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

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

where a0 is a nonnegative integer, and a_1, a_2, \dots are integers satisfying 0\leq a_i\leq 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 a_1, a_2, a_3,\dots are the digits forming the fractional part of r.

Contents

[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 10n. (The fact that rn has a finite decimal representation is easily established.)

[edit] Multiple decimal representations

Main article: Proof that 0.999... equals 1

Some real numbers have two infinite decimal representations. For example, the number 1 may be equally represented by 1.00000... as by 0.99999... (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.

[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 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 10n = 2n5n.

Conversely, if the denominator of x is of the form 2n5m, 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: Recurring 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, Addison-Wesley. 

[edit] External links

  • Plouffe's inverter describes a number given its decimal representation. For instance, it will describe 3.14159265 as π.