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A decimal representation of a non-negative
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
is its expression as a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
of symbols consisting of
decimal digit A numerical digit (often shortened to just digit) is a single symbol used alone (such as "2") or in combinations (such as "25"), to represent numbers in a positional numeral system. The name "digit" comes from the fact that the ten digits ( Lati ...
s traditionally written with a single separator: r = b_k b_\ldots b_0.a_1a_2\ldots Here is the
decimal separator A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form (e.g., "." in 12.45). Different countries officially designate different symbols for use as the separator. The choi ...
, is a
nonnegative integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
, and b_0, \ldots, b_k, a_1, a_2,\ldots are ''digits'', which are symbols representing integers in the range 0, ..., 9. Commonly, b_k\neq 0 if k > 1. The sequence of the a_i—the digits after the dot—is generally infinite. If it is finite, the lacking digits are assumed to be 0. If all a_i are , the separator is also omitted, resulting in a finite sequence of digits, which represents a
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
. The decimal representation represents the infinite sum: r=\sum_^k b_i 10^i + \sum_^\infty \frac. Every nonnegative real number has at least one such representation; it has two such representations (with b_k\neq 0 if k>0)
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
one has a trailing infinite sequence of , and the other has a trailing infinite sequence of . For having a one-to-one correspondence between nonnegative real numbers and decimal representations, decimal representations with a trailing infinite sequence of are sometimes excluded.


Integer and fractional parts

The natural number \sum_^k b_i 10^i, is called the ''integer part'' of , and is denoted by in the remainder of this article. The sequence of the a_i represents the number 0.a_1a_2\ldots = \sum_^\infty \frac, which belongs to the interval [0,1), and is called the ''fractional part'' of (except when all a_i are ).


Finite decimal approximations

Any real number can be approximated to any desired degree of accuracy by
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s with finite decimal representations. Assume x \geq 0. Then for every integer n\geq 1 there is a finite decimal r_n=a_0.a_1a_2\cdots a_n such that: r_n\leq x < r_n+\frac. Proof: Let r_n = \textstyle\frac, where p = \lfloor 10^n x\rfloor. 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.)


Non-uniqueness of decimal representation and notational conventions

Some real numbers x 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 x, an infinite sequence of trailing 0's appearing after the decimal point is omitted, along with the decimal point itself if x is an integer. Certain procedures for constructing the decimal expansion of x will avoid the problem of trailing 9's. For instance, the following algorithmic procedure will give the standard decimal representation: Given x\geq 0, we first define a_0 (the ''integer part'' of x) to be the largest integer such that a_0\leq x (i.e., a_0 = \lfloor x\rfloor). If x=a_0 the procedure terminates. Otherwise, for (a_i)_^ already found, we define a_k inductively to be the largest integer such that: The procedure terminates whenever a_k is found such that equality holds in ; otherwise, it continues indefinitely to give an infinite sequence of decimal digits. It can be shown that x = \sup_k \left\ (conventionally written as x=a_0.a_1a_2a_3\cdots), where a_1,a_2,a_3\ldots \in \, and the nonnegative integer a_0 is represented in
decimal notation The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numer ...
. This construction is extended to x<0 by applying the above procedure to -x>0 and denoting the resultant decimal expansion by -a_0.a_1a_2a_3\cdots.


Types


Finite

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 x=\sum_^n\frac = \sum_^n 10^a_i/10^n 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'', x = \frac=\frac = \frac for some ''p''. While ''x'' is of the form \textstyle\frac, p = \sum_^ 10^i a_i for some ''n''. By x=\sum_^n10^a_i/10^n=\sum_^n\frac, ''x'' will end in zeros.


Infinite


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 In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
(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 \pm 8.123\overline to a fraction one notes the lemma: \begin 0.000\overline & = 4567\times0.000\overline \\ & = 4567\times0.\overline\times\frac \\ & = 4567\times\frac\times\frac \\ & = \frac\times\frac \\ & = \frac& \text \end Thus one converts as follows: \begin \pm 8.123\overline & = \pm \left(8 + \frac + \frac\right) & \text \\ & = \pm \frac & \text\\ & = \pm \frac & \text\\ & = \pm \frac & \text\\ \end If there are no repeating digits one assumes that there is a forever repeating 0, e.g. 1.9 = 1.9\overline, although since that makes the repeating term zero the sum simplifies to two terms and a simpler conversion. For example: \begin \pm 8.1234 & = \pm \left(8 + \frac\right) & \\ & = \pm \frac & \text\\ & = \pm \frac & \text\\ & = \pm \frac & \text\\ \end


See also

*
Decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
*
Series (mathematics) In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
*
IEEE 754 The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point arithmetic established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard addressed many problems found ...
* Simon Stevin


References


Further reading

* * {{Authority control Mathematical notation Articles containing proofs br:Dispakadur dekredel ckb:نواندنی دەدەیی