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''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" />.

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

<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>

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>

====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>