A Formal Theory of Measurement-Based Mathematics

Division by zero is normally banned because the single symbol “0” carries two different meanings: (i) absolute nothingness, and (ii) a quantity that exists but cannot be measured. I tease these notions apart by enlarging the real numbers to

S = ℝ ∪ { 0_bm, 0_m, 1_t },

where 0_bm denotes an absolute void, 0_m a measured-but-missing value, and 1_t is a transient unit that appears when two 0_m ’s are compared. A short list of axioms turns S into a commutative semiring; the usual law “(a / b) · b = a” is relaxed only in the cases that would cause contradictions. This small change lets certain “divide-by-zero” computations return informative results instead of crashing. I contrast the scheme with wheels, projective extensions, and non-standard analysis, and sketch applications to database NULL semantics, sensor fusion below a noise floor, and fault-tolerant programming. A proof-of-concept implementation accompanies the paper.