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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 ≤ ai ≤ 9, called the digits of the decimal representation. The sequence of digits specified may be finite, in which case any further digits ai 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 \( r_n=a_0.a_1a_2\cdots a_n \) such that

\(\( r_n\leq x < r_n+\frac{1}{10^n}.\, \)

Proof:

Let \( r_n = \textstyle\frac{p}{10^n} \), where \(p = \lfloor 10^nx\rfloor \). Then \( p \leq 10^nx < p+1 \), 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.
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 2n5m, where m and n are non-negative integers.

Proof:

If the decimal expansion of x will end in zeros, or \(x=\sum_{i=0}^n\frac{a_i}{10^i}=\sum_{i=0}^n10^{n-i}a_i/10^n \) for some n, then the denominator of x is of the form \(10^n = 2^n5^n. \)

Conversely, if the denominator of x is of the form \( 2^n5^m \), \(x=\frac{p}{2^n5^m}=\frac{2^m5^np}{2^{n+m}5^{n+m}}= \frac{2^m5^np}{10^{n+m}} \) for some p. While x is of the form \( \textstyle\frac{p}{10^k}, p=\sum_{i=0}^{n}10^ia_i \) for some n. By \( x=\sum_{i=0}^n10^{n-i}a_i/10^n=\sum_{i=0}^n\frac{a_i}{10^i}, x \) 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).

See also

Decimal
Series (mathematics)
IEEE 754
Simon Stevin

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

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

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