Fractional part

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 ${\displaystyle \operatorname {frac} ($x)=x-\lfloor x\rfloor }   (Graham, Knuth & Patashnik 1992),^[2] or as the part of the number to the right of the radix point ${\displaystyle \operatorname {frac} ($x)=|x|-\lfloor |x|\rfloor }   (Daintith 2004),^[3] or by the odd function:^[4]

${\displaystyle \operatorname {frac} ($x)={\begin{cases}x-\lfloor x\rfloor &x\geq 0\\x-\lceil x\rceil &x<0\end{cases}}}  

with ${\displaystyle \lceil x\rceil }$  as the smallest integer not less than x, also called the ceilingofx. By consequence, we may get, for example, three different values for the fractional part of just one x: 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

${\displaystyle \operatorname {frac} ($x)=x-\lfloor |x|\rfloor \cdot \operatorname {sgn}(x)}  .

The ${\displaystyle x-\lfloor x\rfloor }$  and the "odd function" definitions permit for unique decomposition of any real number x to the sum of its integer and fractional parts, where "integer part" refers to ${\displaystyle \lfloor x\rfloor }$ or${\displaystyle \lfloor |x|\rfloor \cdot \operatorname {sgn}($x)}   respectively. These two definitions of fractional-part function also provide idempotence.

The fractional part defined via difference from ⌊ ⌋ is usually denoted by curly braces:

${\displaystyle \{x\}:=x-\lfloor x\rfloor .}$ 

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.

• Circle group
• Equidistributed sequence
• One-parameter group
• Pisot–Vijayaraghavan number
• Poussin proof
• Significand