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

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In presenting this thesis in partial fulllment <strong>of</strong> the requirements for a Master's degree at the University <strong>of</strong>Washington, I agree that the Library shall make its copies freely available for inspection. I further agree that extensive copying <strong>of</strong> this thesis is allowable only for scholarly purposes, consistent with \fair use" as prescribed in the U.S. Copyright Law. Any other reproduction for any purposes or by any means shall not be allowed without my written permission. Signature Date
University <strong>of</strong>Washington Abstract A <strong>Calculus</strong> <strong>of</strong> <strong>Number</strong> <strong>Based</strong> on Spatial Forms by Jerey M. James Chairperson <strong>of</strong> Supervisory Committee: Pr<strong>of</strong>essor Judith Ramey Technical Communication A calculus for writing and transforming numbers is dened. The calculus is based on a representational and computational paradigm, called boundary mathematics, in which representation consists <strong>of</strong> making distinctions out <strong>of</strong> the void. The calculus uses three boundary objects to create numbers and covers complex numbers and basic transcendentals. These same objects compose into operations on these numbers. Expressions transform using three spatial match and substitute rules that work in parallel across expressions. From the calculus emerge generalized forms <strong>of</strong> cardinality and inverse that apply identically to addition and multiplication. An imaginary form in the calculus expresses numbers in phase space, creating complex numbers. The calculus attempts to represent computational constraints explictly, thereby improving our ability to design computational machinery and mathematical interfaces. Applications <strong>of</strong> the calculus to computational and educational domains are discussed.
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39 Proof. Rational
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41 For example, a half-cardinality
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BIBLIOGRAPHY [1] B. Banaschewski. O
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75 B.2 Square Root The following sq
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