# Double-Entry Bookkeeping: Mathematical Formulation and Generalization

This is Chapter 6 from my book: Ellerman, David. 1995. Intellectual Trespassing as a Way of Life: Essays in Philosophy, Economics, and Mathematics. Lanham MD: Rowman & Littlefield.

The essay on double-entry bookkeeping (DEB) is intellectually interesting for several reasons in spite of the well-known soporific aspects of bookkeeping.  Several of the essays in the volume explicitly employ the analogy between additive and multiplicative operations (i.e., the common group-theoretic properties of additive groups of numbers and multiplicative groups of nonzero numbers).  For instance, given the system of multiplying whole numbers or integers, there is no operation inverse to multiplication (i.e., there is no division).  But there is a standard method of enlarging the system to allow division.  Consider pairs of whole numbers a/b (with b not 0) and define multiplication in the obvious way: (a/b)(c/d) = (ac)/(bd).  These ordered pairs of integers are the “fractions” and they allow the operation of division (“multiply by the reciprocal”).

Now substitute addition for multiplication.  We start with the additive system of positive numbers along with zero (i.e., the non-negative numbers) where is no inverse operation to addition (i.e., there is no subtraction).  To enlarge the domain of non-negative numbers to include subtraction, consider ordered pairs [a // b] and define addition in the analogous way: [a // b] + [c // d] = [a+c // b+d].  This enlarged system of additive operations on ordered pairs of non-negative numbers allows subtraction (“add on the reversed pair”).  The origin of the intellectual trespassing into DEB was the observation that these ordered pair were simply the T-accounts of DEB.

Aside from illustrating the interplay of additive-multiplicative themes, the essay illustrates one of the most astonishing examples of intellectual insulation between disciplines, in this case, between accounting and mathematics.  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 is 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.

This almost complete lack of cross-fertilization between mathematics and accounting is a topic of some interest for intellectual history and the sociology of knowledge.  The story is rather simple from the mathematics side.  Double-entry bookkeeping is apparently too simple and mundane to inspire any modern mathematician to learn it and then mathematically explicate it.  The real question lies on the accounting side.  How, over the last century, could professional accountants and accounting professors have failed to find the mathematical basis for DEB even though it was part of undergraduate algebra?  The mathematical treatment of double entry bookkeeping (and generalization to multidimensional vectors [Ellerman 1982, 1985]) will take years, if not decades, to become known and understood in the field of accounting.