Realnumbreid

Realnumbreid is a hypothetical extension of the real numbers designed to couple each value with a secondary component that records auxiliary information such as uncertainty, measurement error, or alternate representations. A realnumbreid is typically represented as a pair (x, e) with x in the real numbers and e in a designated error space E, often a real vector space. A projection map π:(x,e) -> x recovers the conventional real value, while the second component preserves nontrivial information about the value's representation.

Several algebraic schemes exist for Realnumbreid. In the simplest first-order model, E is used to propagate error under standard arithmetic: addition and subtraction combine errors additively, while multiplication uses a first-order rule such as (x,e)*(y,f) = (xy, x f + y e). Division and inversion are defined when denominators are nonzero, with analogous first-order error propagation. This mirrors ideas from dual numbers and interval arithmetic, but Realnumbreid emphasizes a persistent error term tied to the value itself rather than an interval bound.

Origins: Realnumbreid arose in speculative discussions of numeric precision and in thought experiments about extending number systems to encode uncertainty natively. It is not part of mainstream algebra or numerical analysis and has no standard conventions, leading to many variant definitions.

Applications and significance: used primarily as a conceptual tool for modeling how errors propagate through arithmetic, and as a didactic device to illustrate how extra information can be attached to numbers. It shares affinities with dual numbers, complexified numbers, and interval representations, but is distinct in treating the second component as an intrinsic attribute rather than a separate object.