Numbriline

Numbriline is a theoretical framework in mathematics that models numbers as points on a line enhanced by a family of base-dependent transformations called numbrilines. These transformations translate numbers between scales while preserving order and addition, enabling comparisons and base-change analyses within a single structure. The concept provides a language for studying how linear relations behave under changes of numerical base or scale.

Formal aspects: A Numbriline consists of a set N with a total order ≤ and an addition operation +, together with a family of order-preserving maps {λ_b} parameterized by bases b drawn from a fixed base set B. Each λ_b maps N into itself and satisfies compatibility conditions such as λ_b(x + y) = λ_b(x) + λ_b(y). The collection {λ_b} acts like a scaling action on the line; when B contains a single base, the map reduces to the identity, yielding the standard real line as a simple Numbriline.

Intuition and examples: In practice, Numbriline recovers familiar structures when B has one element. When B contains multiple bases, it provides a controlled framework for analyzing how numbers and linear relations transform under base changes, while preserving additive structure. This makes it useful for exploring compatibility between numeral systems and linear arithmetic.

Applications and relation to other fields: The concept serves as a pedagogical tool for teaching about number lines, base representation, and scaling. In theoretical contexts, it offers a language for studying order-preserving morphisms and their interaction with addition, with potential links to ordered algebra, model theory, and data encoding where scale-agnostic arithmetic is relevant.

Etymology: The term Numbriline blends 'number' and 'line', signaling the aim of integrating numeric values with linear geometry.