finite decimal

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_{i=0}^infty frac{a_i}{10^i}

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{1}{10^n}.,


Let r_n = textstylefrac{p}{10^n}, 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.


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

See also


  • Tom Apostol (1974). Mathematical analysis. Second edition, Addison-Wesley.

External links

  • Plouffe's inverter describes a number given its decimal representation. For instance, it will describe 3.14159265 as π.

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