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{{Short description|Expression of numbers as sequences of digits}}
{{about|decimal expansion of real numbers|finite decimal representation|Decimal}}
{{More citations needed|date=January 2022}}

A '''decimal representation''' of a [[non-negative]] [[real number]] {{mvar|r}} is its expression as a [[sequence]] of symbols consisting of [[decimal digit]]s traditionally written with a single separator: 
<math display="block">r = b_k b_{k-1}\cdots b_0.a_1a_2\cdots</math>
Here {{char|.}} is the [[decimal separator]], {{mvar|k}} is a [[nonnegative integer]], and <math>b_0, \cdots, b_k, a_1, a_2,\cdots</math> are ''digits'', which are symbols representing integers in the range 0, ..., 9.

Commonly, <math>b_k\neq 0</math> if <math>k \geq 1.</math> The sequence of the <math>a_i</math>—the digits after the dot—is generally [[finite sequence|infinite]]. If it is finite, the lacking digits are assumed to be 0. If all <math>a_i</math> are {{char|0}}, the separator is also omitted, resulting in a finite sequence of digits, which represents a [[natural number]].

The decimal representation represents the [[series (mathematics)|infinite sum]]:
<math display="block"> r=\sum_{i=0}^k b_i 10^i + \sum_{i=1}^\infty \frac{a_i}{10^i}.</math>

Every nonnegative real number has at least one such representation; it has two such representations (with <math>b_k\neq 0</math> if <math>k>0</math>) [[if and only if]] one has a trailing infinite sequence of {{char|0}}, and the other has a trailing infinite sequence of {{char|9}}. For having a one-to-one correspondence between nonnegative real numbers and decimal representations, decimal representations with a trailing infinite sequence of {{char|9}} are sometimes excluded.<ref name="Knuth_1973"/>

==Integer and fractional parts==
The natural number <math display="inline">\sum_{i=0}^k b_i 10^i</math>, is called the ''integer part'' of {{mvar|r}}, and is denoted by {{math|''a''<sub>0</sub>}} in the remainder of this article. The sequence of the <math>a_i</math> represents the number 
<math display="block">0.a_1a_2\ldots = \sum_{i=1}^\infty \frac{a_i}{10^i},</math>
which belongs to the [[interval (mathematics)|interval]] <math>[0,1),</math> and is called the ''fractional part'' of {{mvar|r}} (except when all <math>a_i</math> are equal to {{char|9}}).

==Finite decimal approximations==

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

Assume <math>x \geq 0</math>. Then for every integer <math>n\geq 1</math> there is a finite decimal <math>r_n=a_0.a_1a_2\cdots a_n</math> such that:

<math display="block">r_n\leq x < r_n+\frac{1}{10^n}.</math>

'''Proof''':
Let <math>r_n = \textstyle\frac{p}{10^n}</math>, where <math>p = \lfloor 10^n x\rfloor</math>.
Then <math>p \leq 10^nx < p+1</math>, and the result follows from dividing all sides by <math>10^n</math>.
(The fact that <math>r_n</math> has a finite decimal representation is easily established.)

==Non-uniqueness of decimal representation and notational conventions==
{{Main|0.999...}}
Some real numbers <math>x</math> have two infinite decimal representations. For example, the number 1 may be equally represented by 1.000... as by [[0.999...]] (where the infinite sequences of trailing 0's or 9's, respectively, are represented by "..."). Conventionally, the decimal representation without trailing 9's is preferred.  Moreover, in the ''standard decimal representation'' of <math>x</math>, an infinite sequence of trailing 0's appearing after the [[Decimal mark|decimal point]] is omitted, along with the decimal point itself if <math>x</math> is an integer.

Certain procedures for constructing the decimal expansion of <math>x</math> will avoid the problem of trailing 9's.  For instance, the following algorithmic procedure will give the standard decimal representation:  Given <math>x\geq 0</math>, we first define <math>a_0</math> (the ''integer part'' of <math>x</math>) to be the largest integer such that <math>a_0\leq x</math> (i.e., <math>a_0 = \lfloor x\rfloor</math>).  If <math>x=a_0</math> the procedure terminates.  Otherwise, for <math display="inline">(a_i)_{i=0}^{k-1}</math> already found, we define <math>a_k</math> inductively to be the largest integer such that:
{{NumBlk||<math display="block">a_0+\frac{a_1}{10}+\frac{a_2}{10^2}+\cdots+\frac{a_k}{10^k}\leq x.</math>|{{EqRef|<nowiki>*</nowiki>}}}}
The procedure terminates whenever <math>a_k</math> is found such that equality holds in {{EqNote|*}}; otherwise, it continues indefinitely to give an infinite sequence of decimal digits.  It can be shown that <math display="inline">x = \sup_k \left\{\sum_{i=0}^{k} \frac{a_i}{10^i}\right\}</math><ref name="Walter_1976"/> (conventionally written as <math>x=a_0.a_1a_2a_3\cdots</math>), where <math>a_1,a_2,a_3\ldots \in \{0,1,2,\ldots, 9\},</math> and the nonnegative integer <math>a_0</math> is represented in [[decimal notation]]. This construction is extended to <math>x<0</math> by applying the above procedure to <math>-x>0</math> and denoting the resultant decimal expansion by <math>-a_0.a_1a_2a_3\cdots</math>.

==Types==
===Finite ===

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<sup>''n''</sup>5<sup>''m''</sup>, where ''m'' and ''n'' are non-negative integers.

'''Proof''':

If the decimal expansion of ''x'' will end in zeros, or <math display="inline">x=\sum_{i=0}^n\frac{a_i}{10^i} = \sum_{i=0}^n 10^{n-i}a_i/10^n</math>
for some ''n'', then the denominator of ''x'' is of the form 10<sup>''n''</sup> = 2<sup>''n''</sup>5<sup>''n''</sup>.

