Editing Fractional part (section)

==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 <math>\operatorname{frac} (x)=x-\lfloor x \rfloor</math> {{harv|Graham|Knuth|Patashnik|1992}},<ref>{{citation | title=Concrete mathematics: a foundation for computer science | first1=Ronald L. | last1=Graham | author-link1=Ronald Graham | first2=Donald E. | last2=Knuth | author-link2=Donald Knuth | first3=Oren | last3=Patashnik | author-link3=Oren Patashnik | publisher=Addison-Wesley | isbn=0-201-14236-8 | year=1992 | page=70 }}</ref> or as the part of the number to the right of the radix point <math>\operatorname{frac} (x)=|x|-\lfloor |x| \rfloor</math> {{harv|Daintith|2004}},<ref>{{citation|title=A Dictionary of Computing|first=John|last=Daintith|date=2004|publisher=Oxford University Press}}</ref> or by the [[odd function]]:<ref>[http://mathworld.wolfram.com/FractionalPart.html Weisstein, Eric W. "Fractional Part." From MathWorld--A Wolfram Web Resource]</ref>

:<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 {{mvar|x}}, also called the [[ceiling function|ceiling]] of {{mvar|x}}. By consequence, we may get, for example, three different values for the fractional part of just one {{mvar|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
:<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 {{mvar|x}} to the [[addition|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 [[floor function|⌊ ⌋]] is usually denoted by [[curly brace]]s:
:<math>\{ x \} := x-\lfloor x \rfloor.</math>