Getting a specific digit from a ratio expansion in any base (nth digit of x/y)

Question

Is there an algorithm that can calculate the digits of a repeating-decimal ratio without starting at the beginning?

I\'m looking for a solution that doesn\'t use ar

Answer 1

Ad hoc I have no good idea. Maybe continued fractions can help. I am going to think a bit about it ...

UPDATE

From Fermat's little theorem and because 39 is prime the following holds. (= indicates congruence)

10^39 = 10 (39)

Because 10 is coprime to 39.

10^(39 - 1) = 1 (39)

10^38 - 1 = 0 (39)

[to be continued tomorow]

I was to tiered to recognize that 39 is not prime ... ^^ I am going to update and the answer in the next days and present the whole idea. Thanks for noting that 39 is not prime.

The short answer for a/b with a < b and an assumed period length p ...

• calculate k = (10^p - 1) / b and verify that it is an integer, else a/b has not a period of p
• calculate c = k * a
• convert c to its decimal represenation and left pad it with zeros to a total length of p
• the i-th digit after the decimal point is the (i mod p)-th digit of the paded decimal representation (i = 0 is the first digit after the decimal point - we are developers)

Example

a = 3
b = 7
p = 6

k = (10^6 - 1) / 7
  = 142,857

c = 142,857 * 3
  = 428,571

Padding is not required and we conclude.

3     ______
- = 0.428571
7