The **fractional part** or **decimal part**^{[1]} of a non‐negative real number 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 , 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 corresponds to the digits appearing after the radix point.

## Contents

## 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 (Graham, Knuth & Patashnik 1992),^{[2]} or as the part of the number to the right of the radix point, (Daintith 2004),^{[3]} finally, by the odd function ^{[4]}

with as the smallest integer not less than *x*, also called the ceiling of *x*. 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

- .

## Unique decomposition into integer and fractional parts

Under the first definition all real numbers can be written in the form , where is the number to the left of the radix point, 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.

## 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.

## See also

- Floor and ceiling functions, the main article on fractional parts
- Equidistributed sequence
- One-parameter group
- Pisot–Vijayaraghavan number
- Significand
- Quotient space (linear algebra)

## References

**^**"Decimal part". OxfordDictionaries.com. Retrieved 2018-02-15.**^**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.**^**Weisstein, Eric W. "Fractional Part." From MathWorld--A Wolfram Web Resource