DECIMAL REPRESENTATION

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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
:
where $a\_0$ is a nonnegative integer, and $a\_1,$
a_2, dots are integers satisfying $0leq\; a\_ileq\; 9$; this is usually 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 $r.$

Contents

Finite decimal approximations

Multiple decimal representations

Finite decimal representations

Recurring decimal representations

See also

External links

Finite decimal approximations

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
:
Proof:
Let , where $p\; =\; lfloor\; 10^nx$
floor.
Then , and the result follows from dividing all sides by $10^n$.
(The fact that $r\_n$ has a finite decimal representation is easily established.)

Multiple decimal representations

Main articles: Proof that 0.999... equals 1

Some real numbers have two infinite decimal representations. For example, the number 1 may be equally represented by 1.00000... as by 0.99999... (where for the sake of brevity the infinite sequences of digits 0 and 9, respectively, have been replaced by "..."). Conventionally, the version with zero digits is preferred; by omitting the infinite sequence of zero digits, removing any final zero digits and a possible final decimal point, a normalized finite decimal representation is obtained.

Finite decimal representations

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
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'',

rac{2^m5^np}{10^{n+m}}
for some ''p''.
While ''x'' is of the form ,
$p=sum\_\{i=0\}^\{n\}10^ia\_i$ for some ''n''.
By ,
''x'' will end in zeros.

Recurring decimal representations

Main articles: Recurring decimal

Some real numbers have a decimal expansion that eventually gets into a loop, endlessly repeating a sequence of one or more digits:
:¹/₃ = 0.33333...
:¹/₇ = 0.142857142857...
:¹³¹⁸/₁₈₅ = 7.1243243243...
This happens precisely when the number is a rational number. A special case of this phenomenon is where the expansion ends in all zeros (or nines).

See also

★ Decimal

★ Series (mathematics)

External links

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