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

r=b_{k}b_{k-1}\ldots b_{0}.a_{1}a_{2}\ldots \,,

where $k$ is a nonnegative integer and $b_{0},\ldots ,b_{k},a_{1},a_{2},\ldots$ are integers in the range 0, ..., 9, which are called the digits of the representation.

This expression represents the infinite sum

r=\sum _{i=0}^{k}b_{i}10^{i}+\sum _{i=1}^{\infty }{\frac {a_{i}}{10^{i}}}.

The sequence of the $a_{i}$ —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 $\sum _{i=0}^{k}b_{i}10^{i}$ , denoted by $a 0$ in the remainder of this article, is called the integer part of $r$ , and 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 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 $x\geq 0$ . 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

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 ${\textstyle (a_{i})_{i=0}^{k-1}}$ already found, we define $a_{k}$ inductively to be the largest integer such that

$a_{0}+{\frac {a_{1}}{10}}+{\frac {a_{2}}{10^{2}}}+\cdots +{\frac {a_{k}}{10^{k}}}\leq x.\quad \quad (*)$

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 ${\textstyle x=\sup _{k}\{\sum _{i=0}^{k}{\frac {a_{i}}{10^{i}}}\}}$ [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 in decimal notation. This construction is extended to $x<0$ by applying the above procedure to $-x>0$ and denoting the resultant decimal expansion by $-a_{0}.a_{1}a_{2}a_{3}\cdots$ .

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 2ⁿ5^m, where m and n are non-negative integers.

Proof:

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

Conversely, if the denominator of x 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 some p. While x is of the form $\textstyle {\frac {p}{10^{k}}}$ , $p=\sum _{i=0}^{n}10^{i}a_{i}$ for some n. 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.

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 summing up the integer, non-repeating and repeating parts as in the example below

\pm 8.123{\overline {4567}}=\pm \left(8+{\frac {123}{10^{3}}}+{\frac {4567}{(10^{4}-1)\cdot 10^{3}}}\right)=\pm {\frac {8123\,(10000-1)+4567}{9999000}}=\pm {\frac {20306611}{2499750}}

where the exponents in the denominators are 3 (number of non-repeating digits after decimal point) and 4 (number of repeating digits). If there are no repeating digits assume there is a forever repeating 0, that is $1.9=1.9{\overline {0}}$ .

References

Knuth, Donald Ervin (1973). The Art of Computer Programming. Volume 1: Fundamental Algorithms. Addison-Wesley. p. 21.
Rudin, Walter (1976). Principles of Mathematical Analysis. New York: McGraw-Hill. p. 11. ISBN 0-07-054235-X.