- 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
where is a nonnegative integer, and are integers satisfying ; this is often written more briefly as
That is to say, is the integer part of , not necessarily between 0 and 9, and are the digits forming the fractional part of
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
Let , where .
Then , and the result follows from dividing all sides by .
(The fact that 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
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.
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).
- Tom Apostol (1974). Mathematical analysis. Second edition, Addison-Wesley.
- Plouffe's inverter describes a number given its decimal representation. For instance, it will describe 3.14159265 as π.