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-So what is sliced binary?- ------------------------------------------------------------------------- Sliced binary is a symbolic number system derived from standard binary, with the core idea of representing numbers using non-overlapping, uniquely valued ''slices'' of binary strings, each with only 1, followed by any number of 0s. Unlike standard binary, where each digit represents a power of 2, Sliced Binary redefines the value of each slice using a custom growth pattern. Core concept ------------------------------------------------------------------------ Every valid symbol in Sliced Binary contains exactly one 1 followed by zero or more 0s. These slices represent standalone units. They cannot be combined into multi-1 binary strings. Ex: 11, 111 Although there are exceptions such as 110. Even though it has 2 consecutive 1s, there is no other way to represent it using the given atomic units. The system is built to enforce uniqueness of representation, no value has more than one symbolic representation. How it works ------------------------------------------------------------------------- These are the only allowed atomic units: 1=1 10=2 1,00=3 10,00=6 1,00,00=9 etc. Value Growth Pattern ------------------------------------------------------------------------ Starting from 1=1, values grow by alternating multiplication. - Multiply by 2, then by 1.5, and repeat. 1 = 1 (initial) 10 = 1x2 =2 1,00 = 2x1.5 =3 10,00 = 3x2 =6 1,00,00 = 6x1.5 =9 etc. Why is it named sliced binary? ------------------------------------------------------------------------ The name ''Sliced Binary'' refers to: A restriction of standard binary, you only use ''slices'' (single 1s in a binary position), like cutting individual pieces from a continuous tape. Values grow less rapidly than exponential binary, it's like slicing off part of the traditional doubling behavior. This makes the number line denser, with unique but irregular spacing. Invalid Representations ------------------------------------------------------------------------ Any representation that: Contains more than one or two 1s. Ex: 11=3 Combines valid symbols into a composite string. (Ex: 111, 1011, 1100, etc.) Or repeats a valid value through a different combination. ...all are invalid. Examples Of Invalid Forms: ------------------------------------------------------------------------ 11=3 100 is already equal to 3 111=7 6 is 1000 so it can't exist ------------------------------------------------------------------------ Uniqueness Rule ------------------------------------------------------------------------. A representation is valid only if it maps to a value not reachable by any sum of smaller valid Sliced Binary symbols. This prevents: Duplicate values Multi-1 forms that conflict with cleaner forms (like 10,00 vs 1,10) Fractions in sliced binary: ------------------------------------------------------------------------ You can also express fractional values using repeating forms, just like in decimal. Examples: 0.1 = 0.666... 0.01 = 0.333... 0.010101... = 0.5 Because of the irregular spacing of units, fractional forms tend to behave non-linearly, yet they are still expressible via repeating patterns of valid digits. Features: ------------------------------------------------------------------------ Symbolic- Each valid representation is a unit not a place value. Sparce representations- Only single 1s allow very restricted. Alternating growth- Values grow via x2, x1.5, pattern Unique mapping- Each number has exactly one valid representation. Anomaly-aware Multi-1 forms (Ex: 110) are excluded unless proven unique (none found so far) Custom fractional logic- Supports repeating patterns like decimal but on its own scale Mathematical Implications ------------------------------------------------------------------------ Denser than binary- Value growth is slower than binary doubling, more values are representable at low bit lengths. Well defined and deterministic- You can define algorithms to convert to from decimal Hard boundaries- Cleanly avoids confusion by banning most multi-1 forms. Potential base for numerical research- Can be used to explore alternative radix systems, compression, or symbolic logic. c2a
