Numeration & Arithmetic Representations

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Abstractions about numeral systems, digital constants, logarithmic and fixed-precision arithmetic, signed representations, and alternative ways of encoding or computing numbers.

15 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Alpha max plus beta min algorithm — A low-cost approximation to two-dimensional Euclidean magnitude using a weighted maximum plus weighted minimum of component magnitudes.
• Base36 — A positional numeral and text-encoding convention using 36 digit values, commonly represented by 0–9 and A–Z, to express nonnegative integers compactly.
• Bijective numeration — A numeral system giving every nonnegative integer exactly one finite digit string, with no zero digit or leading-zero ambiguity in its positional form.
• Champernowne constant — The real number formed by concatenating the positive integers in order in a chosen base, with the base-ten version proved normal.
• Common logarithm — The base-ten logarithm, the inverse of raising ten to a power and historically central to decimal calculation tables, scientific notation and orders of magnitude.
• Cyclic Number — Encode a repeating unit fraction as a digit block whose consecutive nonzero multiples appear as cyclic rotations of that block in a fixed base.
• Decimal representation — A positional base-ten digit expansion of a nonnegative real number, with finite integer part and finite or infinite fractional part.
• Fixed-precision arithmetic — Arithmetic performed in a numeric format with a fixed finite number of digits or bits, requiring rounding, overflow and exceptional-value rules.
• Logarithmic number system — A numerical representation encoding a real number by sign and the logarithm of its magnitude.
• Lunar arithmetic — Replace digit addition by maximum and digit multiplication by minimum, extending them positionally without carries so nonnegative base-b numerals form a closed idempotent arithmetic with altered sums, products, factors, and primes.
• Prouhet–Thue–Morse constant — The real number whose binary expansion is the Thue–Morse sequence.
• Residue number system — A numeral representation encoding an integer as its residues modulo a set of pairwise coprime moduli.
• Signed number representations — Encoding schemes that map positive, zero and negative numbers into fixed-width digit patterns without an external sign symbol.
• Steinhaus–Moser notation — A recursive polygon notation for extremely large integers in which additional polygon sides encode iterated nesting of the preceding operation.
• Symmetric level-index arithmetic — A numerical representation and arithmetic system using signed nested logarithmic levels to span extremely large and small magnitudes.