Etymology
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Words related to impromptu

*en 
Proto-Indo-European root meaning "in."

It forms all or part of: and; atoll; dysentery; embargo; embarrass; embryo; empire; employ; en- (1) "in; into;" en- (2) "near, at, in, on, within;" enclave; endo-; enema; engine; enoptomancy; enter; enteric; enteritis; entero-; entice; ento-; entrails; envoy; envy; episode; esoteric; imbroglio; immolate; immure; impede; impend; impetus; important; impostor; impresario; impromptu; in; in- (2) "into, in, on, upon;" inchoate; incite; increase; inculcate; incumbent; industry; indigence; inflict; ingenuous; ingest; inly; inmost; inn; innate; inner; innuendo; inoculate; insignia; instant; intaglio; inter-; interim; interior; intern; internal; intestine; intimate (adj.) "closely acquainted, very familiar;" intra-; intricate; intrinsic; intro-; introduce; introduction; introit; introspect; invert; mesentery.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit antara- "interior;" Greek en "in," eis "into," endon "within;" Latin in "in, into," intro "inward," intra "inside, within;" Old Irish in, Welsh yn, Old Church Slavonic on-, Old English in "in, into," inne "within, inside."
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pro- 

word-forming element meaning "forward, forth, toward the front" (as in proclaim, proceed); "beforehand, in advance" (prohibit, provide); "taking care of" (procure); "in place of, on behalf of" (proconsul, pronoun); from Latin pro (adv., prep.) "on behalf of, in place of, before, for, in exchange for, just as," which also was used as a first element in compounds and had a collateral form por-.

Also in some cases from cognate Greek pro "before, in front of, sooner," which also was used in Greek as a prefix (as in problem). Both the Latin and Greek words are from PIE *pro- (source also of Sanskrit pra- "before, forward, forth;" Gothic faura "before," Old English fore "before, for, on account of," fram "forward, from;" Old Irish roar "enough"), extended form of root *per- (1) "forward," hence "in front of, before, toward, near," etc.

The common modern sense of "in favor of, favoring" (pro-independence, pro-fluoridation, pro-Soviet, etc.) was not in classical Latin and is attested in English from early 19c.

*em- 

Proto-Indo-European root meaning "to take, distribute." 

It forms all or part of: assume; consume; emption; example; exemplar; exemplary; exemplify; exempt; exemption; impromptu; peremptory; pre-emption; premium; presume; presumption; prompt; pronto; ransom; redeem; redemption; resume; sample; sejm; subsume; sumptuary; sumptuous; vintage.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit yamati "holds, subdues;" Latin emere "buy," originally "take," sumere "to take, obtain, buy;" Old Church Slavonic imo "to take;" Lithuanian imu, imti "to take."

For the sense shift from "take" to "buy" in the Latin verbs, compare Old English sellan "to give," source of Modern English sell "to give in exchange for money;" Hebrew laqah "he bought," originally "he took;" and colloquial English I'll take it for "I'll buy it." 

*per- (1)

Proto-Indo-European root forming prepositions, etc., meaning "forward," and, by extension, "in front of, before, first, chief, toward, near, against," etc.

It forms all or part of: afford; approach; appropriate; approve; approximate; barbican; before; deprive; expropriate; far; first; for; for-; fore; fore-; forefather; foremost; former (adj.); forth; frame; frau; fret; Freya; fro; froward; from; furnish; furniture; further; galore; hysteron-proteron; impervious; improbity; impromptu; improve; palfrey; par (prep.); para- (1) "alongside, beyond; altered; contrary; irregular, abnormal;" paradise; pardon; paramount; paramour; parvenu; pellucid; per; per-; percent; percussion; perennial; perestroika; perfect; perfidy; perform; perfume; perfunctory; perhaps; peri-; perish; perjury; permanent; permeate; permit; pernicious; perpendicular; perpetual; perplex; persecute; persevere; perspective; perspire; persuasion; pertain; peruse; pervade; pervert; pierce; portray; postprandial; prae-; Prakrit; pre-; premier; presbyter; Presbyterian; preterite; pride; priest; primal; primary; primate; primavera; prime; primeval; primitive; primo; primogenitor; primogeniture; primordial; primus; prince; principal; principle; prior; pristine; private; privilege; privy; pro (n.2) "a consideration or argument in favor;" pro-; probably; probe; probity; problem; proceed; proclaim; prodigal; produce; profane; profess; profile; profit; profound; profuse; project; promise; prompt; prone; proof; proper; property; propinquity; prophet; prose; prostate; prosthesis; protagonist; Protean; protect; protein; Proterozoic; protest; proto-; protocol; proton; protoplasm; Protozoa; proud; prove; proverb; provide; provoke; prow; prowess; proximate; Purana; purchase; purdah; reciprocal; rapprochement; reproach; reprove; veneer.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit pari "around, about, through," parah "farther, remote, ulterior," pura "formerly, before," pra- "before, forward, forth;" Avestan pairi- "around," paro "before;" Hittite para "outside of," Greek peri "around, about, near, beyond," pera "across, beyond," paros "before," para "from beside, beyond," pro "before;" Latin pro "before, for, on behalf of, instead of," porro "forward," prae "before," per "through;" Old Church Slavonic pra-dedu "great-grandfather;" Russian pere- "through;" Lithuanian per "through;" Old Irish ire "farther," roar "enough;" Gothic faura "before," Old English fore (prep.) "before, in front of," (adv.) "before, previously," fram "forward, from," feor "to a great distance, long ago;" German vor "before, in front of;" Old Irish air- Gothic fair-, German ver-, Old English fer-, intensive prefixes.