A decimal representation of a
non-negative
In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it ...
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 real ...
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 calle ...
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 (Latin ...
s traditionally written with a single separator:
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 cho ...
, 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 n ...
, and
are ''digits'', which are symbols representing integers in the range 0, ..., 9.
Commonly,
if
The sequence of the
—the digits after the dot—is generally
infinite
Infinite may refer to:
Mathematics
* Infinite set, a set that is not a finite set
*Infinity, an abstract concept describing something without any limit
Music
*Infinite (group), a South Korean boy band
*''Infinite'' (EP), debut EP of American m ...
. If it is finite, the lacking digits are assumed to be 0. If all
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 n ...
.
The decimal representation represents the
infinite sum
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 ...
:
Every nonnegative real number has at least one such representation; it has two such representations (with
if
)
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 bicondi ...
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
, is called the ''integer part'' of , and is denoted by in the remainder of this article. The sequence of the
represents the number
which belongs to the
interval and is called the ''fractional part'' of (except when all
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 ration ...
s 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.)
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
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 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
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
by applying the above procedure to
and denoting the resultant decimal expansion by
.
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
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.
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 ration ...
(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.
For example:
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 i ...
*
Simon Stevin
Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician, scientist and music theorist. He made various contributions in many areas of science and engineering, both theoretical and practical. He also translated vario ...
References
Further reading
*
*
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Mathematical notation
Articles containing proofs
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