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]

That’s really cool.

Isn’t it!? =)

I often play this with algebra classes… lots of material, lots to think and talk about, very very cool.

The length of the repeating unit is less than or equal to one less than the denominator. That’s cool.

Besides telling us which fractions will terminate, the factors of the denominators can tell us lots about other aspects of the expansions. The 22nds behave like the 11ths. And look at those 7ths and 21sts…

I love them getting it through long division, but calculator kids can appreciate almost as much.

Not only is it always less than or equal to d-1; for prime denominators, it’s a divisor of d-1. (The quotient is equal to the number of different patterns–13, for example has 12/6 or two 6-digit patterns.)

Delayed repetition always indicates that the denominator has a factor of 2 or 5 in it, to the power of the number of digits it’s delayed.

Its crazy how many different patterns crop up from simple fractions

Brent, this is fascinating to see and to think about!

i think this is good because it really works. sooooooooo cool.

this helps alot

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omg!

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OMG!!

This help me sooo much!!

Thanks!

YAll nerds no effensce ^^^^^^^^^^^^^^^^^^^^^vvvvvvvvv i juss came here for h.w answers guess wat i aint find it so im tight….. anyway yyyy spend ya time doin all diss shiz yyy not go do fun stuff instead of bein math nerds no effensce im not insulting any way… but i do agree that the patterns look cute srry if any one got hurt buhh i like to speak mii feelins out

omg thanks soooo much i’m doing this thing called mathathon 4 st. judes hospital for school. . . This is helping allloootttt!!!!!!!!!!!!!!!!!