Etymology
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better (adj., adv.)

Middle English bettre, from Old English bettra, earlier betera "of superior quality or excellence," from Proto-Germanic *batizo-, perhaps from PIE *bhad- "good," but Boutkan finds no good IE etymology. For etymology and evolution, see best. Cognate words also have become the comparative adjective of good in the older Germanic languages (Old Frisian betera, Old Saxon betiro, Old Norse betr, Danish bedre, Old High German bezziro, German besser, Gothic batiza). All are comparatives of a positive (Proto-Germanic *bat) which is not in use.

In Middle English the adverbial form commonly was bet, sometimes also used an adjective; bet was displaced by c. 1600. Better is attested from late Old English as "improved in health, more healthy" (adv.); from late 12c. as "more useful or desirable." Better half "wife" is attested from 1570s.

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half- 

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

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better (n.1)

late 12c., "that which is better," from better (adj.). The specific meaning "one's superior" is from early 14c. The better "improvement" (as in for the better) is from 1690s. To get the better of someone "obtain mastery or victory over" is from 1650s, from better in a sense of "superiority, mastery," which is recorded from mid-15c. Related: Betters.

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better (v.)

Old English *beterian "improve, amend, make better," from Proto-Germanic *batizojan (source also of Old Frisian beteria, Dutch beteren, Old Norse betra, Old High German baziron, German bessern), from *batizo- (see better (adj.)). The meaning "exceed, surpass, outdo" is from 1540s. Related: Bettered; bettering.

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half-and-half (n.)

1756 as "ale and porter;" as a mixture of milk and cream, by 1946.

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half-eagle (n.)

U.S. $5 gold coin minted from 1795 to 1929, authorized in the 1786 resolution for a new monetary system; see half + eagle in the coinage sense.

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half-measure (n.)

"incomplete effort," 1798, from half + measure (n.).

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half-life (n.)

also half life, 1864, "unsatisfactory way of living," from half + life; the sense in physics, "amount of time it takes half a given amount of radioactivity to decay" is first attested 1907.

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