Decimal representation

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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 of 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

Main article: 0.999...

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 digits 0 and 9, respectively, are represented by "..."). Conventionally, the version with zero digits is preferred; by omitting the infinite sequence of zero digits, removing any final zero digits and a possible final decimal point, a normalized finite decimal representation is obtained. ^{[citation needed]}

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).

References

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

↑ Knuth, D. E. (1973), "Volume 1: Fundamental Algorithms", The Art of Computer Programming, Addison-Wesley, p. 21

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