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

Abstract

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