Definitions

# Decimal representation

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

$r=sum_\left\{i=0\right\}^infty frac\left\{a_i\right\}\left\{10^i\right\}$

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 $r.$

## 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

$r_nleq x < r_n+frac\left\{1\right\}\left\{10^n\right\}.,$

Proof:

Let $r_n = textstylefrac\left\{p\right\}\left\{10^n\right\}$, 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.)

## Multiple decimal representations

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 the infinite sequences of digits 0 and 9, respectively, are represented 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 2n5m, where m and n are non-negative integers.

Proof:

If the decimal expansion of x will end in zeros, or $x=sum_\left\{i=0\right\}^nfrac\left\{a_i\right\}\left\{10^i\right\}=sum_\left\{i=0\right\}^n10^\left\{n-i\right\}a_i/10^n$ 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, $x=frac\left\{p\right\}\left\{2^n5^m\right\}=frac\left\{2^m5^np\right\}\left\{2^\left\{n+m\right\}5^\left\{n+m\right\}\right\}= frac\left\{2^m5^np\right\}\left\{10^\left\{n+m\right\}\right\}$ for some p. While x is of the form $textstylefrac\left\{p\right\}\left\{10^k\right\}$, $p=sum_\left\{i=0\right\}^\left\{n\right\}10^ia_i$ for some n. By $x=sum_\left\{i=0\right\}^n10^\left\{n-i\right\}a_i/10^n=sum_\left\{i=0\right\}^nfrac\left\{a_i\right\}\left\{10^i\right\}$, 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 a non-negative and a positive integer).