Etymology
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Words related to become

be- 

word-forming element of verbs and nouns from verbs, with a wide range of meaning: "about, around; thoroughly, completely; to make, cause, seem; to provide with; at, on, to, for;" from Old English be- "about, around, on all sides" (the unstressed form of bi "by;" see by (prep.)). The form has remained by- in stressed positions and in some more modern formations (bylaw, bygones, bystander).

The Old English prefix also was used to make transitive verbs and as a privative prefix (as in behead). The sense "on all sides, all about" naturally grew to include intensive uses (as in bespatter "spatter about," therefore "spatter very much," besprinkle, etc.). Be- also can be causative, or have just about any sense required. The prefix was productive 16c.-17c. in forming useful words, many of which have not survived, such as bethwack "to thrash soundly" (1550s) and betongue "to assail in speech, to scold" (1630s).

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come (v.)

elementary intransitive verb of motion, Old English cuman "to move with the purpose of reaching, or so as to reach, some point; to arrive by movement or progression;" also "move into view, appear, become perceptible; come to oneself, recover; arrive; assemble" (class IV strong verb; past tense cuom, com, past participle cumen), from Proto-Germanic *kwem- (source also of Old Saxon cuman, Old Frisian kuma, Middle Dutch comen, Dutch komen, Old High German queman, German kommen, Old Norse koma, Gothic qiman), from PIE root *gwa- "to go, come."

The substitution of Middle English -o- for Old English -u- before -m-, -n-, or -r- was a scribal habit before minims to avoid misreading the letters in the old style handwriting, which jammed them together. The practice similarly transformed some, monk, tongue, worm. Modern past tense form came is Middle English, probably from Old Norse kvam, replacing Old English cuom.

Meaning "to happen, occur" is from early 12c. (come to pass "happen, occur" is from 1520s). As an invitation to action, c. 1300; as a call or appeal to a person (often in expanded forms: "come, come," "come, now"), mid-14c. Come again? as an off-hand way of asking "what did you say?" is attested by 1884. For sexual senses, see cum.

Remarkably productive with prepositions (NTC's "Dictionary of Phrasal Verbs" lists 198 combinations); consider the varied senses in come to "regain consciousness," come over "possess" (as an emotion), come at "attack," come on (interj.) "be serious," and come off "occur, have some level of success" (1864). Among other common expressions are:

To come down with "become ill with" (a disease), 1895; come in, of a radio operator, "begin speaking," 1958; come on "advance in growth or development," c. 1600; come out, of a young woman, "make a formal entry into society," 1782; come round "return to a normal state or better condition," 1841; come through "act as desired or expected," 1914; come up "arise as a subject of attention," 1844; come up with "produce, present," 1934.

To have it coming "deserve what one suffers" is from 1904. To come right down to it "get to fundamental facts" is from 1875.

became 
past tense of become (q.v.).
becoming (adj.)
"looking well, aesthetically befitting," 1560s, from earlier sense of "fitting, proper" (early 13c.), present-participle adjective from become. Related: Becomingly; becomingness.
*gwa- 

*gwā-, also *gwem-, Proto-Indo-European root meaning "to go, come."

It forms all or part of: acrobat; adiabatic; advent; adventitious; adventure; amphisbaena; anabasis; avenue; base (n.) "bottom of anything;" basis; become; circumvent; come; contravene; convene; convenient; convent; conventicle; convention; coven; covenant; diabetes; ecbatic; event; eventual; hyperbaton; hypnobate; intervene; intervenient; intervention; invent; invention; inventory; juggernaut; katabatic; misadventure; parvenu; prevenient; prevent; provenance; provenience; revenant; revenue; souvenir; subvention; supervene; venire; venue; welcome.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit gamati "he goes," Avestan jamaiti "goes," Tocharian kakmu "come," Lithuanian gemu, gimti "to be born," Greek bainein "to go, walk, step," Latin venire "to come," Old English cuman "come, approach," German kommen, Gothic qiman.

misbecome (v.)

"suit ill, be unfitting," 1520s, from mis- (1) "badly, wrongly" + become.