I'd always been interested in this kind of number I had discovered. What you do is take any ordinary positive integer, most typical here is 7, and you look at all of its fractions under 1. That is, look at the fractions 1/7, 2/7, ... all the way up to 6/7. Expand those fractions, and then check out their decimal expansions.
Here, we see each respective decimal expansion in sequence. The most interesting thing here is that there are patterns to be found in each expansion. The pattern for the number 7 seems to be "142857". This pattern cyclically repeats itself all throughout each expansion for the number 7. The needed shift to match the cyclic pattern is also shown below, and the pattern being matched is indexed at the end of each line.
0.142857142857 , shift= 0,pattern#= 1,*
0.285714285714 , shift= -2,pattern#= 1,*
0.428571428571 , shift= -1,pattern#= 1,*
0.571428571429 , shift= 2,pattern#= 1,*
0.714285714286 , shift= 1,pattern#= 1,*
0.857142857143 , shift= 3,pattern#= 1,*
Some numbers have more than one pattern. For example, the number 8 has 7 unique patterns.
0.125 , shift= 0,pattern#= 1,*
0.25 , shift= 0,pattern#= 2,**
0.375 , shift= 0,pattern#= 3,***
0.5 , shift= 0,pattern#= 4,****
0.625 , shift= 0,pattern#= 5,*****
0.75 , shift= 0,pattern#= 6,******
0.875 , shift= 0,pattern#= 7,*******
However some numbers are a little more interesting. There are identifiable patterns in the pattern index itself! For example, check out number 11, where the stars at the end form a symmetric mountain of sorts.
0.09090909090909090909 , shift= 0,pattern#= 1,*
0.18181818181818181818 , shift= 0,pattern#= 2,**
0.27272727272727272727 , shift= 0,pattern#= 3,***
0.36363636363636363636 , shift= 0,pattern#= 4,****
0.45454545454545454545 , shift= 0,pattern#= 5,*****
0.54545454545454545455 , shift= 1,pattern#= 5,*****
0.63636363636363636364 , shift= 1,pattern#= 4,****
0.72727272727272727273 , shift= 1,pattern#= 3,***
0.81818181818181818182 , shift= 1,pattern#= 2,**
0.90909090909090909091 , shift= 1,pattern#= 1,*
And for number 13, we see again another symmetric pattern in the stars:
0.076923076923076923076923 , shift= 0,pattern#= 1,*
0.153846153846153846153846 , shift= 0,pattern#= 2,**
0.230769230769230769230769 , shift= 2,pattern#= 1,*
0.307692307692307692307692 , shift= 1,pattern#= 1,*
0.384615384615384615384615 , shift= 4,pattern#= 2,**
0.461538461538461538461538 , shift= 2,pattern#= 2,**
0.538461538461538461538462 , shift= 5,pattern#= 2,**
0.615384615384615384615385 , shift= 1,pattern#= 2,**
0.692307692307692307692308 , shift= 4,pattern#= 1,*
0.769230769230769230769231 , shift= 5,pattern#= 1,*
0.846153846153846153846154 , shift= 3,pattern#= 2,**
0.923076923076923076923077 , shift= 3,pattern#= 1,*
And perhaps, others have no pattern at all, such as for the number 37. And others, have some truly odd quirks, such as in 26 when you look at the expansions for 22/26 and 23/26. In fact, you see this a lot.
There's much we can learn and marvel at just by examining expansion sequences in this way. You can try it yourself with this python script that I developed, located at: http://pastebin.com/a5UY0F0i. To run the script, use "python krall_numbers.py 7" to run the test on the number 7. Simply replace the number 7 with any number you want to experiment with.
