numberis

Numberis is a term used in a hypothetical mathematical framework to describe numbers according to a digit-based evaluation rule. In this context, a numberis is both the rule applied to a natural number and the value produced by that rule. The central object of study is the rule R, which operates on the decimal digits of n to produce an integer R(n). Researchers then investigate the properties of n with respect to R, including fixed points (numbers for which n = R(n)) and the behavior of R under iteration.

Definition and construction. Fix a digit-based rule R that can be applied to any natural number in a given base. The numberis of n is defined as R(n). The study typically emphasizes fixed points and cycles: numbers n for which R(n) = n, and sequences n, R(n), R(R(n)), … which may settle into a repeating cycle. The base of representation may be varied, producing variant numberis across different numeral systems.

Examples. A common example is the digit-sum function s(n), where R(n) equals the sum of the decimal digits of n. Under this rule, every fixed point is a single-digit number (0 through 9). Other rules—such as the digit-product or the count of nonzero digits—yield different fixed points and cycles, illustrating how the choice of R shapes the numberis landscape.

Variants and applications. Beyond fixed points, researchers study convergence, cycles, and sensitivity to initial conditions. While numberis is a fictional construct, it serves as a pedagogical tool in recreational math and number theory to explore digit-based invariants, base dependence, and simple iterative processes.

See also. Digit sum, digit product, digital root, fixed point, iterated function.