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Difference between revisions of "Floor function"

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''entier function, greatest integer function, integral part function''
 
''entier function, greatest integer function, integral part function''
  
The function of a real variable that assigns to a real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130150/f1301501.png" /> the largest integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130150/f1301502.png" />. The modern notation is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130150/f1301503.png" />; the classical notation is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130150/f1301504.png" />. In computer science and computer languages it is often denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130150/f1301505.png" />.
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The function of a real variable that assigns to a real number $x$ the largest integer $\leq x$. The modern notation is $\lfloor x\rfloor$; the classical notation is $[x]$. In computer science and computer languages it is often denoted by $\operatorname{int}(x)$.
  
The related ceiling function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130150/f1301506.png" /> gives the smallest integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130150/f1301507.png" />. The fractional part function is defined as
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The related ceiling function $\lceil x\rceil$ gives the smallest integer $\geq x$. The fractional part function is defined as
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130150/f1301508.png" /></td> </tr></table>
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$$\operatorname{frac}(x)=\begin{cases}x-\lfloor x\rfloor&\text{for }x\geq0,\\x-\lfloor x\rfloor-1&\text{for }x<0.\end{cases}$$
  
 
The nearest integer function is
 
The nearest integer function is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130150/f1301509.png" /></td> </tr></table>
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$$\operatorname{nint}(x)=\operatorname{round}(x)=x-\operatorname{frac}(x).$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.L. Graham,  D.E. Knuth,  O. Patashnik,  "Concrete mathematics: a foundation for computer science" , Addison-Wesley  (1990)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Wolfram,  "Mathematica: Version 3" , Addison-Wesley  (1996)  pp. 718–719</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.L. Graham,  D.E. Knuth,  O. Patashnik,  "Concrete mathematics: a foundation for computer science" , Addison-Wesley  (1990)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Wolfram,  "Mathematica: Version 3" , Addison-Wesley  (1996)  pp. 718–719</TD></TR></table>

Latest revision as of 12:37, 27 August 2014

entier function, greatest integer function, integral part function

The function of a real variable that assigns to a real number $x$ the largest integer $\leq x$. The modern notation is $\lfloor x\rfloor$; the classical notation is $[x]$. In computer science and computer languages it is often denoted by $\operatorname{int}(x)$.

The related ceiling function $\lceil x\rceil$ gives the smallest integer $\geq x$. The fractional part function is defined as

$$\operatorname{frac}(x)=\begin{cases}x-\lfloor x\rfloor&\text{for }x\geq0,\\x-\lfloor x\rfloor-1&\text{for }x<0.\end{cases}$$

The nearest integer function is

$$\operatorname{nint}(x)=\operatorname{round}(x)=x-\operatorname{frac}(x).$$

References

[a1] R.L. Graham, D.E. Knuth, O. Patashnik, "Concrete mathematics: a foundation for computer science" , Addison-Wesley (1990)
[a2] S. Wolfram, "Mathematica: Version 3" , Addison-Wesley (1996) pp. 718–719
How to Cite This Entry:
Floor function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Floor_function&oldid=33154
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article