Decimal representation

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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 in the form of a series, traditionally written as a sum

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

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:


Finite decimal approximations[edit]

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


Let , where . Then , and the result follows from dividing all sides by . (The fact that has a finite decimal representation is easily established.)

Non-uniqueness of decimal representation and notational conventions[edit]

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 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 , an infinite sequence of trailing 0's appearing after the decimal point is omitted, along with the decimal point itself, if is an integer.

Certain procedures for constructing the decimal expansion of will avoid the problem of trailing 9's. For instance, the following procedure will give the standard decimal representation: Given , we can first define (the integer part of ) to be the largest integer such that (i.e., ). Then, for already found, we define inductively to be the largest integer such that

The procedure terminates when is found such that equality holds in ; otherwise, it continues indefinitely to give an infinite sequence of decimal digits. It can be shown that [2] (conventionally written as ), with and . This construction is extended to by applying the above procedure to and denoting the resultant decimal expansion by .

Finite decimal representations[edit]

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.


If the decimal expansion of x will end in zeros, or 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, for some p. While x is of the form , for some n. By , x will end in zeros.

Recurring decimal representations[edit]

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.

See also[edit]


  • Tom Apostol (1974). Mathematical analysis (Second ed.). Addison-Wesley. 
  1. ^ Knuth, D. E. (1973), "Volume 1: Fundamental Algorithms", The Art of Computer Programming, Addison-Wesley, p. 21 
  2. ^ Rudin, Walter (1976). Principles of Mathematical Analysis. New York: McGraw-Hill. p. 11. ISBN 0-07-054235-X.