abrunden

Abrunden is a rounding operation used in mathematics, science, and finance to reduce a number to a specified precision by replacing it with the greatest value that is not greater than the original number. In German usage, it is the counterpart to aufrunden (round up) and to the general term runden (to round).

To abrunden to n decimal places, multiply the number by 10^n, apply the floor function (the greatest integer less than or equal to the result), and then divide by 10^n. For rounding to integers, abrunden uses the floor function directly. The main effect is that the result is never larger than the original value, and for negative numbers the result can be strictly smaller (more negative).

Examples: 3.879 abrunden to two decimal places is 3.87. -7.856 abrunden to two decimal places is -7.86. Another example: 7.1 abrunden to one decimal place remains 7.1, while 7.14 abrunden to one decimal place becomes 7.1.

Applications include measurement and data processing in science and engineering, as well as financial calculations where conservative estimates are preferred. In computing, the operation is commonly implemented via the floor function, sometimes implemented as a combination of multiplication, flooring, and division to achieve the desired decimal precision. In everyday language, abrunden conveys the idea of “rounding down” to a chosen level of precision.