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- This article gives a mathematical definition. For a more accessible article see Decimal.

A decimal representation of a non-negative real number r is an expression of the form

- $r=sum\_\{i=0\}^infty\; frac\{a\_i\}\{10^i\}$

where $a\_0$ is a nonnegative integer, and $a\_1,\; a\_2,\; dots$ are integers satisfying $0leq\; a\_ileq\; 9$; this is often written more briefly as

- $r=a\_0.a\_1\; a\_2\; a\_3dots.$

That is to say, $a\_0$ is the integer part of $r$, not necessarily between 0 and 9, and $a\_1,\; a\_2,\; a\_3,dots$ are the digits forming the fractional part of $r.$

Any real number can be approximated to any desired degree of accuracy by rational numbers with finite decimal representations.

Assume $xgeq\; 0$. Then for every integer $ngeq\; 1$ there is a finite decimal $r\_n=a\_0.a\_1a\_2cdots\; a\_n$ such that

- $r\_nleq\; x\; <\; r\_n+frac\{1\}\{10^n\}.,$

Proof:

Let $r\_n\; =\; textstylefrac\{p\}\{10^n\}$, where $p\; =\; lfloor\; 10^nxrfloor$. Then $p\; leq\; 10^nx\; <\; p+1$, and the result follows from dividing all sides by $10^n$. (The fact that $r\_n$ has a finite decimal representation is easily established.)

The decimal expansion of non-negative real number x will end in zeros (or in nines) if, and only if, x is a rational number whose denominator is of the form 2^{n}5^{m}, where m and n are non-negative integers.

Proof:

If the decimal expansion of x will end in zeros, or $x=sum\_\{i=0\}^nfrac\{a\_i\}\{10^i\}=sum\_\{i=0\}^n10^\{n-i\}a\_i/10^n$
for some n,
then the denominator of x is of the form 10^{n} = 2^{n}5^{n}.

Conversely, if the denominator of x is of the form 2^{n}5^{m},
$x=frac\{p\}\{2^n5^m\}=frac\{2^m5^np\}\{2^\{n+m\}5^\{n+m\}\}=\; frac\{2^m5^np\}\{10^\{n+m\}\}$
for some p.
While x is of the form $textstylefrac\{p\}\{10^k\}$,
$p=sum\_\{i=0\}^\{n\}10^ia\_i$ for some n.
By $x=sum\_\{i=0\}^n10^\{n-i\}a\_i/10^n=sum\_\{i=0\}^nfrac\{a\_i\}\{10^i\}$,
x will end in zeros.

^{1}/_{3}= 0.33333...

^{1}/_{7}= 0.142857142857...

^{1318}/_{185}= 7.1243243243...

- Tom Apostol (1974).
*Mathematical analysis*. Second edition, Addison-Wesley.

- Plouffe's inverter describes a number given its decimal representation. For instance, it will describe 3.14159265 as π.

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Last updated on Monday April 14, 2008 at 15:05:25 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Monday April 14, 2008 at 15:05:25 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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