Decimal representation

This article gives a mathematical definition. For a more accessible article see Decimal.

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 conventions

Main article: 0.999...

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; by omitting an infinite sequence of 0's appearing after the decimal point, and the decimal point itself if $x$ is an integer, the standard decimal representation of $x$ is obtained.

Certain procedures for constructing the decimal expansion of $x$ will avoid the problem of trailing 9's. For instance, given $x\geq 0$ , we can first define $a_{0}$ to be the largest integer such that $a_{0}\leq x$ (i.e., $a_{0}=\lfloor x\rfloor$ ). Then, for $\{a_{i}\}_{i=0}^{k-1}$ already found, define $a_{k}$ inductively to be the largest integer such that $a_{0}+{\frac {a_{1}}{10}}+\cdots +{\frac {a_{k}}{10^{k}}}\leq x$ . Then, it is easily shown that $x=\sup _{k}\{\sum _{i=0}^{k}{\frac {a_{i}}{10^{i}}}\}$ ,^[2] with decimal digits $a_{0}.a_{1}a_{2}\cdots$ such that $a_{0}\in \mathbb {N} _{0}$ and $a_{1},a_{2},\ldots \in \{0,1,2,\ldots ,9\}$ . This procedure gives a unique decimal representation of $x$ , possibly with an infinite sequence of trailing 0's (but not trailing 9's) whenever equality holds for the inequality above. This construction is then extended to $x<0$ by applying the above procedure to $-x$ and denoting the resultant decimal expansion $-a_{0}.a_{1}a_{2}\cdots$ .

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

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 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

Tom Apostol (1974). Mathematical analysis (Second ed.). Addison-Wesley.

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

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