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_^\infty \frac]

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

[r=a_0.a_1 a_2 a_3\dots.]

That is to say, [a_0] 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.]

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.\,]

Proof:

Let [r_n = 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.)

Multiple decimal representations

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 (where for the sake of brevity the infinite sequences of digits 0 and 9, respectively, have been replaced by "...").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.

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_^n\frac=\sum_^n10^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=\frac5^}=\frac}] for some p. While x is of the form p/10^k, [p=\sum_^10^ia_i] for some n. By [x=\sum_^n10^a_i/10^n=\sum_^n\frac], x will end in zeros.

See also

• Decimal

External links

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

 

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