Frumefnum

Frumefnum is a fictional mathematical construct used in speculative discussions of arithmetic that incorporates both a numerical value and a measure of uncertainty. It serves as a thought experiment to explore how value and confidence might be represented together in a single object.

Definition and representation. A frumefnum is an ordered pair (v, r) where v is a real value and r ≥ 0 is called the reliability radius. The set of all frumefnums is denoted F. Real numbers embed into F via (x, 0), recovering ordinary arithmetic when there is no uncertainty.

Arithmetic and uncertainty propagation. The standard operations are extended to carry the reliability parameter. For two frumefnums a = (v1, r1) and b = (v2, r2):

- Addition: a ⊕ b = (v1 + v2, max(r1, r2)).

- Multiplication: a ⊗ b = (v1 · v2, v1 · r2 + v2 · r1 + r1 · r2).

These rules ensure that a result reflects the greatest contributing uncertainty from its operands, while preserving the baseline value.

Order and embedding. Frumefnums are often ordered lexicographically: (v, r) < (u, s) if v < u or (v = u and r < s). When r = 0, the ordering and arithmetic reduce to those of standard real numbers, making frumefnum a context-sensitive extension of the real line.

Uses and reception. In fiction and theoretical exploration, frumefnum models computations where value and confidence must be tracked simultaneously, enabling discussions of uncertainty propagation, decision making under doubt, and nonstandard arithmetic ideas. It is not part of formal mathematics and is primarily a tool for inquiry and storytelling.

See also. Interval arithmetic, fuzzy numbers, hyperreal numbers, nonstandard analysis, uncertainty reasoning.