Francais | English | Espanõl

From Wikipedia, the free encyclopedia

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

<math> r=\sum_{i=0}^\infty \frac{a_i}{10^i}</math>

where <math>a_0</math> is a nonnegative integer, and <math>a_1, a_2, \dots</math> are integers satisfying <math>0\leq a_i\leq 9</math>; this is usually written more briefly as

<math>r=a_0.a_1 a_2 a_3\dots.</math>

That is to say, <math>a_0</math> is the integer part of <math>r</math>, not necessarily between 0 and 9, and <math>a_1, a_2, a_3,\dots</math> are the digits forming the fractional part of <math>r.</math>

Contents 1 Finite decimal approximations 2 Multiple decimal representations 3 Finite decimal representations 4 Recurring decimal representations 5 See also 6 External links

[edit] Finite decimal approximations

Any real number can be approximated to any desired degree of accuracy by rational numbers with finite decimal representations.

Assume <math>x\geq 0</math>. Then for every integer <math>n\geq 1</math> there is a finite decimal <math>r_n=a_0.a_1a_2\cdots a_n</math> such that

<math>r_n\leq x < r_n+\frac{1}{10^n}.\,</math>

Proof:

Let <math>r_n = p / 10^n</math>, where <math>p = \lfloor 10^nx\rfloor</math>. Then <math>p \leq 10^nx < p+1</math>, and the result follows from dividing all sides by <math>10^n</math>. (The fact that <math>r_n</math> 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 (where for the sake of brevity the infinite sequences of digits 0 and 9, respectively, have been replaced by "...").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 2ⁿ5^m, where m and n are non-negative integers.

Proof:

If the decimal expansion of x will end in zeros, or <math>x=\sum_{i=0}^n\frac{a_i}{10^i}=\sum_{i=0}^n10^{n-i}a_i/10^n</math> 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, <math>x=\frac{p}{2^n5^m}=\frac{2^m5^np}{2^{n+m}5^{n+m}}= \frac{2^m5^np}{10^{n+m}}</math> for some p. While x is of the form p/10^k, <math>p=\sum_{i=0}^{n}10^ia_i</math> for some n. By <math>x=\sum_{i=0}^n10^{n-i}a_i/10^n=\sum_{i=0}^n\frac{a_i}{10^i}</math>, x will end in zeros.

[edit] Recurring decimal representations

Main article: Recurring decimal

Some real numbers have a decimal expansion that eventually gets into a loop, endlessly repeating a sequence of one or more digits:

1/3 = 0.33333...

1/7 = 0.142857142857...

1318/185 = 7.1243243243...

This happens precisely when the number is a rational number. A special case of this phenomenon is where the expansion ends in all zeros (or nines).

[edit] See also

• Decimal

[edit] External links

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