# Double Entry Multidimensional Accounting

Double-entry bookkeeping was developed during the fifteenth century and was first recorded as a system by the Italian mathematician Luca Pacioli in 1494.  Double-entry bookkeeping has been used as the accounting system in market-based enterprises of any size throughout the world for several centuries.  Incredibly, however, the mathematical basis for DEB was not known, at least not in the field of accounting.  The mathematical basis behind DEB (algebraic operations on ordered pairs of numbers) was developed in the nineteenth century by Sir William Rowan Hamilton as an abstract mathematical construction to deal with complex numbers and fractions.  The particular example of the ordered pairs construction that is relevant to DEB (“group of differences” in technical terms) is the one used in undergraduate algebra courses to construct a number system with subtraction by using operations on ordered pairs of non-negative numbers.  All that is required to see the connection with DEB is to identify these ordered pairs with the two-sided T-accounts of DEB (debits on the left side and credits on the right side).  Yet with the exception of a paragraph in a semipopular book by D.E. Littlewood, the author has not been able to find a single mathematics book, elementary or advanced, popular or esoteric, which notes that the group of differences construction has been used in the business world for about five centuries. And the mathematical basis for DEB is totally unknown in the separate world of accounting.  The mathematical formulation of DEB also allows it to be generalized to vectors of incommensurate physical quantities so the DEB can finally be “priceless.”

This is a scan of the first journal publication of the mathematical formulation and generalization to vectors of double-entry bookkeeping. [Reprint from: Ellerman, David 1986. Double Entry Multidimensional Accounting. Omega. 14 (1 (January 1986)): 13-22.] The complete book-length treatment is in my 1982 book: Ellerman, David 1982. Economics, Accounting, and Property Theory. Lexington MA: D.C. Heath, which can be downloaded