integrity (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).

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."