Editing Fractional part

{{short description|Excess of a non-negative real number beyond its integer part}}
[[File:Parte_fraccionaria.png|thumb|right|Graph of the fractional part of real numbers]]
The '''fractional part''' or '''decimal part'''<ref>{{cite web|url=https://en.oxforddictionaries.com/definition/decimal_part|archive-url=https://web.archive.org/web/20180215143935/https://en.oxforddictionaries.com/definition/decimal_part|url-status=dead|archive-date=February 15, 2018|title=Decimal part|publisher=[[OxfordDictionaries.com|Oxford Dictionaries]]|access-date=2018-02-15}}</ref> of a non‐negative [[real number]] <math>x</math> is the excess beyond that number's [[integer part]]. The latter is defined as the largest integer not greater than {{mvar|x}}, called ''[[floor function|floor]]'' of {{mvar|x}} or <math>\lfloor x\rfloor</math>. Then, the fractional part can be formulated as a [[difference (mathematics)|difference]]:

:<math>\operatorname{frac} (x)=x - \lfloor x \rfloor,\; x > 0</math>.

The fractional part of [[Logarithm|logarithms]],<ref>{{Cite book |last=Ashton |first=Charles Hamilton |url=https://books.google.com/books?id=Yls6AQAAMAAJ |title=Five Place Logarithmic Tables: Together with a Four Place Table of Natural Functions |date=1910 |publisher=C. Scribner's Sons |pages=iv |language=en}}</ref> specifically, is also known as the [[Mantissa (logarithm)|'''mantissa''']]; by contrast with the mantissa, the integral part of a logarithm is called its ''characteristic''.<ref>{{Cite book |last=Magazines |first=Hearst |url=https://books.google.com/books?id=Ad4DAAAAMBAJ&pg=PA291 |title=Popular Mechanics |date=February 1913 |publisher=Hearst Magazines |pages=291 |language=en}}</ref><ref>{{Cite book |last=Gupta |first=Dr Alok |url=https://books.google.com/books?id=IVTvDwAAQBAJ |title=Business Mathematics by Alok Gupta: SBPD Publications |date=2020-07-04 |publisher=SBPD publications |isbn=978-93-86908-16-2 |pages=140 |language=en}}</ref> The word ''mantissa'' was introduced by [[Henry Briggs (mathematician)|Henry Briggs]].<ref>{{Cite book |last=Schwartzman |first=Steven |url=https://books.google.com/books?id=PsH2DwAAQBAJ |title=The Words of Mathematics: An Etymological Dictionary of Mathematical Terms in English |date=1994-12-31 |publisher=American Mathematical Soc. |isbn=978-1-61444-501-2 |pages=131 |language=en}}</ref>

For a [[positive number]] written in a conventional [[positional numeral system]] (such as [[binary numeral system|binary]] or [[decimal]]), its fractional part hence corresponds to the digits appearing after the [[radix point]], such as the [[decimal point]] in English. The result is a real number in the half-open [[Interval (mathematics)|interval]] [0, 1).

==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>
==Relation to continued fractions==
Every real number can be essentially uniquely represented as a [[simple continued fraction]], namely as the sum of its integer part and the [[reciprocal (mathematics)|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==
* [[Circle group]]
* [[Equidistributed sequence]]
* [[One-parameter group]]
* [[Pisot–Vijayaraghavan number]]
* [[Poussin proof]]
* [[Significand]]

==References==
{{Reflist}}

[[Category:Arithmetic]]
[[Category:Unary operations]]