Working on and with Division in Early China, Third Century BCE—Seventh Century CE

This chapter draws on the earliest extant ChineseChina [Chinese] mathematical sourcesMathematical text [mathematical tablet, manuscript, classic, work, source] to address history of executionHistory execution (of an operation) [historicity operations] divisionDivisionexecution division. The sources are two kinds: third- and second-century manuscripts classics, completed from first century CE onwards. Chemla argues that part classic (The Gnomon Zhou [Dynasty]) share a way prescribing division with term chu 除, which, when used alone in these documents, means subtraction 除subtraction, orRepeated repeated 除repeated subtraction, intention finding remainder. These expressions using prescribe divisions also refer subtractions, each yielding remainder dividendDividend, unit quotient make clear. core which documents attest has following specificities: Dividend divisor, being both measurement valuesMeasurement value, transformed into decimal expansionsDecimaldecimal expansion without unitsUnit Measurement [measurement unit]. Throughout subtractionsRepeated neither nor order magnitude divisor change. serves as pattern detach dividend parts other units, thereby sequence integral numbers successive at time, no place-value system was used. In brief, use is correlated situation, argues, changes radically later Classics. There, chu, seldom regularly employed terminological change meshes execution. Now, dividendDividend appear be written rodsCalculating rods (筭 ) [rods, counting rods, for computation, counting-rods] according systemPlace-value numeration [Place-value systems, Place-value number system, positional system]decimal [decimal notation]. yields orderExecution operations, arithmetical operations) [Execution Execution operation, Executing, Execute]order by order, is, digitDigit digit. this new execution, shifted backwards forwards correspond production units attached given power tenPowers ten. conclusion, suggests radical meaning division, systems author decomposes four types phasePhase (in execution). Phase 1 transforms expansions. Phases 2 3, respectively, yield quotient, adjust subsequent units. 4 fractional result. decomposition highlights several procedures deal 2–4. It enables us interpret devoted 1. show actors’ aim shape short possible divisor. Classics key transformations all phasesPhase For 1, Classic Nine Chapters, CE, attests theorization relates procedures, basis common fundamental operationsOperation (on integers)fundamental designated highly theoretical words. Chapters further completion promotion shows another type had taken root since it displaysDisplay [displayed] shaping set operationsStructure operations [set refined structure] structure, plays central part. developmentEvolution (linear) [evolutionary, improvement, development] practicePracticemathematical practice practices] grants role positions calculating surfaceCalculating surface. cannot severed network issues practices embedded.