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Expression of every real number as a sequence of digits
The sequence of the --the digits after the dot--may be finite, in which case the lacking digits are assumed to be 0.
Every nonnegative real number has at least one such representation; it has two such representations if and only one has a trailing infinite sequence of zeros, and the other has a trailing infinite sequence of nines. Some authors forbid decimal representations with a trailing infinite sequence of nines because this allows a one-to-one correspondence between nonnegative real numbers and decimal representations.
The integer , denoted by a0 in the remainder of this article, is called the integer part of r, and the sequence of the represents the number
which is called 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 . 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
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 algorithmic procedure will give the standard decimal representation: Given , we first define (the integer part of ) to be the largest integer such that (i.e., ). If the procedure terminates. Otherwise, for already found, we define inductively to be the largest integer such that
The procedure terminates whenever is found such that equality holds in ; otherwise, it continues indefinitely to give an infinite sequence of decimal digits. It can be shown that  (conventionally written as ), where and the nonnegative integer is represented in decimal notation. This construction is extended to by applying the above procedure to and denoting the resultant decimal expansion by .
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.
Repeating 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 an integer and a positive integer).
Also the converse is true: The decimal expansion of a rational number is either finite, or endlessly repeating.
Conversion to fraction
Every decimal representation of a rational number can be converted to a fraction by converting it into a sum of the integer, non-repeating, and repeating parts and then converting that sum to a single fraction with a common denominator.
For example to convert to a fraction one notes the lemma:
Thus one converts as follows:
If there are no repeating digits one assumes that there is a forever repeating 0, e.g. , although since that makes the repeating term zero the sum simplifies to two terms and a simpler conversion.