Namespaces
Variants
Actions

Difference between revisions of "Integral part"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
Line 1: Line 1:
''entier, integer part of a (real) number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051580/i0515801.png" />''
+
{{TEX|done}}
 +
''entier, integer part of a (real) number $x$''
  
The largest integer not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051580/i0515802.png" />. It is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051580/i0515803.png" /> or by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051580/i0515804.png" />. It follows from the definition of an integer part that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051580/i0515805.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051580/i0515806.png" /> is an integer, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051580/i0515807.png" />. Examples: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051580/i0515808.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051580/i0515809.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051580/i05158010.png" />. The integral part is used in the factorization of, for example, the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051580/i05158011.png" />, viz.
+
The largest integer not exceeding $x$. It is denoted by $[x]$ or by $E(x)$. It follows from the definition of an integer part that $[x]\leq x<[x]+1$. If $x$ is an integer, $[x]=x$. Examples: $[3.6]=3$; $[1/3]=0$, $[-13/5]=-5$. The integral part is used in the factorization of, for example, the number $n!=1\dots n$, viz.
  
<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/i/i051/i051580/i05158012.png" /></td> </tr></table>
+
$$n!=\prod_{p\leq n}p^{\alpha(p)},$$
  
where the product consists of all primes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051580/i05158013.png" /> not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051580/i05158014.png" />, and
+
where the product consists of all primes $p$ not exceeding $n$, and
  
<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/i/i051/i051580/i05158015.png" /></td> </tr></table>
+
$$\alpha(p)=\left[\frac np\right]+\left[\frac{n}{p^2}\right]+\mathinner{\ldotp\ldotp\ldotp\ldotp}$$
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051580/i05158016.png" /> of the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051580/i05158017.png" /> is piecewise continuous (a step function) with jumps at the integers. Using the integral part one defines the fractional part of a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051580/i05158018.png" />, denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051580/i05158019.png" /> and given by
+
The function $y=[x]$ of the variable $x$ is piecewise continuous (a step function) with jumps at the integers. Using the integral part one defines the fractional part of a number $x$, denoted by the symbol $\{x\}$ and given by
  
<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/i/i051/i051580/i05158020.png" /></td> </tr></table>
+
$$x-[x];\quad0\leq\{x\}<1.$$
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051580/i05158021.png" /> is a periodic and piecewise continuous.
+
The function $y=\{x\}$ is a periodic and piecewise continuous.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Vinogradov,  "Elements of number theory" , Dover, reprint  (1954)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Vinogradov,  "Elements of number theory" , Dover, reprint  (1954)  (Translated from Russian)</TD></TR></table>

Revision as of 12:44, 27 August 2014

entier, integer part of a (real) number $x$

The largest integer not exceeding $x$. It is denoted by $[x]$ or by $E(x)$. It follows from the definition of an integer part that $[x]\leq x<[x]+1$. If $x$ is an integer, $[x]=x$. Examples: $[3.6]=3$; $[1/3]=0$, $[-13/5]=-5$. The integral part is used in the factorization of, for example, the number $n!=1\dots n$, viz.

$$n!=\prod_{p\leq n}p^{\alpha(p)},$$

where the product consists of all primes $p$ not exceeding $n$, and

$$\alpha(p)=\left[\frac np\right]+\left[\frac{n}{p^2}\right]+\mathinner{\ldotp\ldotp\ldotp\ldotp}$$

The function $y=[x]$ of the variable $x$ is piecewise continuous (a step function) with jumps at the integers. Using the integral part one defines the fractional part of a number $x$, denoted by the symbol $\{x\}$ and given by

$$x-[x];\quad0\leq\{x\}<1.$$

The function $y=\{x\}$ is a periodic and piecewise continuous.

References

[1] I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)
How to Cite This Entry:
Integral part. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_part&oldid=33155
This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article