Numeral Bases & Arithmetic Functions

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Abstractions about nonstandard numeral bases, division, Euclidean domains, radical notation, inequalities, and special integer functions or number classes.

8 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.

• Complex-base system — Represent real or complex numbers positionally using a nonreal radix and a finite digit alphabet, allowing a single unsigned expansion to encode multiple dimensions when admissibility, convergence, and uniqueness conditions hold.
• Division (mathematics) — Recover a quotient q from dividend a and nonzero divisor b by solving bq=a, with exact, remainder, rational, field and algorithmic meanings determined by the ambient number system.
• Euclidean domain — An integral domain equipped with a Euclidean function that supports division with remainder of strictly smaller value and therefore the Euclidean algorithm.
• Kaprekar number — Classify a base-b natural number whose square can be split at a declared digit position into two parts that sum back to the number.
• Khintchine inequality — Two-sided bounds comparing the Lp norm of a Rademacher random sum with the ℓ2 norm of its coefficients, using constants depending only on p.
• Non-integer base of numeration — Represent numbers positionally with a real or complex radix that is not an integer, making admissible digits, expansion algorithms, and nonuniqueness depend on the radix.
• Radical symbol — The √ notation indicating a principal square root or, with an index, an nth-root operation, whose branch and domain conventions determine the represented value.
• Sum of squares function — Count ordered signed integer k-tuples whose squared coordinates sum to n, yielding the arithmetic function r_k(n) and its divisor-sum, theta-series, and local-obstruction structure.