integer

Integer is a Latin adjective meaning 'whole, complete, untouched, intact,' from in- (not) and the root tag-/tang- (to touch), from tangere (to touch). The root tangere gave Latin tangent (touching), contact (touching together), tangible (able to be touched), contaminate (to touch harmfully), and intact (untouched). An integer, in the original Latin sense, was something that had not been touched, that remained whole and undivided — the same idea expressed in 'integrity' (the quality of being whole) and 'integrate' (to make whole again). The mathematical application of the word to whole numbers — numbers that have not been divided, fractured into fractions — preserves the original image exactly. An integer is a number nobody has cut.

Latin integer was used in everyday speech to mean 'fresh, whole, untampered': a physician might speak of an integer wound (one not yet affected by infection), a lawyer of an integer case (one not yet judged). Cicero uses integer to mean 'unbiased, clean, impartial' — a person whose judgment has not been 'touched' by prior commitment. The root's physical meaning (not touched) extended readily to moral and legal domains (not corrupted, not prejudiced). The mathematical use — a whole number, as opposed to a fraction — is a Latin application of the same image: fractions are numbers that have been divided, cut; integers have not been.

The formal mathematical definition of integers includes not just the positive whole numbers but zero and the negative whole numbers: …, −3, −2, −1, 0, 1, 2, 3, …. The set of integers is denoted ℤ, from the German word Zahlen (numbers). This extension to include negatives and zero was a slow process — negative numbers were treated with suspicion by European mathematicians well into the seventeenth century, and the integers as a unified set including negatives did not become standard notation until the nineteenth century. The Latin word for 'untouched and complete' was eventually extended to include numbers that the Romans would not have recognized as numbers at all.

Integrity, integrate, and integer are the same word applied to ethics, mathematics, and civic life. When a politician is said to have 'integrity,' the word means they are unbroken — untouched by corruption, whole in the sense of being internally consistent. When engineers 'integrate' systems, they make them whole. When a mathematician works with integers, they work with uncut numbers. The Latin root tag- (to touch) runs through all of them: the common idea is that some things should remain untouched, and that wholeness is the sign of the untouched thing.