numeronsa

Numeronsa is a fictional class of real numbers used in speculative mathematics and puzzle contexts to illustrate how simple generation rules can produce a wide range of numeric objects.

Definition and construction: One way to define a numeronsa is to choose a deterministic rule that outputs a binary sequence (c_n) in {0,1}. Interpret c_n as the nth digit in the binary expansion of a real number in [0,1], giving x = sum_{n=1}^\infty c_n 2^{-n}. For example, let c_1 = 1 and, for n ≥ 1, let c_{n+1} = 1 if the number of 1s in the binary representation of n is odd, and 0 otherwise. The resulting x is a numeronsa.

Variants and properties: Different rules generate different numeronsa. Some yield very regular, ultimately periodic binary expansions, while others produce highly irregular, quasi-random-looking sequences. In all cases, the numbers exist and are contained in [0,1]. The construction highlights how a simple rule on the natural numbers can map to a continuum of real values.

Relation to mathematics and uses: In this fictional framing, numeronsa serve as thought experiments about computability, information content, and base-n numeral representations. They are not part of standard mathematical definitions and have no established theorems outside the context of the article in which the concept is used.

See also: binary expansion, normal number, computability, pseudo-random sequence.