Fractional part

The fractional part of a non‐negative real number ${\displaystyle x}$ is the excess beyond that number's integer part. If the latter is defined as the largest integer not greater than x, called floor of x or ${\displaystyle \lfloor x\rfloor }$, its fractional part can be written as:

For a positive number written in a conventional positional numeral system (such as binary or decimal), its fractional part hence equals the digits appearing after the radix point.

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), or as the part of the number to the right of the radix point, ${\displaystyle \operatorname {frac} (x)=|x|-\lfloor |x|\rfloor }$ (Daintith 2004), finally, by the odd function

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