also co-efficient, c. 1600, "that which unites in action with something else to produce a given effect," from co- + efficient. Probably influenced by Modern Latin coefficiens, which was used in mathematics in 16c., introduced by French mathematician François Viète (1540-1603). As an adjective, "acting in union to the same end," from 1660s. Related: Coefficiency.

in mathematics, "the result of the process of division, quantity resulting from the division of one number by another, number of times one quantity is contained in another," mid-15c., quocient, from Latin quotiens "how often? how many times?; as often as," pronominal adverb of time, from quot "how many?" (from PIE root *kwo-, stem of relative and interrogative pronouns). The Latin adverb quotiens was mistaken in Middle English for a present participle of quot in -ens.

1540s, in mathematics, "that term of a fraction which indicates the value of the fractional unit" (commonly the number written below the numerator or dividend), from Medieval Latin denominator, agent noun from past-participle stem of denominare "to name," from de- "completely" (see de-) + nominare "to name," from nomen "name" (from PIE root *no-men- "name"). As "one who or that which gives a name," 1570s.

"arithmetic, music, geometry, astronomy" (the four branches of mathematics, according to the Pythagoreans), by 1751, from Latin quadrivium, which meant "place where four roads meet, crossroads," from quadri- "four" (from PIE root *kwetwer- "four") + via "way, road, channel, course" (see via). Compare liberal arts, and also see trivium.

The adjective **quadrivial** is attested from mid-15c. in English with the sense of "belonging to the quadrivium," late 15c. with the sense of "having four roads, having four ways meeting in a point."

1650s, "square," with -ic + obsolete quadrate "a square; a group of four things" (late 14c.), from Latin quadratum, noun use of neuter adjective quadratus "square, squared," past participle of quadrare "to square, make square; put in order," related to quadrus "a square," quattuor "four" (from PIE root *kwetwer- "four"). In mathematics by 1660s; the algebraic **quadratic equations** (1680s) are so called because they involve the square and no higher power of x.

"art of computation, the most elementary branch of mathematics," mid-13c., arsmetike, from Old French arsmetique (12c.), from Latin arithmetica, from Greek arithmetikē (tekhnē) "(the) counting (art)," fem. of arithmetikos "of or for reckoning, arithmetical," from arithmos "number, counting, amount" (from PIE *erei-dhmo-, suffixed variant form of root *re- "to reason, count").

The form arsmetrik was based on folk-etymology derivation from Medieval Latin ars metrica; the spelling was corrected early 16c. in English (though arsmetry is attested from 1590s) and French. The native formation in Old English was tælcræft, literally "tell-craft."

1690s, "Arabic system of computation," from French algorithme, refashioned (under mistaken connection with Greek arithmos "number") from Old French algorisme "the Arabic numeral system" (13c.), from Medieval Latin algorismus, a mangled transliteration of Arabic al-Khwarizmi "native of Khwarazm" (modern Khiva in Uzbekistan), surname of the mathematician whose works introduced sophisticated mathematics to the West (see algebra). The earlier form in Middle English was algorism (early 13c.), from Old French. The meaning broadened to any method of computation; from mid-20c. especially with reference to computing.

1530s, "one's proper work or purpose; power of acting in a specific proper way," from French fonction (16c.) and directly from Latin functionem (nominative functio) "a performance, an execution," noun of action from funct-, past-participle stem of fungi "perform, execute, discharge," from PIE *bhung- "be of use, be used" (source also of Sanskrit bhunjate "to benefit, make benefit, atone," Armenian bowcanem "to feed," Old Irish bongaid "to break, harvest"), which is perhaps related to root *bhrug- "to enjoy." Meaning "official ceremony" is from 1630s, originally in church use. Use in mathematics probably was begun by Leibnitz (1692). In reference to computer operations, 1947.