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

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Distribution <strong>of</strong> multiplication over addition follows as a special case <strong>of</strong> the distribution
axiom.
Identity
x +0=x x=x
1x=x ([][x])=([x])=x
The additive identity is implicit in spatial collection as the void. The multiplicative
identity reduces to void within the outer instance boundary and the resulting
unary multiplication reduces to identity.
A.5 Formulas
Common algebraic formulas can be deduced directly from the boundary axioms. Some
are listed below. Many addition formulas have corresponding ones in multiplication.
The pairs are listed together to demonstrate their similarities in boundary form.
Cardinality
x + :::+x = nx x:::x=([x][:::])
x :::x= x n ([x]:::[x])=(([[x]][:::]))
Cardinality counts repeated instances in the same space, whether a collection <strong>of</strong>
x for addition or a collection <strong>of</strong> [x] for multiplication.
Inversion
x +(,x)=0 x=
x=x =1 ([x])=()
Inversion cancels an item and its inverse. It acts on the additiveinverse, cancelling
x and , and it acts on the multiplicative, cancelling [x] and .
Inverse Collection
(,x)+(,y)=,(x+y) =
1=x 1=y =1=xy ([()][()])=()
Inverse collection accumulates multiple objects within the same inverse boundary.
It carries the additive inverse over an sum and it carries the multiplicative inverse