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"a whole number" (opposed to fraction), 1570s, from Latin integer (adj.) "whole, complete," figuratively, "untainted, upright," literally "untouched," from in- "not" (see in- (1)) + root of tangere "to touch" (see tangent). The word was used earlier in English as an adjective meaning "whole, entire" (c.1500).
The whole numbers, plus their counterparts less than zero, and zero. The negative integers are those less than zero (–1, –2, –3, and so on); the positive integers are those greater than zero (1, 2, 3, and so on).