## Topic overview

- Reasons why it can happen in rare cases in Alasco that the calculated gross amount or the totalling of amounts deviates from manually calculated values.

#### Note:

These are **not** calculation or rounding errors, but **unavoidable rounding differences**.

## There are many reasons for rounding differences

## 1. Technical reasons

Even if it often looks like it, computers cannot store infinitely long numbers, as each digit of a number requires a small amount of memory. For this reason, there are various data types with different numbers of decimal places in order to be able to carry out calculations as accurately as possible. The selection of a particular data type for storing values results from a trade-off between accuracy and computing speed.

In Alasco, we store the numbers with **twelve decimal places** in order to guarantee a high level of accuracy for the calculated values.

Rounding differences usually occur with values that are the result of calculations in the system. For this reason, gross amounts, which are calculated from the net amount and the tax rate, and totals in lists are particularly affected.

If two numbers that already have decimal places are multiplied together, this can often result in many more decimal places. However, these are automatically rounded by the system to the 12th decimal place, which can result in minimal but unavoidable rounding differences. The following example illustrates the issue once again:

- 127.955181 becomes 127.9552 (Difference +0.000019)
- 127.454443 becomes 127.4544 (Difference -0.000043)

If such figures are now used for further calculations, previous rounding differences may be propagated, as they have already been rounded. At Alasco, we try to round our intermediate results as rarely as possible during calculations to ensure the greatest possible accuracy of your data.

Another reason for rounding differences is the subsequent commercial representation of the calculated figures in Alasco with two decimal places after symmetrical rounding:

- 127.9552 becomes 127.96 € (Difference +0.0048)
- 127.4544 becomes 127.45 € (Difference -0.0044)

However, these decimal places of calculated values that are no longer displayed are still saved in the background and are included in further calculations in order to provide you with a high level of accuracy for subsequent values.

For this reason, it can happen in rare cases that a sum of €0.01 is displayed in Alasco even for apparent amounts of €0.00, such as in the cash outflow plan shown below.

If we were to simply delete these hidden decimal places from the system and only calculate with the displayed values, unavoidable rounding differences would occur elsewhere in the system (e.g. in the budget distribution).

## 2. Rounding rule applied

**Business rounding**

Commercial rounding is defined by the standard DIN 1333 “Numerical data” and is the rounding rule that is also taught in elementary school. Accordingly, a positive or negative amount is rounded according to the following two rules:

- If the digit in the first decimal place to be omitted is 0, 1, 2, 3 or 4, it is rounded down.
- If the figure in the first decimal place is 5, 6, 7, 8 or 9, it is rounded up.

The following examples are intended to illustrate the two rules with amounts rounded to two decimal places:

- 12.3749… € ≈ 12.37 €
- 12.3750… € ≈ 12.38 €
- −12.3749… € ≈ −12.37 €
- −12.3750… € ≈ −12.38 €

With commercial rounding, it can happen that certain values are positively distorted. This means that you can sometimes save a cent on the payout with negative amounts (deductions).

**Symmetrical rounding**

Symmetrical rounding differs from commercial rounding only in the decision as to how a number whose first decimal place is exactly halfway between 4 (rounding down) and 5 (rounding up) is rounded.

The following rules define symmetrical rounding:

- If the digit in the first omitted decimal place is a 0, 1, 2, 3 or 4, it is rounded down in the same way as for commercial rounding.
- If the digit in the first omitted decimal place is a 5, 6, 7, 8 or a 9 (followed by further digits that are not all zero), the number is rounded up.
- If the digit in the first decimal place to be omitted is only a 5 (or a 5 followed only by zeros), the number is rounded in such a way that the last digit to be retained is even (“
**even number rule**”).

The following examples serve to illustrate the three rules, whereby the amounts are rounded to two decimal places:

- 2.12499 ≈ 2.12 (according to rule 1)

2.12501 ≈ 2.13 (according to rule 2)

2.12500 ≈ 2.12 (rounded to an even number according to rule 3)

2.13500 ≈ 2.14 (rounded to an even number according to rule 3)

Commercial rounding produces small systematic errors, as rounding up by 0.5 occurs, but rounding down by 0.5 never does, which can lead to a slight distortion in statistics. Symmetrical rounding always rounds up or down from the exact middle between two digits to the next even digit.

This type of rounding is also used in **Alasco**, as symmetrical rounding helps to **minimize the deviation from the unrounded sum** when adding many rounded values.

## 3. Adding columns or rows (item level or grand total)

When calculating the gross amount of an invoice, you basically have two different options. Either you calculate the tax according to the specified tax rate line by line for each item of the invoice and then add them up, or you first calculate the net total and use this to calculate the tax and the total gross amount.

The following example should illustrate the differences between the two procedures and show you that there may be unavoidable differences.

An invoice contains three invoice items, each with a net amount of €0.99. In this example, the gross amount is calculated once line by line and once based on the total net amount. As you can see, the two different procedures alone result in a difference of €0.01.

Quantity | Net amount | Value added tax | Gross amount | Rounded off |

1 | 0.99 | 19% | 1.1781 | 1.18 |

1 | 0.99 | 19% | 1.1781 | 1.18 |

1 | 0.99 | 19% | 1.1781 | 1.18 |

Total | 3.54 |

Quantity | Net amount | Value added tax | Gross amount | Rounded off |

1 | 0.99 | |||

1 | 0.99 | |||

1 | 0.99 | |||

Total | 2.97 | 19% | 3.5343 | 3.53 |

## What rules and decisions are there in the construction industry on this topic?

In the area of quantity determination, there are several sets of rules that contain regulations regarding the decimal places to be specified and the rounding rules to be applied.

- The current procurement and contract handbook for federal construction measures (VHB 2017; as of July 2019) does not contain any information on decimal places for quantities. Decimal places are only specified for prices. A
**maximum of three decimal places**is possible here.

- The collection of regulations for electronic construction invoicing (REB-VB 23.003) “General quantity calculation” regulates quantity calculations and data exchange. Section 2.4 Rounding also requires quantities to be specified with three decimal places.

- The standard DIN 1333 “Numerical data” defines the scope and purpose of the standard as specifying how numbers should be written in everyday life in business, technology and science. The last sentence of this section states:

“This standard does not apply to the internal numerical representation of computers.”

DIN 1333 also admits in chapter 4.2 “Specifying the rounding point”:

“The rounding point may ... result from technical reasons (last digit of the result storage in the memory of a data processing system).”

#### Tip:

**How do you avoid discussions about cent deviations and rounding differences?**

Agree with your clients and contractors in advance on sensible, required or permissible accuracies in the measurement and rounding and point out rounding differences due to technical reasons. Don't just think about individual values, but also about the project totals. Experience has shown that rounding differences tend to cancel each other out, for example due to symmetrical rounding, so that there is little or no difference in the total.