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

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Chapter 1 INTRODUCTION <strong>Number</strong>s are simpler than they appear. This thesis presents a calculus <strong>of</strong> number that demonstrates this simplicity. We commonly represent and manipulate numbers with a centuries-old notation comprised <strong>of</strong> Arabic digits, decimal point, plus sign, minus sign, times sign, fraction bar, a few other operators, and some constants. Typical numbers are 0, 1, 2, 10, ,1, 3=7, 2:9, p 2, e 2 , i. The standard Arabic notation represents ten numbers directly as digits and a few more as symbolic constants. All others are constructed out <strong>of</strong> these using arithmetic operations. Many <strong>of</strong> the operations used to build numbers are implicit in the representation, such as the magnitude gain in place value. Hiding information in this manner achieves conciseness but leaves the forms abstract and dissociated from their own behavior. These traditional representations are but one embodiment <strong>of</strong>numbers. Formal alternatives exist but generally lack the scope and conciseness <strong>of</strong> the standard form. Other representations are useful because they display interesting properties <strong>of</strong> numbers. Conway <strong>Number</strong>s, for instance, build real numbers out <strong>of</strong> a single object, the bi-set [12]. This thesis presents a representation <strong>of</strong> numbers, called boundary numbers, in which numbers and arithmetic operations are built out <strong>of</strong> three objects. 1.1 Minimalism The calculus in this thesis progresses representation <strong>of</strong> number towards a minimal basis. It begins with the most fundamental acts <strong>of</strong> representation. By starting from scratch, the calculus avoids the conceptual baggage that has been introduced into standard notation throughout its development. The calculus is an attempt to minimize the implicit constraints on form manipulation to just those necessary to dene the system <strong>of</strong> mathematics. Other constraints should be explicit in the forms. Commutativity, for example, would be explicit if commutative functions displayed symmetry between their arguments and
2 Solution Loop application 6 ? conceptual objects 6 Activity problem representation translation and control ? 6 computational objects calculation Figure 1.1: Stages <strong>of</strong> Mathematical Representation. non-commutative functions did not. By making constraints explicit, much \knowledge" <strong>of</strong> mathematics can be embodied in the representation. Standard notation has abstracted away the behavior <strong>of</strong> its forms so that knowledge <strong>of</strong> their behavior has become implicit. For example, the notation abstracts away magnitudes <strong>of</strong> the digits, making 1 + 1 = 2 not altogether obvious but requiring memorization. A model <strong>of</strong> this disembodiment is shown in Figure 1.1. The standard notation represents the conceptual objects <strong>of</strong> numbers and functions but not the computational objects upon which constraints are imposed. When using math, we work through the solution loop shown in Figure 1.1. We represent elements <strong>of</strong> a problem as conceptual objects in a mathematical notation. We manipulate these objects according to calculation rules, usually through separate computational objects. Guided by solution techniques, we manipulate the conceptual objects into forms that lend insights into the problem. Layers <strong>of</strong> mathematical objects are necessary because most mathematical forms cannot be \directly" manipulated with a concise set <strong>of</strong> rules. Comprehensively dened manipulation <strong>of</strong>ten requires intermediate forms. Moreover, these intermediate forms can be completely disparate from the original form, making the translation itself complicated. For example, with digit magnitude implicit, numbers are <strong>of</strong>ten added by use <strong>of</strong> separate representations. Many computers use bits as computational objects because bits can be physically added by digital circuitry. Many people learn to add by using
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Page 40 and 41: 27 Multiplication of</stron
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Page 46 and 47: 33 Inversion: A = : Equivalent Expr
Page 48 and 49: 35 Boundary integers add by collect
Page 50 and 51: 37 Multiplication requires manageme
Page 52 and 53: 39 Proof. Rational
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63 dominion A theorem stating that
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69 A.2 Functions The standard algeb
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75 B.2 Square Root The following sq
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