The fractional part of a non‐negative real number is the excess beyond that number integer part: .
There are various conflicting ways to extend the fractional part function to negative numbers. It is either defined as (Graham, Knuth & Patashnik 1992), as the part of the number to the right of the radix point, (Daintith 2004), or as the odd function:
For example, the number −1.3 has a fractional part of 0.7 according to the first definition, 0.3 according to the second definition and −0.3 according to the third definition.
For a positive number written in a conventional positional numeral system (such as binary or decimal), the fractional part equals the digits appearing after the radix point. Equivalently, it equals the original number with the digits before the radix point substituted with 0.
Under the first definition all real numbers can be written in the form where is the and the remaining fractional part is a nonnegative real number less than one. If is a positive rational number, then the fractional part of can be expressed in the form , where and are integers and . For example, if x = 1.05, then the fractional part of x is 0.05 and can be expressed as 5 / 100 = 1 / 20.
- Floor and ceiling functions, the main article on fractional parts
- Equidistributed sequence
- One-parameter group
- Pisot–Vijayaraghavan number
- Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1992), Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, p. 70, ISBN 0-201-14236-8
- John Daintith (2004). A Dictionary of Computing. Oxford University Press.
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