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

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35 Boundary integers add by collection, as ab; and are closed under addition. Pro<strong>of</strong>. Elements a; b 2 I assume one <strong>of</strong> three forms: the void, a positive integer, or a negative integer. 1. If either is the void, the sum reduces to the other value, which isinI. 2. If both are positive integers, the sum is comprised <strong>of</strong> all units, which isinI. 3. If both are negativeintegers, the sum is <strong>of</strong> the form which equals byinverse collection. This form is in I. 4. If a and b are <strong>of</strong> opposite sign and the positive number has fewer units, the negative number can be split by inverse collection and that part cancelled out, e.g. :::=:::=: This result is in I. 5. Otherwise, if a and b must be <strong>of</strong> opposite sign with the positive number having more units. In this case, the negative number cancels directly with part <strong>of</strong> the positive value, e.g. ::::::=:::: This result is in I. Addition is commutative because spatial collection is unordered. Addition is associative because spatial collection makes no grouping distinctions for multiple additions. The additive identity is still void. Every element inIhas an additive inverse. Pro<strong>of</strong>. The inverse <strong>of</strong> the void is the void, = ; derived directly by inversion. All successors to the void, the natural numbers <strong>of</strong> the form :::; have as additive inverses the same value wrapped by inverse, : All predecessors to the void, the natural numbers wrapped by the inverse boundary as ; have as additive inverses the same value wrapped again by the inverse ; which reduces to the uninverted natural number <strong>of</strong> the form ::: byinverse cancellation. Integers a; b 2 I multiply with the same form used for natural numbers, ([a][b]): The integers are closed under multiplication.
36 Pro<strong>of</strong>. Elements a; b 2 I each assume one <strong>of</strong> three forms: the void, a positive integer, or a negative integer. Let c; d 2 N be the magnitude <strong>of</strong> a; b respectively. 1. When either is the void, the product reduces to the void by void multiplication, which isinI. 2. When both are positive, the product is in I because natural numbers are closed under multiplication. 3. When one <strong>of</strong> them is negative, as ([c][]); the product is the inverse <strong>of</strong> the product <strong>of</strong> their magnitudes, ; by inverse promotion. Since the product <strong>of</strong> two natural numbers is a natural number and the inverse <strong>of</strong> a natural number is an integer, this result is an integer. 4. When both numbers are negative, as ([][]); the product is the inverseinverse <strong>of</strong> the product <strong>of</strong> their magnitudes, ; by inverse promotion applied twice. This reduces to ([c][d]); the product <strong>of</strong> two natural numbers, which is a natural number. Multiplication is commutative because [a] and [b] are unordered within the outer instance. Multiplication is associative bytwo applications <strong>of</strong> involution: ([([a][b])][c])=([a][b][c])=([a][([b][c])]): 5.3.3 Calculation Calculation with the additiveinverse is straightforward, seeking to eliminate abstract and multiple inverses. Subtraction requires matching <strong>of</strong> positive and negative quantities: 3 , 5 Given , <strong>Number</strong> Rewrite Subtraction Rewrite Inverse Collection Inversion
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Page 9 and 10: LIST OF TABLES 1.1 Simple Calculati
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Page 14 and 15: Chapter 1 INTRODUCTION Numb
Page 16 and 17: 3 Table 1.1: Simple Calculations in
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Page 22 and 23: Chapter 2 PRIOR WORK Boundary mathe
Page 24 and 25: 11 3 2 Given Number</stro
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Page 28 and 29: 15 Table 2.3: Denition of</
Page 30 and 31: 17 Figure 3.1: The Void. Figure 3
Page 32 and 33: 19 connectivities denes the system
Page 34 and 35: 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 50 and 51: 37 Multiplication requires manageme
Page 52 and 53: 39 Proof. Rational
Page 54 and 55: 41 For example, a half-cardinality
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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 86 and 87: 73 Radians 1+e i =0 ([]) = Euler's
Page 88 and 89: 75 B.2 Square Root The following sq
Page 90 and 91: 77 The a coecient can now be remove
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