Numeral Systems & Digit Algorithms

← Back to Domain-Specific Families

Abstractions about representing numbers and manipulating their digits — positional and fixed-point encodings (Q number format, Z-order curve, division algorithm), digit-based number properties (cyclic number, normal number, multiplicative digital root, sum-product number), and arithmetic procedures exploiting digit structure such as the Bailey-Borwein-Plouffe formula.

11 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.

• Bailey–Borwein–Plouffe Formula — A base-16 rational series for π whose radix alignment lets modular exponentiation recover hexadecimal or binary digits at a distant position without first generating the intervening expansion.
• BIT predicate — A binary relation that returns the zero-based j-th binary digit of a nonnegative integer i, enabling finite-set membership tests by bit position.
• 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.
• Division Algorithm — A terminating integer-arithmetic procedure that returns quotient and remainder satisfying a declared Euclidean division contract.
• Multiplicative Digital Root — The terminal single base-b digit reached by repeatedly replacing a nonnegative integer with the product of its digits, paired with multiplicative persistence as the number of iterations required to reach that fixed point.
• Multiply–accumulate operation — A computational recurrence that multiplies two operands and adds the product into a running accumulator, with numeric behavior determined by precision, rounding, overflow, and update semantics.
• Normal Number — A real number whose base-b expansion gives every finite length-k digit block its uniform limiting frequency b^-k, for every k, with the base kept explicit.
• Perfect digit-to-digit invariant — A perfect digit-to-digit invariant is a positive integer equal to the sum obtained by raising each of its decimal digits to the power of that same digit.
• Q Number Format — Encode binary fixed-point values as stored integers with an implicit power-of-two scale, using a Q notation that declares fractional bits, integer width, and signedness while keeping vendor sign-bit conventions explicit.
• Sum-product number — Sum-product number denotes a number equal to the product of the sum and product of its digits within recreational number theory.
• Z-Order Curve — Linearize a quantized multidimensional grid by interleaving coordinate bits into a Morton code, making every quadtree or octree cell a contiguous one-dimensional key interval while preserving locality imperfectly.