Decimal representation

Finite decimal approximations

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

Assume x ≥ 0 . Then for every integer n ≥ 1 there is a finite decimal r n = a 0 . a 1 a 2 ⋯ a n such that

Proof:

Let r n = p 10 n , where p = ⌊ 10 n x ⌋ . Then p ≤ 10 n x < 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 and notational conventions

Some real numbers x 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 trailing 0's or 9's, respectively, are represented by "..."). Conventionally, the decimal representation without trailing 9's is preferred. Moreover, in the standard decimal representation of x , an infinite sequence of trailing 0's appearing after the decimal point is omitted, along with the decimal point itself, if x is an integer.

Certain procedures for constructing the decimal expansion of x will avoid the problem of trailing 9's. For instance, the following procedure will give the standard decimal representation: Given x ≥ 0 , we can first define a 0 (the integer part of x ) to be the largest integer such that a 0 ≤ x (i.e., a 0 = ⌊ x ⌋ ). Then, for ( a i ) i = 0 k − 1 already found, we define a k inductively to be the largest integer such that

a 0 + a 1 10 + a 2 10 2 + ⋯ + a k 10 k ≤ x . ( ∗ )

The procedure terminates when a k is found such that equality holds in ( ∗ ) ; otherwise, it continues indefinitely to give an infinite sequence of decimal digits. It can be shown that x = sup k { ∑ i = 0 k a i 10 i } (conventionally written as x = a 0 . a 1 a 2 a 3 ⋯ ), with a 0 ∈ N 0 and a 1 , a 2 , a 3 … ∈ { 0 , 1 , 2 , … , 9 } . This construction is extended to x < 0 by applying the above procedure to − x and denoting the resultant decimal expansion by − a 0 . a 1 a 2 a 3 ⋯ .

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 = ∑ i = 0 n a i 10 i = ∑ i = 0 n 10 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 = p 2 n 5 m = 2 m 5 n p 2 n + m 5 n + m = 2 m 5 n p 10 n + m for some p. While x is of the form p 10 k , p = ∑ i = 0 n 10 i a i for some n. By x = ∑ i = 0 n 10 n − i a i / 10 n = ∑ i = 0 n a i 10 i , x will end in zeros.

Recurring decimal representations

Some real numbers have decimal expansions that eventually get into loops, endlessly repeating a sequence of one or more digits:

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). Also the converse is true: The decimal expansion of a rational number is either finite, or endlessly repeating.