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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

$r=\sum_{i=0}^\infty \frac{a_i}{10^i}$

where a₀ is a nonnegative integer, and a₁, a₂, … are integers satisfying 0 ≤ a_i ≤ 9 this 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:

$r=\lim_{n\to \infty} \sum_{i=0}^n \frac{a_i}{10^i}$ .

[edit] 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.)

[edit] 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]}

[edit] 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 $x=\sum_{i=0}^n\frac{a_i}{10^i}=\sum_{i=0}^n10^{n-i}a_i/10^n$ 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, $x=\frac{p}{2^n5^m}=\frac{2^m5^np}{2^{n+m}5^{n+m}}= \frac{2^m5^np}{10^{n+m}}$ 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.

[edit] 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 a non-negative and a positive integer).

[edit] See also

[edit] References

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

[edit] External links

Plouffe's inverter tries to identify a number given the start of its decimal representation. For instance, given 3.14159265 it will say that your input probably came from one of the following and list π as the simplest.

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