Differenz

Differenz is the German noun for difference and is used across mathematics, logic, and everyday language to denote how far apart two values or concepts are. In general terms, a Differenz describes a relation of disparity or inequality between two items. The word is closely related to the English term “difference” and appears in many technical phrases.

In arithmetic and algebra, the difference of two numbers a and b is written as a − b. The absolute difference, denoted |a − b|, measures distance on the number line and is always nonnegative. For example, 7 − 3 = 4 and |7 − 10| = 3. Differenzen are the basis of subtraction, and, in more abstract settings, they extend to expressions involving variables and functions.

In set theory, the difference (or relative complement) of sets A and B, written A \ B or A − B, consists of elements that are in A but not in B. For instance, if A = {1, 2, 3} and B = {2, 3, 4}, then A \ B = {1}.

In calculus and numerical analysis, finite differences approximate derivatives. The forward difference Δf(x) = f(x + h) − f(x), backward difference Δf(x) = f(x) − f(x − h), and central difference [f(x + h) − f(x − h)]/(2h) are discrete analogs of rates of change. The difference quotient (f(x + h) − f(x)) / h is a precursor to the derivative.

Etymologically, Differenz derives from Latin differre, with the German term Differenz and related cognates used to express disparity or separation in various technical contexts.