Mathematics and the Accounting Equation

Today's accounting practice is primitive in its use of modern mathematics. It has the potential to greatly expand its horizons by easily applying the following mathematical tools to its double-entry system:

    Linear Algebra and its application of vectors and matrices;

    Graph Theory to related one account to another;

    Statistics to visualize change and forecast trends;

    Multi-dimensional analysis to provide deep research;

    Signed numbers to allow for a true arithmetic.

Modern accounting practices have not really changed in 500 years, in which time mathematics has expanded exponentially. Accounting is a profession that requires a lot of skill but, unfortunately, very little imagination. All of the above mathematical tools could be utilized without changing any of the traditional practices of the profession. It would supplement the practice without challenging it.

The Income Statement as a Matrix

 

In the matrix version of the Balance Sheet, shown below and in the previous post, all of the credits cancel out all of the debits, bringing the balance to zero, as shown by the matrix elements in the lower right hand corner. This, of course indicates a balance of the double-entry bookkeeping.

 

We can make another matrix based on the same data, but only showing a subset of the accounts (the accounts in the matrix are shown as A, B, C, D, and E to keep the demonstration at an abstraction level). The accounts that we would include are the Revenue accounts and the Expense Accounts. And the matrix would represent a linear algebra view of an Income Statement.” 

 

Accounts	A Credits	B Credits	C Credits	D Credits	E Credits	Debit Totals
A Debits	0	to A from B	to A from C	to A from D	to A from E	A Debit Total
B Debits	to B from A	0	to B from C	to B from D	to B from E	B Debit Total
C Debits	to C from A	to C from B	0	to C from D	to C from E	C Debit Total
D Debits	to D from A	to D from B	to D from C	0	to D from E	D Debit Total
E Debits	to E from A	to E from B	to E from C	to E from D	0	E Debit Total
Credit Totals	A Credit Total	B Credit Total	C Credit Total	D Credit Total	E Credit Total	0

 

Since only a subset of the accounts are shown, the summary that we find in a Balance Sheet would very likely no longer show a balance of zero. The matrix element in the lower right corner would now show a debit balance, indicating a profitable period, or a credit balance showing net losses. The debits to the Revenue accounts would offset and exceed the credits shown in the expense accounts where a profitable Income Statement is given. Otherwise, the credits would be in excess.

 

Again this matrix Income Statement is much easier to read and contains an order of magnitude more information. The future of financial analysis belongs to matrix arithmetic and linear algebra.

Linear Algebra as a Balance Sheet

 

The matrix below is the accounting balance sheet of the future. It is a two-dimensional table like those found in the linear algebra branch of mathematics.. Each element of the matrix is a vector from an account that is credited to an account that is debited (the use of capital letters is preferred for this abstract demonstration over the more concrete names such as “Cash” or “Accumulated Depreciation”). Each vector is consistent with the use of a credited account as the origin of the transferred resource and the a debited account as its destination. This modern balance sheet contains all the information found in the traditional balance sheet, but it also contains an order of magnitude more information. This matrix of vectors is explained further in my book, “The Tao of Financial Information.” 

 

Accounts	A Credits	B Credits	C Credits	D Credits	E Credits	Debit Totals
A Debits	0	to A from B	to A from C	to A from D	to A from E	A Debit Total
B Debits	to B from A	0	to B from C	to B from D	to B from E	B Debit Total
C Debits	to C from A	to C from B	0	to C from D	to C from E	C Debit Total
D Debits	to D from A	to D from B	to D from C	0	to D from E	D Debit Total
E Debits	to E from A	to E from B	to E from C	to E from D	0	E Debit Total
Credit Totals	A Credit Total	B Credit Total	C Credit Total	D Credit Total	E Credit Total	0

 

The current conventional balance sheet is very primitive and is based on the traditions and mathematics of the historical medieval period, a time when western civilization was first using Arabic numbers and did not believe that negative numbers existed. The accounting analysis of the future can be found in the practice of linear algebra.

Relational Matrix as the Ultimate Financial Statement

The accounting profession has lost sight of the true nature of double-entry bookkeeping and this gravely limits the analytical power of today's financial record keeping. Double-entry bookkeeping is far more than providing a duplicated check on our recording accuracy. Each of the two entries provides unique and non-duplicated information about the transaction. The credit entry tells us the origin of the resources being transferred and the debit entry tells us the destination of those resources.

Once we realize the importance of each entry we can begin to harness the tremendous analytical power that it provides. The real financial statement for providing business behavior is not the income statement or the balance sheet; the real tool for analysis is the relational matrix.

The relational matrix has a column for all the accounts and a row for all of those accounts. The columns can hold the record of the debits and the rows can hold the record of the credits. Each cell of the table represents a transfer from the row that is credited to the column that is debited. A given cell tells us the sum that has gone from the corresponding row's account to the receiving column's account.

The sums of the rows and columns contains the information in the balance sheet. The individual cells represent the flow of resources from the one account to another (for graphical examples see my book, The Tao of Financial Information).

Accounting as Data Warehousing

The accounting journal is nothing more than a simple data warehouse. The direction of the transfer or resources in a transaction (from a credited account to a debited account) is only one dimension of a potentially multi-dimensional database, allowing the analyst to to view financial behavior through many different critical factors.

Another connection that needs to be made between the accounting world and the data processing industry is the recognition that the accounting journal is a powerful and complete form of event processing. Each transaction recorded in the journal is an event that is part of the history of a business organization.

The ability to do financial analysis is stunted by the inability of the accounting and IT professions to cross-fertilize their respective skills.

Vector Power

To fully appreciate the elegance of double-entry accounting, one needs to realize that each set of double entries defines an economic vector, tracking the movement of financial resources from one place to another. The credit entry identifies the origin of the transfer and the corresponding debit entry identifies the destination of the resources.

The insight that the credit/debit entry is a vector can greatly empower (and simplify) financial analysis and can allow us to enhance the record-keeping potential of the accounting journal. Accounting is the tracking of resource allocation through time. The economic vectors recorded by double-entry accounting can track the allocation of resources through any number of dimensions (for example; geography, business units, corporate organization, and consumer demographics).

For further explanation of the vector approach, obtain a free copy of the book, Banking on the Past, by adding a comment here or emailing this author at rob.meldahl@gmail.com.

Vector Accounting

I want to thank Irus, who added a comment to an earlier post, for the use of the term "vector" to describe the approach that I am using to clarify the process of accounting. A vector is a quantity that has both a magnitude and a direction. This is exactly what a debit or credit entry is. The magnitude is the monetary amount of the entry and the direction is either "in" (debit) or "out" (credit). Accountants have been recording vectors into data warehouses for five centuries; they just did not know it and have failed to use the resulting powers.

For more on vector accounting, see the blog at http://taofinancial.blogspot.com/. Also, get a free electronic copy of my two books, The Tao of Financial Information and Banking the Past, by sending me an email at rob.meldah@gmail.com.