# Mathematical basis

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For the user of Banana's Double-entry accounting, it may seem strange that the revenue/income (sales) and the liabilities (start-up capital) appear in negative, while normally one expects that the sales appear in positive and the expenses in negative.

The double-entry accounting is based on the debit and credit principle that usually appear in two seperate columns. If the amounts in debit and in credit appear in one column, the mathematics of the Double-entry accounting anticipates the use of the minus sign (-), so that the amounts in debit can be easily distinguished from the amounts in credit.

Banana goes completely with the mathematical logic of the Double-entry accounting and thus indicates the amounts with the minus sign (and in red print to let the user distinguish even better the amounts in debit and in credit). In the Double-entry accounting the amounts in negative (credit) don't have the same meaning as in the accounting that is based on income and expenses.

The indication of the credit amounts in negative is a much more efficient system than other ones that add codes to amounts, such CR, DEB, CR or brackets.

This system is based on simple mathematical equations which assure that the accounting is correct. It is not by accident that the Double-entry accounting method has in fact been encoded by Luca Pacioli in 1494, one of the most illustrous mathematicians of the italien Renaissance (he taught mathematics to Leonardo da Vinci). In his "Summa de arithmetica, geometria, proportioni et proportionalita", Luca Pacioli didn't just deal with the Double-entry accounting, but also with other mathematical questions.

In the Renaissance there were no electronic calculators, and so they used to indicate the amounts in debit and in credit in seperate columns. The result was that it became easier to totalize the amounts. Today, with the computers, no calculations need to be made and the principal issue is to use a user-friendly system which is at the same time easy and correct.

For those interested in accounting mathematics and the equations on which Banana accounting is based, the following explanation can be interesting.

### Debit/Credit

The Double-entry accounting leans on the principle that each movement is noted in debit as well as in credit and the amounts have to match.

Debit = Credit

The balance (the total difference between debit and credit) has to always be equal to zero

Debit - Credit = 0

In order to easily distinguish (very useful while working on the screen), the amounts in credit are indicated in red, preceded by the minus sign

Debit + (-Credit) = 0

 Two transaction examples Debit Credit Debit + (- Credit) Cash income 200 200 for product sales 200 -200 Several cash payments 170 -170 for purchase of merchandise 100 100 for office supplies 50 50 small expenses 20 20 Total 370 370 0 Balance (Debit - Credit) 0 0

### Balance Sheet and Profit/Loss Statement

The mathematics of the Double-entry accounting anticipates the use of two "accounts".

• Balance Sheet Account (Balance Sheet) keeps note of the financial state at a certain moment.
• Debit represents the Assets
• Credit represents the Liabilities (here will always means including the Equity)
• Profit/Loss Account (Profit and Loss) keeps note of the progress.
• Debit represents the Expenses
• Credit represents the Revenue

The transactions are entered in debit and in credit in the appropriate accounts of the Balance Sheet and the Profit/Loss statement.

 Balance Sheet Profit/Loss Statement Registrazioni Assets Liabilities Expenses Revenue Cash income 200 for product sales 200 Several cash payments 170 for purchase of merchandise 100 for office supplies 50 small expenses 20 Total 200 170 170 200 Balance (difference debit - credit) 30 30

 Equations with two columns Debit = Credit 370 = 370 Assets + Expenses = Liabilities + Revenue 200 + 170 = 170 + 200 Assets - Liabilities = Revenue - Expenses 200 - 170 = 200 - 170 Balance = Balance 30 = 30

The same transactions can be represented in a single column (credit in negative)

The result is, of course, the same.

 Transactions Balance  Sheet Profit/Loss  Statement Cash income 200 for product sales -200 Several cash payments -170 for purchase of merchandise 100 for office supplies 50 small expenses 20 Balence (debit - credit) 30 -30

 Equations single column (credit in negative) Debit - Credit = 0 270 - 270 = 0 Assets + Expenses - Liabilities - Revenue = 0 200 + 170 - 170 - 200 = 0 Assets - Liabilities - Revenue + Expenses 200 - 170 - 200 + 170 = 0 Balance - Balance = 0 30 - 30 = 0

## Equations for the business results

The business result is the balance (Debit - Credit) of the Balance Sheet and the Profit/Loss Statement.

 Equations double column Assets - Liabilities = Revenue - Expenses 200 - 170 = 200 - 170 Balance of the Balance Sheet = Balance of Profit/Loss Statement 30 Balance of the Balance Sheet = Assets - Liabilities 30

 Equations for the determination of the result (in two columns) Profit Assets > Liabilities Expenses < Revenue Balance Assets = Liabilities Expenses = Revenue Loss Assets < Liabilities Expenses > Revenue

One has a profit when the assets are higher than the liabilities and the revenue more than the expenses.

One has a loss when the liabilities are higher than the assets and the revenue is lower than the expenses.

 Equations single column (credit in negative) Assets - Liabilities - (Revenue - Expenses) = 0 200 - 170 - (-200 - 170) = 0 Balance of the Balance Sheet - Balance of Profit/Loss Statement = 0 30 - 30 = 0 Balance of the Balance Sheet 30 Balance of Profit/Loss Statement -30

 Equations for the determination of the result in a single column Profit Assets - Liabilities > 0 Expenses - Revenue < 0 Balance Assets - Liabilities = 0 Expenses - Revenue = 0 Loss Assets - Liabilities <0 Expenses - Revenue > 0

One has a profit, when the balance of the Balance Sheet is positive and and the balance of the Profit/Loss Statement negative.

One has a loss, when the balance of the Balance Sheet is negative and and the balance of the Profit/Loss Statement positive.

### Use of noting in a single column

Noting the data in double columns is ideal for transactions in paper books. The amounts in debit and in credit are inscribed in separate columns. The totals are being calculated at the end of the page and the totals and calculation of the results are being done at the end of the period.

Computerized systems offer the possibility to keep the balances of the accounts constantly updated. For the calculations, the softwares therefore normally use noting in a single column, with the amounts in debit in positive and the amounts in credit in negative. In order to use noting in double columns, the softwares would have to register the credit amounts in the Balance sheet in negative, and, in the Profit/loss Statement, register the credit amounts in positive and the debit amounts in negative.

For the programmer's point of view, this solution is more complicated, and thus the majority of the softwares use internally the mathematics of the single column and noting in double columns only for the presentation.

Banana Accounting instead systematically uses the minus sign to indicate the credit amounts. Initially, the user has to get used to this system, but it has the advantage of being linear and allows for a better understanding of the mathematics on which the Double-entry accounting is based, especially when more complicated operations are being required like write-off's, profit at the end of the year or VAT due/recoverable.

For the presentation of the Balance Sheet and the Profit/loss Statement, the logic of the double columns is however more adapted.

Thus the print-outs of the Balance Sheet and the Profit/Loss Statement that are used for the presentation of the results are edited with Assets, Liabilities, Expenses and Revenue always in positive. When the Profit/Loss Statement is presented in a scalar format (Revenue minus Expenses), the Revenue is indicated in positive and the Expenses in negative.