In a comment on a previous post about rational numbers and decimal expansions, Steve Gilberg noted:

I’ve been fascinated at how any multiple of 1/7 that’s not an integer repeats the same digits in decimal expression, only starting at different points in the sequence:

1/7 = .142857…
3/7 = .428571…
2/7 = .285714…
6/7 = .857142…
4/7 = .571428…
5/7 = .714285…

I’ve rearranged the order of the fractions to make the pattern obvious.

This is pretty cool indeed, and probably well-known to many. But there’s nothing particularly special about 7, other than the fact that it is small. In order to talk about the sorts of patterns we find in the expansions of rational numbers—and why—I’m going to start by having a special exhibit: the Decimal Expansion Zoo! Have a look around and see if you notice any patterns. I’ll follow the convention of enclosing repeating portions in [square brackets]. So, for example, 0.23[48] means 0.234848484848… and so on.

Some questions for you to think about while you wander about the zoo: what patterns do you notice? Which decimal expansions terminate, and which are repeating—and how does it relate to the denominator? How are the different cycles for a given denominator related to each other, and why? How are the lengths of the cycles for a given denominator related to the denominator itself? I’ll answer all these questions, and more, in an upcoming post!

1/2 = .5                              1/8 = .125
                                      3/8 = .375
1/3 = .[3]                            5/8 = .625
2/3 = .[6]                            7/8 = .875

1/4 = .25                             1/9 = .[1]
3/4 = .75                             2/9 = .[2]
                                      3/9 = .[3]
1/5 = .2                              4/9 = .[4]
2/5 = .4                              5/9 = .[5]
3/5 = .6                              6/9 = .[6]
4/5 = .8                              7/9 = .[7]
                                      8/9 = .[8]
1/6 = .1[6]
5/6 = .8[3]                           1/10 = .1
                                      3/10 = .3
1/7 = .[142857]                       7/10 = .7
3/7 = .[428571]                       9/10 = .9
2/7 = .[285714]
6/7 = .[857142]                       1/12  = .08[3]
4/7 = .[571428]                       5/12  = .41[6]
5/7 = .[714285]                       7/12  = .58[3]
                                      11/12 = .91[6]
1/11  = .[09]
2/11  = .[18]                         1/13  = .[076923]
3/11  = .[27]                         10/13 = .[769230]
4/11  = .[36]                         9/13  = .[692307]
5/11  = .[45]                         12/13 = .[923076]
6/11  = .[54]                         3/13  = .[230769]
7/11  = .[63]                         4/13  = .[307692]
8/11  = .[72]                         2/13  = .[153846]
9/11  = .[81]                         7/13  = .[538461]
10/11 = .[90]                         5/13  = .[384615]
                                      11/13 = .[846153]
1/17  = .[0588235294117647]           6/13  = .[461538]
10/17 = .[5882352941176470]           8/13  = .[615384]
15/17 = .[8823529411764705]
14/17 = .[8235294117647058]           1/14  = .0[714285]
4/17  = .[2352941176470588]           3/14  = .2[142857]
6/17  = .[3529411764705882]           5/14  = .3[571428]
9/17  = .[5294117647058823]           9/14  = .6[428571]
5/17  = .[2941176470588235]           11/14 = .7[857142]
16/17 = .[9411764705882352]           13/14 = .9[285714]
7/17  = .[4117647058823529]
2/17  = .[1176470588235294]           1/15  = .0[6]
3/17  = .[1764705882352941]           2/15  = .1[3]
13/17 = .[7647058823529411]           4/15  = .2[6]
11/17 = .[6470588235294117]           7/15  = .4[6]
8/17  = .[4705882352941176]           8/15  = .5[3]
12/17 = .[7058823529411764]           11/15 = .7[3]
                                      13/15 = .8[6]
1/18  = .0[5]                         14/15 = .9[3]
5/18  = .2[7]
7/18  = .3[8]                         1/16  = .0625
11/18 = .6[1]                         3/16  = .1875
13/18 = .7[2]                         5/16  = .3125
17/18 = .9[4]                         7/16  = .4375
                                      9/16  = .5625
1/19  = .[052631578947368421]         11/16 = .6875
10/19 = .[526315789473684210]         13/16 = .8125
5/19  = .[263157894736842105]         15/16 = .9375
12/19 = .[631578947368421052]
6/19  = .[315789473684210526]         1/31  = .[032258064516129]
3/19  = .[157894736842105263]         10/31 = .[322580645161290]
11/19 = .[578947368421052631]         7/31  = .[225806451612903]
15/19 = .[789473684210526315]         8/31  = .[258064516129032]
17/19 = .[894736842105263157]         18/31 = .[580645161290322]
18/19 = .[947368421052631578]         25/31 = .[806451612903225]
9/19  = .[473684210526315789]         2/31  = .[064516129032258]
14/19 = .[736842105263157894]         20/31 = .[645161290322580]
7/19  = .[368421052631578947]         14/31 = .[451612903225806]
13/19 = .[684210526315789473]         16/31 = .[516129032258064]
16/19 = .[842105263157894736]         5/31  = .[161290322580645]
8/19  = .[421052631578947368]         19/31 = .[612903225806451]
4/19  = .[210526315789473684]         4/31  = .[129032258064516]
2/19  = .[105263157894736842]         9/31  = .[290322580645161]
                                      28/31 = .[903225806451612]
1/23  = .[0434782608695652173913]     3/31  = .[096774193548387]
10/23 = .[4347826086956521739130]     30/31 = .[967741935483870]
8/23  = .[3478260869565217391304]     21/31 = .[677419354838709]
11/23 = .[4782608695652173913043]     24/31 = .[774193548387096]
18/23 = .[7826086956521739130434]     23/31 = .[741935483870967]
19/23 = .[8260869565217391304347]     13/31 = .[419354838709677]
6/23  = .[2608695652173913043478]     6/31  = .[193548387096774]
14/23 = .[6086956521739130434782]     29/31 = .[935483870967741]
2/23  = .[0869565217391304347826]     11/31 = .[354838709677419]
20/23 = .[8695652173913043478260]     17/31 = .[548387096774193]
16/23 = .[6956521739130434782608]     15/31 = .[483870967741935]
22/23 = .[9565217391304347826086]     26/31 = .[838709677419354]
13/23 = .[5652173913043478260869]     12/31 = .[387096774193548]
15/23 = .[6521739130434782608695]     27/31 = .[870967741935483]
12/23 = .[5217391304347826086956]     22/31 = .[709677419354838]
5/23  = .[2173913043478260869565]
4/23  = .[1739130434782608695652]
17/23 = .[7391304347826086956521]
9/23  = .[3913043478260869565217]
21/23 = .[9130434782608695652173]
3/23  = .[1304347826086956521739]
7/23  = .[3043478260869565217391]