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

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Table 1.4: Axioms <strong>of</strong> the <strong>Calculus</strong>.
Involution ([A]) = : A = : [(A)]
Distribution (A[BC]) = : (A[B])(A[C])
Inversion A =
:
Table 1.5: Theorems <strong>of</strong> the <strong>Calculus</strong>.
Cardinality
A:::A = ([A][:::])
Dominion 2A = 2
Inverse Collection =
Inverse Cancellation = A
Inverse Promotion = (A[])
Phase Independence [] = A[]
J Cancellation [][] =
Its forms can be interpreted so as to extend the calculus to complex and transcendental
numbers. Taking cardinalities <strong>of</strong> the inverse produces radian values that can
act as complex numbers when their manipulation is further restricted. The instance
and abstract boundaries act as exponential and logarithmic functions, respectively.
By interpreting these to be <strong>of</strong> base e, basic transcendental values and trigonometric
functions can be formed. These interpretations are included in Tables 1.2 and 1.3.
1.3 Conclusions
The calculus represents many types <strong>of</strong> numbers but it covers only part <strong>of</strong> number
mathematics. It lacks larger structures for doing mathematics and requires much
practical support to make it useful. There are many additions that can be made
to the calculus and areas to which the content can extend. Future work with the
calculus will expand and enhance the ideas presented here.
The calculus presents a new paradigm <strong>of</strong> number representation that challenges
how mathematics is currently done. This material may impact the way mathematics
is done physically in hardware, logically in s<strong>of</strong>tware, and conceptually in an interface