Boundary Numbers
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Conventional mathematics is dominated by a textual string-based structure. Common numbers use place-value notation in which the linear position of a digit determines the power of its base, and thus its magnitude. The rules of algebra emphasize commutativity (ordering of arguments) and associativity (ordering of operations), left- and right-identities, and operator precedence. From a boundary perspective, the notation of both arithmetic and algebra is dimensionally impoverished; a one-dimensional string imposes structural and computational rules that are not essential to the inherently parallel structure of numerics.

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BOUNDARY NUMBER SYSTEMS –– Conway, Spencer-Brown, Kauffman, Bricken, and James boundary number systems. Conway numbers were popularized by Donald Knuth as surreal numbers. The other four systems use a depth-value notation. Addition is achieved by placing numbers in the same space, on the outside of each other. Multiplication is achieved by placing one number inside another. Computation becomes much simpler when the overhead of maintaining a linear ordering is removed.

The circle enclosure and graph representations that follow include animations of boundary number computation. Reading a depth-value boundary number is a bit more difficult than reading a place-value conventional number; operations on boundary numbers, in contrast, are much easier.

CIRCLE NUMBERS –– Computation with and standardization of Kauffman numbers are animated in two-dimensions, showing the dynamics of counting, reading, addition, subtraction, multiplication, and division.

GRAPH NUMBERS –– This is an earlier attempt to animate computation using Bricken graph numbers. Animations show that numbers are added by stacking them horizontally, while numbers are multiplied by stacking them vertically. Two simple rules standardize compound structures to a minimal form.