Desimalsdelen

Desimalsdelen, or the fractional part of a real number, is the portion of the number that lies after the decimal point in its decimal representation. It is commonly denoted {x} and defined as {x} = x − ⌊x⌋, where ⌊x⌋ is the floor of x (the greatest integer less than or equal to x). As a result, 0 ≤ {x} < 1 for every real x.

This relation means that any real number x can be written as x = ⌊x⌋ + {x}. The desimalsdelen captures the non-integer part of x, and its value is independent of the integer part. For example, if x = 3.75, then desimalsdelen = 0.75. If x = -2.4, the floor is -3 and desimalsdelen = -2.4 − (-3) = 0.6.

In decimal notation, the desimalsdelen corresponds to the digits that appear after the decimal point in the fractional part, expressed as a number between 0 and 1. The fractional part {x} can be computed by various methods depending on context; commonly, it is x − floor(x). In programming, it may be obtained by x % 1 in languages that define modulo with real numbers, with caveats for negative inputs.

Applications of desimalsdelen include rounding and truncation, numerical analysis, and the study of distributions modulo 1 in number theory. It also provides a way to separate an integer quantity from its non-integer remainder, useful in error analysis and measurement.

In Danish mathematical terminology, desimalsdelen is the fractional part of a number, complementary to the integer part (heltalsdelen).