Expanding rational numbers in different bases

Take a look at these amazing facts:

• 0588235294117647 x 7 = 4117647058823529, the same digits in the same sequence, but starting at a different point

• The same happens if you multiply 0588235294117647 by any integer!

• The number x = 076923 has a different behaviour: 2x = 153846, 3x = 230769, 4x = 307692, 5x = 384615, 6x = 461538, 7x = 538461, 8x = 0.615384, 9x = 0.692307, 10x = 769230, 11x = 8461543, 12x = 923076: you get either a cyclic permutation of 076923 or of 153846

• When you expand 1/7 in base 10, the result recurs after 6 digits, 1/17 recurs after 16 digits but 1/13 recurs after 6 digits, not 12

• If you do the same exercise in base 8, 1/7 recurs after 1 digit, 1/17 after 8 and 1/13 after 4 digits

Is there a connection between these facts? Hint: try expanding 1/17 and 1/13 in base 10!

Why do the recurrence lengths of the same fraction differ in different bases?

The downloadable document offers suggestions for exploration along these lines, together with useful links.

There are some files you can download for free here, which will help you experiment more with such numbers!