perhaps from PIE *yag- "to worship, reverence" (source also of Greek agnos "chaste," Sanskrit yajati "reveres (a god) with sacrifices, worships," Old Persian ayadana "temple").
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half- 
in genealogical combinations, "sharing one parent," Middle English, from half.
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halo- 
before vowels hal-, word-forming element meaning "salt, sea," from Greek hals (genitive halos) "a lump of salt, salt generally," in Homer, "the sea," from PIE root *sal- "salt."
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haplo- 

before vowels hapl-, word-forming element meaning "simple, single; simply, once," from Greek haploos, haplous "single, simple" (as opposed to "compound"); "natural, plain," from PIE compound *sm-plo-, from root *sem- (1) "one; as one, together with" + *-plo- "-fold" (from PIE root *pel- (2) "to fold"). Compare simple, which represents the same compound in Latin.

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helio- 

word-forming element meaning "sun," from Greek hēlios "sun" (from PIE root *sawel- "the sun").

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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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hendeca- 
word-forming element meaning "eleven," from Latinized form of Greek hendeka "eleven," from hen, neuter of heis "one," from PIE *hems-, from root *sem- (1) "one; as one, together with" + deka "ten" (from PIE root *dekm- "ten").
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hepta- 
before vowels hept-, word-forming element meaning "seven," from Greek hepta "seven," cognate with Latin septem, Gothic sibun, Old English seofon, from PIE root *septm (see seven).
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hetero- 
before vowels heter-, word-forming element meaning "other, different," from Greek heteros "the other (of two), another, different; second; other than usual." It is a compound; the first element means "one, at one, together," from PIE root *sem- (1) "one; as one, together with;" the second is cognate with the second element in Latin al-ter, Gothic an-þar, Old English o-ðer "other."

Compounds in classical Greek show the range of the word there: Heterokretes "true Cretan," (that is, of the old stock); heteroglossos "of foreign language;" heterozelos "zealous for one side;" heterotropos "of a different sort or fashion," literally "turning the other way;" heterophron "raving," literally "of other mind."
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