Abstract
Investigated is a number system in which the square of a basis number: ( w) 2, and the square of its additive inverse: (− w) 2, are not equal. Termed W space, a vector space over the reals, this number system will be introduced by restating defining relations for complex space C , then changing a defining conjugacy relation from conj( z) + z = 0 in the complexes to conj( z) + z = 1 for W space. This change produces a dual-represented vector space consisting of two dual, isomorphic fields, which are unified under one “context-sensitive” multiplication. Fundamental algebraic and geometric properties will be investigated. W space can be interpreted as a generalization of the complexes but is characterized by an interacting duality which seems to produce two of everything: two representations, two multiplications, two norm values, and two solutions to a linear equation. W space will be compared to a previous suggestion of a similar algebra, and then possible applications will be offered, including a W space fractal.
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