Etymology
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integer (n.)

"a whole number" (as opposed to a fraction), 1570s, from noun use of Latin integer (adj.) "intact, whole, complete," figuratively, "untainted, upright," literally "untouched," from in- "not" (see in- (1)) + root of tangere "to touch" (from PIE root *tag- "to touch, handle"). The word was used earlier in English as an adjective in the Latin sense, "whole, entire" (c. 1500).

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entire (adj.)

mid-14c., of things, "whole, intact," from Old French entier "whole, unbroken, intact, complete," from Latin integrum "completeness" (nominative integer; see integer). Related: Entireness.

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integral (adj.)

late 15c., "of or pertaining to a whole; intrinsic, belonging as a part to a whole," from Old French intégral (14c.), from Medieval Latin integralis "forming a whole," from Latin integer "whole" (see integer). Related: Integrally. As a noun, 1610s, from the adjective.

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entirety (n.)

"wholeness, completeness, state of being entire or whole," also entierty, mid-14c., enterete, intierty, from Anglo-French entiertie, Old French entiereté "totality, entirety; integrity, purity," from Latin integritatem (nominative integritas) "completeness, soundness, integrity," from integer (see integer).

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integrity (n.)
Origin and meaning of integrity

c. 1400, integrite, "innocence, blamelessness; chastity, purity," from Old French integrité and directly from Latin integritatem (nominative integritas) "soundness, wholeness, completeness," figuratively "purity, correctness, blamelessness," from integer "whole" (see integer).

The sense of "wholeness, perfect condition" is attested from mid-15c.; that of "soundness of moral principle and character; entire uprightness or fidelity, especially in regard to truth and fair dealing" is by 1540s.

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*tag- 
Proto-Indo-European root meaning "to touch, handle," with figurative extensions ("border on; taste, partake of; strike, hit; affect, impress; trick, cheat; mention, speak of").

It forms all or part of: attain; contact; contaminate; entire; intact; integer; integrate; integrity; noli me tangere; tact; tactics; tactile; tangent; tangible; task; taste; tax; taxis.

It is the hypothetical source of/evidence for its existence is provided by: Latin tangere "to touch," taxare "to touch, assess," tactus "touch," integer "intact, whole, complete, perfect; honest;" Greek tassein "to arrange," tetagon "having seized;" Old English þaccian "stroke, strike gently."
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integrate (v.)
1630s, "to render (something) whole, bring together the parts of," from Latin integratus, past participle of integrare "make whole," from integer "whole, complete," figuratively, "untainted, upright," literally "untouched," from in- "not" (see in- (1)) + root of tangere "to touch," from PIE root *tag- "to touch, handle."

The meaning "put together parts or elements and combine them into a whole" is from 1802. The "racially desegregate" sense (1940) probably is a back-formation from integration. Related: Integrated; integrating.
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cloud nine (n.)
by 1950, sometimes also cloud seven (1956, perhaps by confusion with seventh heaven), American English, of uncertain origin or significance. Some connect the phrase with the 1895 International Cloud-Atlas (Hildebrandsson, Riggenbach and Teisserenc de Bort), long the basic source for cloud shapes, in which, of the ten cloud types, cloud No. 9, cumulonimbus, was the biggest, puffiest, most comfortable-looking. Shipley suggests the sense in this and other expressions might be because, "As the largest one-figure integer, nine is sometimes used for emphasis." The phrase might appear in the 1935 aviation-based play "Ceiling Zero" by Frank Wilbur Wead.
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