Conversely, if the denominator of ''x'' is of the form 2<sup>''n''</sup>5<sup>''m''</sup>,
<math>x = \frac{p}{2^n5^m}=\frac{2^m5^np}{2^{n+m}5^{n+m}} = \frac{2^m 5^np}{10^{n+m}}</math>
for some ''p''.
While ''x'' is of the form <math>\textstyle\frac{p}{10^k}</math>,
<math>p = \sum_{i=0}^{n} 10^i a_i</math> for some ''n''.
By <math>x=\sum_{i=0}^n10^{n-i}a_i/10^n=\sum_{i=0}^n\frac{a_i}{10^i}</math>, ''x'' will end in zeros.

===Infinite ===
====Repeating decimal representations====
{{Main|Repeating decimal}}
Some real numbers have decimal expansions that eventually get into loops, endlessly repeating a sequence of one or more digits:
:{{frac|1|3}} = 0.33333...
:{{frac|1|7}} = 0.142857142857...
:{{frac|1318|185}} = 7.1243243243...
Every time this happens the number is still a [[rational number]] (i.e. can alternatively be represented as a ratio of an integer and a positive integer).
Also the converse is true:  The decimal expansion of a rational number is either finite, or endlessly repeating.

Finite decimal representations can also be seen as a special case of infinite repeating decimal representations. For example, {{frac|36|25}} = 1.44 = 1.4400000...; the endlessly repeated sequence is the one-digit sequence "0".

====Non-repeating decimal representations====
Other real numbers have decimal expansions that never repeat. These are precisely the [[irrational numbers]], numbers that cannot be represented as a ratio of integers. Some well-known examples are:
:[[Square root of 2|{{sqrt|2}}]] = 1.41421356237309504880...
:&nbsp;&thinsp;[[Euler's number|''e'']]&nbsp; = 2.71828182845904523536...
:&nbsp;&thinsp;[[Pi|''π'']]&nbsp; = 3.14159265358979323846...

==Conversion to fraction==
{{further|Fraction#Arithmetic with fractions}}
Every decimal representation of a rational number can be converted to a fraction by converting it into a sum of the integer, non-repeating, and repeating parts and then converting that sum to a single fraction with a common denominator.

For example, to convert <math display="inline">\pm 8.123\overline{4567}</math> to a fraction one notes the lemma:
<math display="block">
\begin{align}
0.000\overline{4567} 
& = 4567\times0.000\overline{0001} \\
& = 4567\times0.\overline{0001}\times\frac{1}{10^3} \\
& = 4567\times\frac{1}{9999}\times\frac{1}{10^3} \\
& = \frac{4567}{9999}\times\frac{1}{10^3} \\
& = \frac{4567}{(10^4 - 1)\times10^3}& \text{The exponents are the number of non-repeating digits after the decimal point (3) and the number of repeating digits (4).} 
\end{align}
</math>

Thus one converts as follows:
<math display="block">
\begin{align}
\pm 8.123\overline{4567} 
& = \pm \left(8 + \frac{123}{10^3} + \frac{4567}{(10^4 - 1) \times 10^3}\right) & \text{from above} \\
& = \pm \frac{8\times(10^4-1)\times10^3+123\times(10^4-1)+4567}{(10^4 - 1) \times 10^3} & \text{common denominator}\\
& = \pm \frac{81226444}{9999000} & \text{multiplying, and summing the numerator}\\
& = \pm \frac{20306611}{2499750} & \text{reducing}\\
\end{align}
</math>

If there are no repeating digits one assumes that there is a forever repeating 0, e.g. <math>1.9 = 1.9\overline{0}</math>, although since that makes the repeating term zero the sum simplifies to two terms and a simpler conversion.

For example:
<math display="block">
\begin{align}
\pm 8.1234 
& = \pm \left(8 + \frac{1234}{10^4}\right) & \\
& = \pm \frac{8\times10^4+1234}{10^4} & \text{common denominator}\\
& = \pm \frac{81234}{10000} & \text{multiplying, and summing the numerator}\\
& = \pm \frac{40617}{5000} & \text{reducing}\\
\end{align}
</math>

==See also==
*[[Decimal]]
*[[Series (mathematics)]]
*[[IEEE 754]]
*[[Simon Stevin#Decimal fractions|Simon Stevin]]

==References==
{{Reflist|refs=
<ref name="Knuth_1973">{{Cite book |author-last=Knuth |author-first=Donald Ervin |author-link=Donald Ervin Knuth |title=The Art of Computer Programming |title-link=The Art of Computer Programming |volume=1: Fundamental Algorithms |publisher=[[Addison-Wesley]] |date=1973 |pages=21}}</ref>
<ref name="Walter_1976">{{Cite book |title=Principles of Mathematical Analysis |author-last=Rudin |author-first=Walter |author-link=Walter Rudin |publisher=[[McGraw-Hill]] |date=1976 |isbn=0-07-054235-X |location=New York |pages=11}}</ref>
}}

==Further reading==
* {{cite book |author-first=Tom |author-last=Apostol |author-link=Tom Apostol |title=Mathematical analysis |edition=Second |publisher=[[Addison-Wesley]] |date=1974}}
* {{cite web |title=Decimal Representations |author-first=John J. G. |author-last=Savard |date=2018 |orig-year=2006 |work=quadibloc |url=http://www.quadibloc.com/comp/cp0203.htm |access-date=2018-07-16 |url-status=live |archive-url=https://web.archive.org/web/20180716101321/http://www.quadibloc.com/comp/cp0203.htm |archive-date=2018-07-16}}

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[[Category:Mathematical notation]]
[[Category:Articles containing proofs]]

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