On Double-Entry Bookkeeping: The Mathematical Treatment

Abstract

Double-entry bookkeeping (DEB) implicitly uses a specific mathematical construction, the group of differences using pairs of unsigned numbers (‘T-accounts’). That construction was only formulated abstractly in mathematics in the nineteenth century, even though DEB had been used in the business world for over five centuries. Yet the connection between DEB and the group of differences (here called the ‘Pacioli group’) is still largely unknown both in mathematics and accounting. The precise mathematical treatment of DEB allows clarity on certain conceptual questions and it immediately yields the generalization of the double-entry method to multi-dimensional vectors typically representing the different types of property involved in an enterprise or household.

Key Words:

• Double-entry method
• group of differences
• Pacioli group
• T-accounts
• multi-dimensional double-entry bookkeeping

Notes

1 The double-entry system seems to have been first described, but not published, by the Dubrovnik (then Ragusa) merchant-economist Benedetto Cotrugli (or Benedikt Kotruljevic in the Croatian version) in 1458 (Yamey, 1994), and Pacioli was familiar with Cotrugli's manuscript.

2 See Jacobson (1951, p. 10), Littlewood (1960, p. 18), Dubisch (1965, p. 17), Mac Lane and Birkhoff (1967, p. 45), and so forth.

3 The Pacioli group and its use to carry out the operations of DEB is a rational reconstruction, not an effort at historical exegesis of Pacioli or Cotrugli (where the author has no expertise). For instance, it may well be that Pacioli as a mathematician implicitly assumed an underlying equation (so the double-entry principle would always hold), but did not mention it explicitly in his treatise intended for practical business purposes. The explicit recognition and treatment of the balance sheet equation came later (Sangster, 2010).

4 The double-slash notation was suggested by Pacioli. ‘At the beginning of each entry, we always provide “per”, because, first, the debtor must be given, and immediately after the creditor, the one separated from the other by two little slanting parallels (virgolette), thus, //, … .’ (Pacioli, 1914, p. 43).

5 There is, in fact, a whole system of multiplicative DEB (Ellerman, 1982, pp. 58–66) to update multiplicative equations of natural numbers like 216 × 140 = 12 × 63 × 40.

6 We could work with real or rational numbers but for simplicity will stick to the whole numbers.

7 A plus added to a plus is positive and a minus added to a minus is negative but a plus times a plus is positive while a minus times a minus is also positive.

8 There is a whole literature about DEB and negative numbers. While Pacioli as a mathematician knew about negative numbers (Scorgie, 1989), it is uncontroversial to say that the use of DEB by businesses at that time was facilitated by not requiring familiarity with the relatively new notion of a negative number (Heeffer, 2011).

9 In going from x and y in N to x−y or y−x in Z, we are, as usual, identifying the unsigned numbers in N with the non-negative signed numbers in Z.

10 In the context of vectors, single numbers are called ‘scalars’.

11 For simplicity and brevity, we are leaving out the temporary or flow accounts of the income statement and dealing directly with equity, rather than accumulating the changes to equity in the income statement and then closing it into equity at the end of the accounting period. To use income statement accounts, one would simply add a credit-balance revenue account and a debit-balance expenses account to the ledger (with zero initial balances). The two transactions affecting equity would then be posted in the usual way to the expense and revenue accounts. Finally, another transaction or two at the end of the period would ‘close the income statement into the balance sheet’ by making the changes to equity and resetting the zero balance in the revenue and expense accounts.

12 Zero being the additive identity means for any x, 0 + x = x + 0 = x. Hence if 0 + x = 0, then x = 0.

13 Technically a ‘matrix’ is a way of presenting a linear transformation from one space of vectors to another, so the key operation is the multiplication of a matrix times a vector to yield another vector. The so-called ‘transaction matrices’ used in the table presentation of double-entry bookkeeping do not multiply, so it may be impressive but ultimately misleading to call the tables ‘matrices’. There are genuine uses of matrices in business mathematics; see Kemeny et al. (1962) or Shank (1972) for applications to proportional overhead allocation and Markov chains.

14 In spite of being the best treatment of mathematical accounting in the recent literature, there are some infelicities in the treatment of these questions by Nehmer and colleagues (Cruz Rambaud et al., 2010) in addition to casually going back and forth across the debit isomorphism (which muddles the SSS and DSU systems). For instance, they use single-sided accounts but then define ‘T-diagrams’ (Cruz Rambaud et al., 2010, p. 53), which are then used only to represent the transactions and not the accounts themselves. They do consider (Cruz Rambaud et al., 2010, pp. 54–55) the Pacioli group from the author's earlier work (Ellerman, 1982, 1985, 1986). But even that definition is somewhat infelicitous since they allow all the negative numbers of the integers Z to appear in the T-accounts of the Pacioli group, which negates the whole point of the group of differences construction. That is, instead of constructing P(N) ≅ Z from N, they construct P(Z) ≅ Z from Z. More importantly, they consider only the Pacioli group and ignore the whole double-entry method of T-accounts with the balance sheet, income statement, ledger, journal, posting the journal to the ledger, and so forth, not to mention the multi-dimensional generalization. Hence it seems the mathematical treatment of DEB using the group of differences construction is still essentially unknown in the accounting literature.

15 The ‘affects-two-accounts’ mischaracterization of double-entry bookkeeping is far too embedded in the DNA of the accounting profession to realistically expect it to be changed now. At best, on occasions where there is clarity in distinguishing the DEB (or DSU) system from the SSS system, then retronymic phrases will probably be used so that the DEB system will be called the ‘double-entry system with T-accounts’ while the SSS system would be called, by contrast, the ‘double-entry system without T-accounts’.

16 By ‘multi-dimensional DEB’ I do not mean using scalar DEB coupled with ‘multi-dimensional’ database records; instead I mean a multi-dimensional version of the full DEB systems with multi-dimensional (vector) equations and with transactions to update the equations. Hence all the usual machinery of balance sheets, T-accounts, ledgers, journals, flow-statements, and so forth will generalize. Moreover, the property vector accounting system, in a certain sense, underlies the value scalar accounting system.

17 The numbers will later be interpreted as types of property (goods and services) but for now the focus is just on the mathematics.

18 Expanding the transaction amount in scalar accounting in a link to database information such as the account affected, the date, or the counterparty, and then calling the result a ‘vector’, does not create ‘vector accounting’ (Mattessich, 1964, p. 94) in the sense used here. Vectors, unlike database records, can be meaningfully added together.

19 An ‘abuse of language’ is involved in calling the non-negative vector x^– the ‘negative part’ of x.

20 For simplicity, we are valuing the output widgets at their selling price of 100, whereas the usual accounting practice is to conservatively value them at cost until actually sold. That usual practice could be accommodated in property accounting at the cost of treating ‘Produced Widgets’ as a different form of property from ‘Sold Widgets’ (so each type of property has a single unit value), but such complications are not relevant to our mathematical purposes.