Fractional part
The fractional part or decimal part of a non‐negative real number is the excess beyond that number's integer part. The latter is defined as the largest integer not greater than, called floor of or. Then, the fractional part can be formulated as a difference:
The fractional part of logarithms, specifically, is also known as the mantissa; by contrast with the mantissa, the integral part of a logarithm is called its characteristic. The word mantissa was introduced by Henry Briggs.
For a positive number written in a conventional positional numeral system.
For negative numbers
However, in case of negative numbers, there are various conflicting ways to extend the fractional part function to them: It is either defined in the same way as for positive numbers, i.e., by , or as the part of the number to the right of the radix point , or by the odd function:with as the smallest integer not less than, also called the ceiling of. By consequence, we may get, for example, three different values for the fractional part of just one : let it be −1.3, its fractional part will be 0.7 according to the first definition, 0.3 according to the second definition, and −0.3 according to the third definition, whose result can also be obtained in a straightforward way by
The and the "odd function" definitions permit for unique decomposition of any real number to the sum of its integer and fractional parts, where "integer part" refers to or respectively. These two definitions of fractional-part function also provide idempotence.
The fractional part defined via difference from ⌊ ⌋ is usually denoted by curly braces: