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

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over a product.
Inverse Cancellation
,(,x) =x =x
1=(1=x) =x ()=x
The generalized inverse cancels itself out.
Inverse Promotion
,x =(,1)x =([x][])
1=x = x ,1 ()=(([[x]][]))
,(xy) =x(,y) =([x][])
1=x y = x ,y ()=(([[x]][]))
Inverse promotion converts the additiveinverse to a product with negative one and
the multiplicative inverse to a power <strong>of</strong> negative one. It provides a straightforward
means <strong>of</strong> moving the inverse around within an expression.
Powers
x 1 = x (([[x]][]))=x
x 0 =1 (([[x]][]))=()
x m x n = x m+n ([(([[x]][m]))][(([[x]][n]))])
= (([[x]][mn]))
(x m ) n = x mn (([[(([[x]][m]))]][n]))
= (([[x]][([m][n])]))
x m y m =(xy) m ([(([[x]][m]))][(([[y]][m]))])
= (([[([x][y])]][m]))
Powers <strong>of</strong> one and zero reduce in boundary form byinvolution and dominion. The
other power formulas are derived by involution and distribution.
Logarithms
ln(xy) =lnx+lny [([x][y])]=[x][y]
ln x=y =lnx,ln y [([x])]=[x]
ln x y = y ln x [(([[x]][y]))]=([[x]][y])
These logarithmic formulas reduce directly by involution.