Fractionderived

Fractionderived is a neologism used to describe a class of objects produced by applying a fixed transformation to rational numbers. In a given framework, every rational r = p/q with q ≠ 0 is sent to an element δ = F(r) in a target set, and the collection of all such δ is called the fractionderived set. The nature of δ depends on the chosen function F; common templates include F(r) = r^2, F(r) = floor(r), and F(r) = 1/(1 − r) for r ≠ 1. The construction aims to study how properties of fractions translate through F into the target structure.

If F is linear over the rationals and compatible with addition and multiplication, the fractionderived set inherits algebraic structure; in other cases it can encode information about size, order, or density of rationals.

Relation to existing concepts: When F encodes continued fraction expansions, the derived objects capture finite truncations or convergents; when F is a modular embedding, the fractionderived elements interact with residue classes. Consequently, fractionderived can be used as a framework for comparing representations of rationals.

Status: The term fractionderived is not widely standardized; it appears primarily in theoretical discussions or speculative literature as a way to discuss transformations of fractions. No canonical references exist, and definitions vary by context.