Decimal representation

A decimal representation of a non-negative real number r is an expression in the form of a series, traditionally written as a sum

r=\sum _{i=0}^{\infty }{\frac {a_{i}}{10^{i}}}

where a₀ is a nonnegative integer, and a₁, a₂, ... are integers satisfying 0 ≤ a_i ≤ 9, called the digits of the decimal representation. The sequence of digits specified may be finite, in which case any further digits a_i are assumed to be 0. Some authors forbid decimal representations with a trailing infinite sequence of "9"s.^[1] This restriction still allows a decimal representation for each non-negative real number, but additionally makes such a representation unique. The number defined by a decimal representation is often written more briefly as

r=a_{0}.a_{1}a_{2}a_{3}\dots .

That is to say, a₀ is the integer part of r, not necessarily between 0 and 9, and a₁, a₂, a₃, ... are the digits forming the fractional part of r.

Both notations above are, by definition, the following limit of a sequence:

r=\lim _{n\to \infty }\sum _{i=0}^{n}{\frac {a_{i}}{10^{i}}}

.

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 procedure will give the standard decimal representation: Given $x\geq 0$ , we can first define $a_{0}$ (the integer part of $x$ ) to be the largest integer such that $a_{0}\leq x$ (i.e., $a_{0}=\lfloor x\rfloor$ ). Then, 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 when $a_{k}$ 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 $x=a_{0}.a_{1}a_{2}a_{3}\cdots$ ), with $a_{0}\in \mathbb {N} _{0}$ and $a_{1},a_{2},a_{3}\ldots \in \{0,1,2,\ldots ,9\}$ . This construction is extended to $x<0$ by applying the above procedure to $-x$ 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.

Recurring 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.

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.