San José State University

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Thayer Watkins
Silicon Valley
USA

 The Sets of Repeating Decimals

This is an investigation of infinitely repeating decimals such as

#### 1/7 = 0.142857142857…

For convenience such sequences will be denoted as the repeating sequence underlined; e.g., 0.142857.

Only the numbers between 0 and 1 will be considered.

## Proposition: All such repreating decimals represent rational numbers

Let K be a sequence of decimal digits of length k considered as an integer. Then 0.K is an infinite sum

#### 0.K = K/10k + K/102k + … = K/10k(1 + 1/10k + 1/102k + …)

The value of the geometric series within the parentheses is 1/(1-10-k). Therefore

#### 0.K = (K/10k)(1/(1-10-k) = K/(10k−1)

Thus any repeating decimal is a rational number. The form K/(10k−1) may not give the rational number in its simplest form. For example, for K=142857 both 142857 and 999999 are divisible by 142857 so 0.142857=1/7.

Let p and q be such that

#### K/(10k−1) = p/q

where p is less than q.

Then

#### K = p(10k−1)/q

Now consider which values of K are such that p=1. This requires that q be a factor of (10k-1). Take the case of k=6. The number 999,999 has a prime factorization of 3³7·11·13·37. Thus 1/3, 1/7, 1/11, 1/13 and 1/37 have such a repeating form. Note that 1/3=0..33333333… can be considered as a repeition of 333333 as well as a repetition of 3. But also 1/9, 1/21, 1/33, 1/39, 1/111 and so on will have that form, but likewise with 1/27, 1/45 and so forth.

Any multiple of (10k−1)/q for such values of q generated as a product of the prime factors of (10k−1) will be a repeating decimal such as

#### 2/3 = 0.666666 2/7 = 0.285714

These multiples can only go up to (q-1).

## The Relevant Factors

Note that (10k−1) always has 9 as a factor. Therefore the crucial numbers for factorization are 111…111. The table below shows the factorization of integers to the base 10 which are sequences of 1.

 Number Factorization 1 1 11 11 111 3·37 1111 11·101 11111 41·271 111111 3·7·11·13·37 1111111 239·4649 11111111 11·73·101·137