A Prefix-Sum Representation for Division by 9 in Base-10

This work presents a prefix-sum representation for division by 9 in the decimal numeral system. The method expresses any positive integer as a combination of the sum of its decimal prefixes and the sum of its digits. The resulting identity yields an alternative interpretation of division by 9 and provides a simple mental arithmetic procedure. The paper discusses the mathematical derivation of the identity and illustrates its application through examples.