Etymology
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Words related to integer

fraction (n.)
late 14c., originally in the mathematical sense, from Anglo-French fraccioun (Old French fraccion, "a breaking," 12c., Modern French fraction) and directly from Late Latin fractionem (nominative fractio) "a breaking," especially into pieces, in Medieval Latin "a fragment, portion," noun of action from past participle stem of Latin frangere "to break (something) in pieces, shatter, fracture," from Proto-Italic *frang-, from a nasalized variant of PIE root *bhreg- "to break." Meaning "a breaking or dividing" in English is from early 15c.; sense of "broken off piece, fragment," is from c. 1600.
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in- (1)
Origin and meaning of in-
word-forming element meaning "not, opposite of, without" (also im-, il-, ir- by assimilation of -n- with following consonant, a tendency which began in later Latin), from Latin in- "not," cognate with Greek an-, Old English un-, all from PIE root *ne- "not."

In Old French and Middle English often en-, but most of these forms have not survived in Modern English, and the few that do (enemy, for instance) no longer are felt as negative. The rule of thumb in English has been to use in- with obviously Latin elements, un- with native or nativized ones.
*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."
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.

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).

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.

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.