 Decimal representation

 This article gives a mathematical definition. For a more accessible article see Decimal.
A decimal representation of a nonnegative real number r is an expression of the form of a series, traditionally written as a sum
where a_{0} is a nonnegative integer, and a_{1}, a_{2}, … are integers satisfying 0 ≤ a_{i} ≤ 9, called the digits of the decimal representation. The sequence of digits specified may be finite, in which case any further digits a_{i} are assumed to be 0. Some authors forbid decimal representations with an infinite sequence of digits 9.^{[1]} This restriction still allows a decimal representation for each nonnegative real number, but additionally makes such a representation unique. The number defined by a decimal representation is often written more briefly as
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}, … are the digits forming the fractional part of r.
Both notations above are, by definition, the following limit of a sequence:
 .
Contents
Finite decimal approximations
Any real number can be approximated to any desired degree of accuracy by rational numbers with finite decimal representations.
Assume . Then for every integer there is a finite decimal such that
Proof:
Let , where . 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.)
Nonuniqueness of decimal representation
Main article: 0.999...Some real numbers 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 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.^{[citation needed]}
Finite decimal representations
The decimal expansion of nonnegative 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 nonnegative 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}, for some p. While x is of the form , for some n. By , x will end in zeros.
Recurring decimal representations
Main article: Repeating decimalSome 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 nonnegative and a positive integer).
See also
 Decimal
 Series (mathematics)
 IEEE 754
 Simon Stevin
References
 Tom Apostol (1974). Mathematical analysis (Second edition ed.). AddisonWesley.
 ^ Knuth, D. E. (1973), "Volume 1: Fundamental Algorithms", The Art of Computer Programming, AddisonWesley, pp. 21
External links
 Plouffe's inverter tries to identify a number given the start of its decimal representation. For instance, given 3.14159265 it will say that your input probably came from one of the following and list π as the simplest.
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