# Decimal representation

A decimal representation of a non-negative real number r is an expression in the form of a series, traditionally written as a sum

${\displaystyle r=\sum _{i=0}^{\infty }{\frac {a_{i}}{10^{i}}}}$

where a0 is a nonnegative integer, and a1, a2, ... are integers satisfying 0 ≤ ai ≤ 9, called the digits of the decimal representation. The sequence of digits specified may be finite, in which case any further digits ai are assumed to be 0. Some authors forbid decimal representations with a trailing infinite sequence of "9"s.[1] This restriction still allows a decimal representation for each non-negative real number, but additionally makes such a representation unique. The number defined by a decimal representation is often written more briefly as

${\displaystyle r=a_{0}.a_{1}a_{2}a_{3}\dots .}$

That is to say, a0 is the integer part of r, not necessarily between 0 and 9, and a1, a2, a3, ... are the digits forming the fractional part of r.

Both notations above are, by definition, the following limit of a sequence:

${\displaystyle r=\lim _{n\to \infty }\sum _{i=0}^{n}{\frac {a_{i}}{10^{i}}}}$.

## Finite decimal approximations

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

Assume ${\displaystyle x\geq 0}$. Then for every integer ${\displaystyle n\geq 1}$ there is a finite decimal ${\displaystyle r_{n}=a_{0}.a_{1}a_{2}\cdots a_{n}}$ such that

${\displaystyle r_{n}\leq x

Proof:

Let ${\displaystyle r_{n}=\textstyle {\frac {p}{10^{n}}}}$, where ${\displaystyle p=\lfloor 10^{n}x\rfloor }$. Then ${\displaystyle p\leq 10^{n}x, and the result follows from dividing all sides by ${\displaystyle 10^{n}}$. (The fact that ${\displaystyle r_{n}}$ has a finite decimal representation is easily established.)

## Non-uniqueness of decimal representation and notational conventions

Some real numbers ${\displaystyle x}$ 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 ${\displaystyle x}$, an infinite sequence of trailing 0's appearing after the decimal point is omitted, along with the decimal point itself if ${\displaystyle x}$ is an integer.

Certain procedures for constructing the decimal expansion of ${\displaystyle x}$ will avoid the problem of trailing 9's. For instance, the following algorithmic procedure will give the standard decimal representation: Given ${\displaystyle x\geq 0}$, we first define ${\displaystyle a_{0}}$ (the integer part of ${\displaystyle x}$) to be the largest integer such that ${\displaystyle a_{0}\leq x}$ (i.e., ${\displaystyle a_{0}=\lfloor x\rfloor }$). If ${\displaystyle x=a_{0}}$ the procedure terminates. Otherwise, for ${\textstyle (a_{i})_{i=0}^{k-1}}$ already found, we define ${\displaystyle a_{k}}$ inductively to be the largest integer such that

${\displaystyle a_{0}+{\frac {a_{1}}{10}}+{\frac {a_{2}}{10^{2}}}+\cdots +{\frac {a_{k}}{10^{k}}}\leq x.\quad \quad (*)}$

The procedure terminates whenever ${\displaystyle a_{k}}$ is found such that equality holds in ${\displaystyle (*)}$; otherwise, it continues indefinitely to give an infinite sequence of decimal digits. It can be shown that ${\textstyle x=\sup _{k}\{\sum _{i=0}^{k}{\frac {a_{i}}{10^{i}}}\}}$[2] (conventionally written as ${\displaystyle x=a_{0}.a_{1}a_{2}a_{3}\cdots }$), with ${\displaystyle a_{0}\in \mathbb {N} _{0}}$ and ${\displaystyle a_{1},a_{2},a_{3}\ldots \in \{0,1,2,\ldots ,9\}}$. This construction is extended to ${\displaystyle x<0}$ by applying the above procedure to ${\displaystyle -x>0}$ and denoting the resultant decimal expansion by ${\displaystyle -a_{0}.a_{1}a_{2}a_{3}\cdots }$.

## 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 2n5m, where m and n are non-negative integers.

Proof:

If the decimal expansion of x will end in zeros, or ${\displaystyle x=\sum _{i=0}^{n}{\frac {a_{i}}{10^{i}}}=\sum _{i=0}^{n}10^{n-i}a_{i}/10^{n}}$ for some n, then the denominator of x is of the form 10n = 2n5n.

Conversely, if the denominator of x is of the form 2n5m, ${\displaystyle x={\frac {p}{2^{n}5^{m}}}={\frac {2^{m}5^{n}p}{2^{n+m}5^{n+m}}}={\frac {2^{m}5^{n}p}{10^{n+m}}}}$ for some p. While x is of the form ${\displaystyle \textstyle {\frac {p}{10^{k}}}}$, ${\displaystyle p=\sum _{i=0}^{n}10^{i}a_{i}}$ for some n. By ${\displaystyle x=\sum _{i=0}^{n}10^{n-i}a_{i}/10^{n}=\sum _{i=0}^{n}{\frac {a_{i}}{10^{i}}}}$, x will end in zeros.

## Recurring decimal representations

Some real numbers have decimal expansions that eventually get into loops, endlessly repeating a sequence of one or more digits:

1/3 = 0.33333...
1/7 = 0.142857142857...
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.

## Conversion to fraction

Every decimal representation of a rational number can be converted to a fraction by summing up the integer, non-repeating and repeating parts as in the example below

${\displaystyle \pm 8.123{\overline {4567}}=\pm 8\pm {\frac {123}{10^{3}}}\pm {\frac {4567}{(10^{4}-1)\cdot 10^{3}}}=\pm {\frac {81234567-8123}{9999000}}=\pm {\frac {20306611}{2499750}}}$

where the exponents in the denominators are 3 (number of non-repeating digits after decimal point) and 4 (number of repeating digits). If there are no repeating digits assume there's a forever repeating 0 or 9, i.e. ${\displaystyle 1.9=1.9{\overline {0}}=1.8{\overline {9}}}$.