Qnumbers

Qnumbers are a hypothetical generalization of standard numbers designed to encode both numeric value and measurement precision within a single entity. Each Qnumber is an ordered pair (x, p) where x is a real number and p is a precision parameter between 0 and 1. The parameter p represents the confidence or uncertainty attached to x; p = 1 denotes exact knowledge, while smaller values indicate greater uncertainty.

Arithmetic on Qnumbers is defined coordinatewise to carry precision information through computations. Addition is (x1, p1) + (x2, p2) = (x1 + x2, min(p1, p2)). Multiplication is (x1, p1) × (x2, p2) = (x1 x2, min(p1, p2)). The use of the minimum preserves the idea that the result’s precision cannot exceed the least precise input. In this system, equality is (x1, p1) = (x2, p2) if and only if x1 = x2 and p1 = p2.

Rational and real numbers embed naturally. Every real number r corresponds to (r, 1), representing exact knowledge. Qnumbers with p < 1 model quantities known only approximately, making this framework useful for discussions that track uncertainty alongside value.

Besides the basic order relation (x1, p1) ≤ (x2, p2) defined by x1 ≤ x2 and p1 ≤ p2, Qnumbers form a flexible structure for reasoning about computations with finite precision, error propagation, and measurement theory. They are not standard numerical systems but serve as a conceptual tool in theoretical discussions and certain educational settings.

See also: real numbers, rational numbers, interval arithmetic, measurement theory.