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Difference between revisions of "Integral part"

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''entier, integer part of a (real) number $x$''
 
''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.
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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: $[3.6]=3$; $[1/3]=0$, $[-13/3]=-5$. The integral part is used in the factorization of, for example, the number $n!=1\cdots n$, viz.
  
 
$$n!=\prod_{p\leq n}p^{\alpha(p)},$$
 
$$n!=\prod_{p\leq n}p^{\alpha(p)},$$
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$$\alpha(p)=\left[\frac np\right]+\left[\frac{n}{p^2}\right]+\mathinner{\ldotp\ldotp\ldotp\ldotp}$$
 
$$\alpha(p)=\left[\frac np\right]+\left[\frac{n}{p^2}\right]+\mathinner{\ldotp\ldotp\ldotp\ldotp}$$
  
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
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The function $y=[x]$ of the variable $x$ is piecewise constant (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
  
 
$$x-[x];\quad0\leq\{x\}<1.$$
 
$$x-[x];\quad0\leq\{x\}<1.$$
  
 
The function $y=\{x\}$ is a periodic and piecewise continuous.
 
The function $y=\{x\}$ is a periodic and piecewise continuous.
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====Comments====
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The notation $\lfloor x \rfloor$ ("floor") is also in use.  The smallest integer not less than $x$ is denoted $\lceil x \rceil$ ("ceiling").
  
 
====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>
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<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Vinogradov,  "Elements of number theory" , Dover, reprint  (1954)  (Translated from Russian) {{ZBL|0057.28201}}</TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren "Concrete mathematics: a foundation for computer science" (2nd ed.) Addison-Wesley (1994) {{ISBN|0201558025}} {{ZBL|0836.00001}}
 +
</TD></TR>
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</table>

Latest revision as of 16:50, 23 November 2023

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: $[3.6]=3$; $[1/3]=0$, $[-13/3]=-5$. The integral part is used in the factorization of, for example, the number $n!=1\cdots n$, 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 constant (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.

Comments

The notation $\lfloor x \rfloor$ ("floor") is also in use. The smallest integer not less than $x$ is denoted $\lceil x \rceil$ ("ceiling").

References

[1] I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian) Zbl 0057.28201
[a1] Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren "Concrete mathematics: a foundation for computer science" (2nd ed.) Addison-Wesley (1994) ISBN 0201558025 Zbl 0836.00001
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