Fraktionale

Fraktionale is a German adjective meaning "fractional" or "relating to a fraction," used to describe quantities, parts, or concepts that are not whole. In mathematical and scientific contexts it often prefixes terms to indicate a fraction of a unit or a non-integer quantity. Example phrases include fraktionale Anteile and fraktionale Teilungen, in contrast to ganzzahlig for integers.

In mathematics, Fraktionalkalkül, or fractional calculus, denotes the extension of differentiation and integration to non-integer orders. Operators of order α (where α is a real number) generalize the standard derivative and integral, and can be defined in several equivalent ways, such as through integral transforms or series representations. These operators are used to model systems with memory and anomalous dynamics.

In the context of numbers, the fractional part refers to the portion of a real number after the decimal point; for example, the fractional part of 3.14 is 0.14. The notation {x} is commonly used for this quantity in mathematical texts.

Applications of fractional concepts appear in physics and engineering, including viscoelastic materials, diffusion processes, signal processing, and control theory, where fractional-order models capture complex, history-dependent behavior more accurately than integer-order models.

Terminology varies by discipline; in German usage, Fraktionalkalkül is the standard term for fractional calculus, and "fraktionale" serves as the descriptive adjective in phrases describing fractional quantities. See also: fraction, fractional calculus, fractional part.