Decimal Representation

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## Finite decimal approximations

## Non-uniqueness of decimal representation and notational conventions

## Finite decimal representations

## Repeating decimal representations

## Conversion to fraction

## See also

## References

## Further reading

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Decimal Representation

A **decimal representation** of a non-negative real number r is an expression in the form of a sequence of decimal digits traditionally written with a single separator

where k is a nonnegative integer and are integers in the range 0, ..., 9, which are called the *digits* of the representation.

This expression represents the infinite sum

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.^{[1]}

The integer , denoted by *a*_{0} 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.

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 . (The fact that has a finite decimal representation is easily established.)

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 ^{[2]} (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 .

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 2^{n}5^{m}, where *m* and *n* are non-negative 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.

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.

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.

For example:

**^**Knuth, Donald Ervin (1973).*The Art of Computer Programming*. Volume 1: Fundamental Algorithms. Addison-Wesley. p. 21.`|volume=`

has extra text (help)**^**Rudin, Walter (1976).*Principles of Mathematical Analysis*. New York: McGraw-Hill. p. 11. ISBN 0-07-054235-X.

- Apostol, Tom (1974).
*Mathematical analysis*(Second ed.). Addison-Wesley. - Savard, John J. G. (2018) [2006]. "Decimal Representations".
*quadibloc*. Archived from the original on 2018-07-16. Retrieved .

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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