A Calculus of Number Based on Spatial Forms - University of ...

Chapter 5
INVERSE
5.1 Introduction
This chapter extends boundary numbers to integers, rationals, and algebraic irrationals
by introducing a third boundary to serve as a generalized inverse.
The rst two boundaries, instance and abstract, remain exactly as dened in
Chapter 4. The rst two axioms, involution and distribution, are also the same.
The third boundary extends this system with a generalized inverse. This boundary
shall be called the inverse boundary and be written as a triangle when empty, \4,"
or as angled brackets otherwise, \:" Aninverse boundary with no content is
equivalent tovoid. The inverse boundary is its own functional inverse, so nesting
inverse inside <strong>of</strong> inverse cancels out.
The characteristic property <strong>of</strong> the inverse boundary is dened by the inversion
axiom. An element and its inverse cancel to the void. Similarly, an element and its
inverse can be introduced from the void. All elements have aninverse except for the
black hole.
Inverse serves directly as the additive inverse (i.e. ,x) and, in a construction
with instance and abstract, as the multiplicative inverse (i.e. 1=x). Reduction <strong>of</strong>
both inverses is handled by the inversion axiom, cancelling a form collected with its
inverse. The context <strong>of</strong> this cancellation unambiguously determines the semantics
as the subtraction (i.e. x , x = 0) or division (i.e. x=x = 1). Because the inverse
boundary serves both addition and multiplication, it is considered to be a generalized
inverse.
This chapter begins by redening the calculus with three boundaries.
5.2 The Three-Boundary <strong>Calculus</strong>
The two-boundary calculus from Chapter 4 is extended with a third boundary by
redening the set <strong>of</strong> elements to include it and by adding a third axiom to dene its