# Decimal representation

A decimal representation of a non-negative real number r is an expression in the form of a sequence of decimal digits traditionally written with a single separator

${\displaystyle r=b_{k}b_{k-1}\ldots b_{0}.a_{1}a_{2}\ldots \,,}$

where k is a nonnegative integer and ${\displaystyle b_{0},\ldots ,b_{k},a_{1},a_{2},\ldots }$ are integers in the range 0, ..., 9, which are called the digits of the representation.

This expression represents the infinite sum

${\displaystyle r=\sum _{i=0}^{k}b_{i}10^{i}+\sum _{i=1}^{\infty }{\frac {a_{i}}{10^{i}}}.}$

The sequence of the ${\displaystyle a_{i}}$—the digits after the dot—may be finite, in which case the lacking digits are assumed to be 0.

Every nonnegative real number has at least one such representation; it has two such representations if and only one has a trailing infinite sequence of zeros, and the other has a trailing infinite sequence of nines. Some authors forbid decimal representations with a trailing infinite sequence of nines because this allows a one-to-one correspondence between nonnegative real numbers and decimal representations.[1]

The integer ${\displaystyle \sum _{i=0}^{k}b_{i}10^{i}}$, denoted by a0 in the remainder of this article, is called the integer part of r, and the sequence of the ${\displaystyle a_{i}}$ represents the number

${\displaystyle 0.a_{1}a_{2}\ldots =\sum _{i=1}^{\infty }{\frac {a_{i}}{10^{i}}},}$

which is called the fractional part of r.

## 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 ${\displaystyle x=a_{0}.a_{1}a_{2}a_{3}\cdots }$), where ${\displaystyle a_{1},a_{2},a_{3}\ldots \in \{0,1,2,\ldots ,9\},}$ and the nonnegative integer ${\displaystyle a_{0}}$ is represented in decimal notation. 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.

## Repeating 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[clarification needed]

${\displaystyle \pm 8.123{\overline {4567}}=\pm \left(8+{\frac {123}{10^{3}}}+{\frac {4567}{(10^{4}-1)\cdot 10^{3}}}\right)=\pm {\frac {8123\,(10000-1)+4567}{9999000}}=\pm {\frac {20306611}{2499750}}}$

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 is a forever repeating 0, that is ${\displaystyle 1.9=1.9{\overline {0}}}$.

1. ^ Knuth, Donald Ervin (1973). The Art of Computer Programming. Volume 1: Fundamental Algorithms. Addison-Wesley. p. 21. |volume= has extra text (help)