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Decimal representation

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Expression of numbers as sequences of digits

This article is about decimal expansion of real numbers. For finite decimal representation, seeDecimal.

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Adecimal representation of anon-negative real numberr is its expression as asequence of symbols consisting ofdecimal digits traditionally written with a single separator: $r=b_{k}b_{k-1}\cdots b_{0}.a_{1}a_{2}\cdots$ Here. is thedecimal separator,k is anonnegative integer, and $b_{0},\cdots ,b_{k},a_{1},a_{2},\cdots$ aredigits, which are symbols representing integers in the range 0, ..., 9.

Commonly, $b_{k}\neq 0$ if $k\geq 1.$ The sequence of the $a_{i}$ —the digits after the dot—is generallyinfinite. If it is finite, the lacking digits are assumed to be 0. If all $a_{i}$ are0, the separator is also omitted, resulting in a finite sequence of digits, which represents anatural number.

The decimal representation represents theinfinite sum: $r=\sum _{i=0}^{k}b_{i}10^{i}+\sum _{i=1}^{\infty }{\frac {a_{i}}{10^{i}}}.$

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 one has a trailing infinite sequence of0, and the other has a trailing infinite sequence of9. For having a one-to-one correspondence between nonnegative real numbers and decimal representations, decimal representations with a trailing infinite sequence of9 are sometimes excluded.^[1]

Integer and fractional parts

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The natural number ${\textstyle \sum _{i=0}^{k}b_{i}10^{i}}$ , is called theinteger part ofr, and is denoted bya₀ in the remainder of this article. The sequence of the $a_{i}$ represents the number $0.a_{1}a_{2}\ldots =\sum _{i=1}^{\infty }{\frac {a_{i}}{10^{i}}},$ which belongs to theinterval $[0,1),$ and is called thefractional part ofr (except when all $a_{i}$ are equal to9).

Finite decimal approximations

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Any real number can be approximated to any desired degree of accuracy byrational numbers with finite decimal representations.

Assume $x\geq 0$ . Then for every integer $n\geq 1$ there is a finite decimal $r_{n}=a_{0}.a_{1}a_{2}\cdots a_{n}$ such that:

$r_{n}\leq x<r_{n}+{\frac {1}{10^{n}}}.$

Proof:Let $r_{n}=\textstyle {\frac {p}{10^{n}}}$ , where $p=\lfloor 10^{n}x\rfloor$ .Then $p\leq 10^{n}x<p+1$ , 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

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Main article:0.999...

Some real numbers $x {\displaystyle x}$ have two infinite decimal representations. For example, the number 1 may be equally represented by 1.000... as by0.999... (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 thestandard decimal representation of $x {\displaystyle x}$ , an infinite sequence of trailing 0's appearing after thedecimal point is omitted, along with the decimal point itself if $x {\displaystyle x}$ is an integer.

Certain procedures for constructing the decimal expansion of $x {\displaystyle x}$ 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 $a_{0}$ (theinteger part of $x {\displaystyle x}$ ) 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 ${\textstyle (a_{i})_{i=0}^{k-1}}$ already found, we define $a_{k}$ inductively to be the largest integer such that:

a_{0}+{\frac {a_{1}}{10}}+{\frac {a_{2}}{10^{2}}}+\cdots +{\frac {a_{k}}{10^{k}}}\leq x.

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 ${\textstyle x=\sup _{k}\left\{\sum _{i=0}^{k}{\frac {a_{i}}{10^{i}}}\right\}}$ ^[2] (conventionally written as $x=a_{0}.a_{1}a_{2}a_{3}\cdots$ ), where $a_{1},a_{2},a_{3}\ldots \in \{0,1,2,\ldots ,9\},$ and the nonnegative integer $a_{0}$ is represented indecimal notation. This construction is extended to $x<0$ by applying the above procedure to $-x>0$ and denoting the resultant decimal expansion by $-a_{0}.a_{1}a_{2}a_{3}\cdots$ .

Types

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Finite

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The decimal expansion of non-negative real numberx will end in zeros (or in nines) if, and only if,x is a rational number whose denominator is of the form 2ⁿ5^m, wherem andn are non-negative integers.

Proof:

If the decimal expansion ofx will end in zeros, or ${\textstyle x=\sum _{i=0}^{n}{\frac {a_{i}}{10^{i}}}=\sum _{i=0}^{n}10^{n-i}a_{i}/10^{n}}$ for somen, then the denominator ofx is of the form 10ⁿ = 2ⁿ5ⁿ.

Conversely, if the denominator ofx is of the form 2ⁿ5^m, $x={\frac {p}{2^{n}5^{m}}}={\frac {2^{m}5^{n}p}{2^{n+m}5^{n+m}}}={\frac {2^{m}5^{n}p}{10^{n+m}}}$ for somep.Whilex is of the form $\textstyle {\frac {p}{10^{k}}}$ , $p=\sum _{i=0}^{n}10^{i}a_{i}$ for somen.By $x=\sum _{i=0}^{n}10^{n-i}a_{i}/10^{n}=\sum _{i=0}^{n}{\frac {a_{i}}{10^{i}}}$ ,x will end in zeros.

Infinite

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Repeating decimal representations

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Main article:Repeating decimal

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 arational 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.

Finite decimal representations can also be seen as a special case of infinite repeating decimal representations. For example,36⁄25 = 1.44 = 1.4400000...; the endlessly repeated sequence is the one-digit sequence "0".

Non-repeating decimal representations

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Other real numbers have decimal expansions that never repeat. These are precisely theirrational numbers, numbers that cannot be represented as a ratio of integers. Some well-known examples are:

√2 = 1.41421356237309504880...

e = 2.71828182845904523536...

π = 3.14159265358979323846...

Conversion to fraction

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Further information:Fraction § Arithmetic with fractions

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 ${\textstyle \pm 8.123{\overline {4567}}}$ to a fraction one notes the lemma: ${\begin{aligned}0.000{\overline {4567}}&=4567\times 0.000{\overline {0001}}\\&=4567\times 0.{\overline {0001}}\times {\frac {1}{10^{3}}}\\&=4567\times {\frac {1}{9999}}\times {\frac {1}{10^{3}}}\\&={\frac {4567}{9999}}\times {\frac {1}{10^{3}}}\\&={\frac {4567}{(10^{4}-1)\times 10^{3}}}&{\text{The exponents are the number of non-repeating digits after the decimal point (3) and the number of repeating digits (4).}}\end{aligned}}$

Thus one converts as follows: ${\begin{aligned}\pm 8.123{\overline {4567}}&=\pm \left(8+{\frac {123}{10^{3}}}+{\frac {4567}{(10^{4}-1)\times 10^{3}}}\right)&{\text{from above}}\\&=\pm {\frac {8\times (10^{4}-1)\times 10^{3}+123\times (10^{4}-1)+4567}{(10^{4}-1)\times 10^{3}}}&{\text{common denominator}}\\&=\pm {\frac {81226444}{9999000}}&{\text{multiplying, and summing the numerator}}\\&=\pm {\frac {20306611}{2499750}}&{\text{reducing}}\\\end{aligned}}$

If there are no repeating digits one assumes that there is a forever repeating 0, e.g. $1.9=1.9{\overline {0}}$ , although since that makes the repeating term zero the sum simplifies to two terms and a simpler conversion.