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

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Table 6.1: Transcendentals in the <strong>Calculus</strong>.
Transcendental Standard Boundary
J i []
p
i
,1 (([J]))
3:1415 ::: (J[J]([J]))
e 2:7183 ::: ()
From J and i, a radian interpretation <strong>of</strong> can be algebraically constructed.
= ,1 i i
= ([][(([J]))][J])
= (J[J]([J]))
This construction <strong>of</strong> treats it not as a real number but as the radian unit. Its
semantic value comes out <strong>of</strong> this algebraic construction and has no relation to its
numerical value <strong>of</strong> =3:14159 :::.
The numerical values <strong>of</strong> e and in standard mathematics are based on criteria
independent <strong>of</strong> the manipulations used here (e.g. geometry) and are not essential
to the algebraic behavior <strong>of</strong> the transcendentals. In the calculus, they are devoid <strong>of</strong>
numerical value and remain incommensurable with other quantities. The boundary
representations <strong>of</strong> these basic transcendental values are given in Table 6.1.
6.5.3 Trigonometric Functions
In the calculus, trigonometric functions can be computed symbolically using the three
boundary rules. The function for cosine in standard and boundary forms appears as:
cos x = eix + e ,ix
2
versus
cosx=([(([x]([J])))(([]([J])))])
The cosine function behaves as expected, reducing to real results. For instance,
cos = ,1. Although the derivation can be lengthy and complicated, it is based
entirely on the three axioms and the theorems that follow from them. Accordingly,
trigonometric formulas can be derived from these axioms.