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

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73 Radians 1+e i =0 ([]) = Euler's formula works because the form [] was dened that way.
Appendix B EXAMPLES B.1 Multiplication Here is an example <strong>of</strong> multiplication <strong>of</strong> two integers. To visually simplify the forms, a few denitions will be made rst. Let b be the base radix, b = : : Let fAg be the base function on A, dened as fAg =([b][A]): : The radix carries into the base function, bfAg=fAg: The base function collects, fAgfBg=fABg; and promotes, f(A[B])g=(A[fBg]); just like inverse: fAgfBg=([b][A])([b][B])=([b][AB])=fABg; f(A[B])g=([b][(A[B])])=([b]A[B])=(A[([b][B])])=(A[fBg]): Using these denitions, the multiplication 23 114 can be computed by making copies at each magnitude, collecting magnitudes, and doing a carry operation. 23 114 Given fg ffgg <strong>Number</strong> Rewrite ([fg][ffgg]) Function Rewrite ([fg][ffgg])([][ffgg]) Distribution f([][ffgg])g([][ffgg]) Promotion fffggffgggffggffggffgg Cardinality (2x) fffg fgg fg fg fgg Collection fff g g g Collection fffggbgb Replacement fffggg Carry 2622 Rewrite
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A Calculus <strong
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University ofWashi
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4.2 The Two-Boundary Calcul
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A.2 Functions : : : : : : : : : : :
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LIST OF TABLES 1.1 Simple Calculati
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as an undergraduate at the Universi
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Chapter 1 INTRODUCTION Numb
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3 Table 1.1: Simple Calculations in
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5 contents, they are also drawn as
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7 Table 1.4: Axioms of</str
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Chapter 2 PRIOR WORK Boundary mathe
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11 3 2 Given Number</stro
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13 this system uses string rewrite
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15 Table 2.3: Denition of</
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17 Figure 3.1: The Void. Figure 3
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19 connectivities denes the system
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Chapter 4 INSTANCE AND ABSTRACT 4.1
Page 36 and 37: 23 The third rule creates new eleme
Page 38 and 39: 25 Distribution: (A[BC]) : =(A[B])(
Page 40 and 41: 27 Multiplication of</stron
Page 42 and 43: 29 Table 4.1: The Two-Boundary <str
Page 44 and 45: 31 transformations. 5.2.1 Elements
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
Page 54 and 55: 41 For example, a half-cardinality
Page 56 and 57: Chapter 6 PHASE 6.1 Introduction Th
Page 58 and 59: 45 Using J, two concise and useful
Page 60 and 61: 47 The ambiguous sign makes the car
Page 62 and 63: 49 Table 6.1: Transcendentals in th
Page 64 and 65: Chapter 7 FUTURE WORK 7.1 Introduct
Page 66 and 67: 53 ematics problems could be phrase
Page 68 and 69: 55 boundary form, in search <strong
Page 70 and 71: 57 Technology interface sof
Page 72 and 73: 59 such asintegrals and Fourier tra
Page 74 and 75: 61 mulas specify how they transform
Page 76 and 77: 63 dominion A theorem stating that
Page 78 and 79: BIBLIOGRAPHY [1] B. Banaschewski. O
Page 80 and 81: 67 [28] L. Lamport. LaTeX: A Docume
Page 82 and 83: 69 A.2 Functions The standard algeb
Page 84 and 85: 71 Distribution of
Page 88 and 89: 75 B.2 Square Root The following sq
Page 90 and 91: 77 The a coecient can now be remove
Page 92: 79 f = (([[(([][]) [(([[b]][]))])]]
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