A repeating decimal such as 0.142857142857... is necessarily equal to a rational number, a ratio of two integers. Click here for the proof. However, going on with the example, consider 0.142857142857... which is equal to 1/7. Multiplying by this repeating decimal is equivalent to dividing by 7. The digit sum of the quotient for dividing a number by 7 is equal to the digit sum of the product of that number by 4. Therefore the digit sum of the repeating decimal 0.142857142857... can be defined as being equal to 4, even though that value cannot be computed from the definition of digit sum. Consider 19/7. The quotient is 2+5/7 and its digit sum would be 2+5*DigitSum(1/7). With the DigitSum of 1/7 this would give 2+5*4=22, whose digit sum is 4. The digit sum of 19 is 1 and this multiplied by 4 gives 4.

This logic can be extended to 1/14. The digit sum of 1/14 should be one half of the digit sum of 1/7 or 4/2=2. But the digit sum of 1/14 should also be the digit sum of 1/DigitSum(14) or 1/5. The digit sum of 1/5=0.2 is 2.

So at least some non-terminating but repeating decimal do have well defined digit sums. In fact any fraction such as 1/n has a digit sum equal to the multiplicative inverve of DigitSum(n) providing DigitSum(n) is not a multiple of 3. For example, DigitSum(1/13)=DigitSum(1/4)=7. Consider 130/13=10. The digit sum of the quotient is 1. The digit sum of the dividend is 4 and 4 multiplied by 7 (the equivalent of 1/4) is 4*7=28, whose digit sum is 1.

Consider 1/17. This quotient is a nonterminating decimal but the digit sum of 1/17 is the same as the digit sum of 1/8 which is 8. Thus 35/17=2+1/17. The digit sum of this is 2+8=10, whose digit sum is 1. The digit sum of the dividend is 8 and this multiplied by 8, the digit sum of 1/8, is the digit sum of 8*8=64 which is 1.

For another example take 1/11. The digit sum of this is the digit sum of 1/2 which is 5. Consider 46/11=4+2/11. The digit sum of this is 4+2*5=14, whose digit sum is 5. On the other hand the digit sum of the dividend is 1 and this multiplied by 5 is 5.

The digit sums of 1/3, 1/6 and 1/9 cannot be defined consistently. Aside from these exceptions however the digit sum of other fractions can be defined consistently. The digit sum of 5/13 would be the digit sum of 5*7, which is 8.

The nonexistence of an equivalent Digitsum for 1/3=0.333…, 1/6=0.1666… and 1/9=0.111… can be justified by the fact that Digitsum arithmetic is equivalent to arithmetic modulo 9. Modulo 9 arithmetic is the arithmetic of remainders after division by nine. In modulo 9 arithmetic 9 is equivalent to 0. For 9 to have a multiplicative inverse is equivalent to 0 having an inverse. The fact that 0*n=0 for any number n is equivalent to Digitsum(9*n)=9 for any number n.

Since 3*3=9, 3 is in the nature of the square root of zero. Thus if 3 were to have a multiplicative inverse then its square would be the multiplicative inverse of 9, which cannot exist. Since 6=2*3, if 6 were to have a multiplicative inverse then half of it would be the multiplicative inverse of 3, which from the above cannot exist. Thus there can be no Digitsum representations for the repeating decimals 0.333…, 0.1666… and 0.111…

Some mathematicians formulated a mathematical structure called the extended real numbers which involved adjoining +∞ and −∞ to the real numbers and thus defining 1/0 as ±∞. This however results in the multiplication funcion not being single-valued because ∞*0 can be any number. And likewise the addition function is not necessarily single-valued either because ∞−∞=∞+(−∞) can be any number. The loss of single-valuednesss is probably more of a mathematical inconvenience than the nonexistence of a multiplicative inverse for 0.

Proof that a Repeating Decimal is the Representation of a Rational Number

Let x be of the form 0.qqqqq… where q is a sequence of n digits. For example q might equal 142857 and x would be 0.142857142857…. This means that x is equal to (q/10ⁿ)+(q/10²ⁿ)+… and hence

x = (q/10ⁿ)[1 + (1/10ⁿ)+ (1/10²ⁿ)+ …]
and since the expression in the brackets is a geometric series
x = (q/10ⁿ)[1/(1−(1/10ⁿ)] = (q/10ⁿ)/(1−(1/10ⁿ)]
and with multiplication of the numerator
and denominator by 10ⁿ this reduces to
x = q/(10ⁿ−1)
 

This is a ratio of two integers and thus x is a rational number. The form q/(10ⁿ−1) may, however, not be in the simplest form. For example in the case of q=142857, x = 142857/999999 = 1/7.

Consider the repeating decimal representation of 1/13, which is 0.076923076923…. In this case q=076923 and the fractional representation, as derived above, is x = 76923/999999, which reduces to 1/13.

The surprising element of the above is that the integer 999999 has such an interesting factorization. It is 7*11*13*111.

(To be continued.)