The fractional part or decimal part[1] of a non‐negative real number <math>x</math> is the excess beyond that number's integer part. The latter is defined as the largest integer not greater than Шаблон:Mvar, called floor of Шаблон:Mvar or <math>\lfloor x\rfloor</math>. Then, the fractional part can be formulated as a difference:
<math>\operatorname{frac} (x)=x - \lfloor x \rfloor,\; x > 0</math>.
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 <math>\operatorname{frac} (x)=x-\lfloor x \rfloor</math> Шаблон:Harv,[2] or as the part of the number to the right of the radix point <math>\operatorname{frac} (x)=|x|-\lfloor |x| \rfloor</math> Шаблон:Harv,[3] or by the odd function:[4]
<math>\operatorname{frac} (x)=\begin{cases}
x - \lfloor x \rfloor & x \ge 0 \\
x - \lceil x \rceil & x < 0
\end{cases}</math>
with <math> \lceil x \rceil</math> as the smallest integer not less than Шаблон:Mvar, also called the ceiling of Шаблон:Mvar. By consequence, we may get, for example, three different values for the fractional part of just one Шаблон:Mvar: 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
<math>\operatorname{frac} (x)= x - \lfloor |x| \rfloor \cdot \sgn(x)</math>.
The <math>x - \lfloor x \rfloor</math> and the "odd function" definitions permit for unique decomposition of any real number Шаблон:Mvar to the sum of its integer and fractional parts, where "integer part" refers to <math>\lfloor x \rfloor</math> or <math>\lfloor |x| \rfloor \cdot \sgn(x)</math> respectively. These two definitions of fractional-part function also provide idempotence.
The fractional part defined via difference from ⌊ ⌋ is usually denoted by curly braces:
<math>\{ x \} := x-\lfloor x \rfloor.</math>
Relation to continued fractions
Every real number can be essentially uniquely represented as a continued fraction, namely as the sum of its integer part and the reciprocal of its fractional part which is written as the sum of its integer part and the reciprocal of its fractional part, and so on.
