Decimal expansion and notation

Decimal expansion and notation

##Decimal expansion and notation

###Background

For David's response on: 

http://math.stackexchange.com/questions/1371400/four-mathematical-notations-for-fraction-1-999-and-how-to-show-present-they-ar/1372077#1372077 

I want to explain background of the numerical musings, which I was already pondering on other topics like: 

http://math.stackexchange.com/questions/1365506/doubling-sequences-of-the-cyclic-decimal-parts-of-the-fraction-numbers

http://math.stackexchange.com/questions/1368968/how-to-notate-all-integers-gt-1-except-products-of-2-3-5

http://math.stackexchange.com/questions/1369596/are-there-more-convenient-ways-of-getting-the-number-of-digits-of-a-positive-int

First I looked up reciprocals of 7-rough numbers up to 100:

**1/7 142857**
1/11 09
1/13 076923
1/17 0588235294117647
1/19 052631578947368421
1/23 0434782608695652173913
1/29 0344827586206896551724137931
1/31 032258064516129
1/37 027
1/41 02439
1/43 023255813953488372093
1/47 0212765957446808510638297872340425531914893617
**1/49 020408163265306122448979591836734693877551**
1/53 0188679245283
1/59 0169491525423728813559322033898305084745762711864406779661
1/61 016393442622950819672131147540983606557377049180327868852459
1/67 014925373134328358208955223880597
**1/71 01408450704225352112676056338028169**
1/73 01369863
1/77 012987
1/79 0126582278481
**1/83 01204819277108433734939759036144578313253**
**1/89 01123595505617977528089887640449438202247191**

noticing some of the sequences were doubling, tripling, quadring, hexing and such, but some of them seemed to be arbitrary. So I wanted to know, what is going on behind. I decided to look up numbers up to 1000:

1/989 001011122345803842264914054600606673407482305358948432760364004044489383215369059656218402426693629929221435793731041456016177957532861476238624873609706774519716885743174924165824064711830131445904954499494438827098078867542972699696663296258847320525783619817997977755308392315470171890798786653185035389282103134479271991911021233569261880687563195146612740141557128412537917087967644084934277047522750252780586450960566228513650151668351870576339737108190091

1/991  001009081735620585267406659939455095862764883955600403632694248234106962663975782038345105953582240161453077699293642785065590312815338042381432896064581231079717457114026236125126135216952573158425832492431886982845610494450050454086781029263370332996972754793138244197780020181634712411705348133198789101917255297679112008072653884964682139253279515640766902119071644803229061553985872855701311806256306760847628657921291624621594349142280524722502522704339051463168516649848637739656912209889

1/997 0010030090270812437311935807422266800401203610832497492477432296890672016048144433299899699097291875626880641925777331995987963891675025075225677031093279839518555667

Then I looked not only reciprocals of 7-rought numbers:

**1/999 001001001...**

**1/99 010101...**

**1/3 333333...**


### Realizing the pattern

And I realized there is an arithmetic pattern indeed. Every sequence starts with an exponent of 1 and depends on 10th multitude. Calculation would be something like this: 

For $1/x$ take the full power of x (pardon my inability to phrase this well) and subtract x from it. for $1/9$ it means 9 -> 10 so $10 - 9 = 1$. Or for $1/49$ it means 49 -> 100 so $100 - 49 = 51$. Or for $1/997$, $1000 - 997 = 3$. Now take the result (r) and start adding it likewise: $r^k/full product^k+1$. For $1/49$ it means $(100-49)^k / 100^k+1$. And so on for any reciprocal prime number. Eventually I found that number can be any whole number, not just prime or 7-rough.

### Generalization

So that's how I ended up with the complicated summation: $$\sum_{k=1}^{\infty} \frac{(10^{\lfloor\strut\log_{10}(p)\rfloor+1} - p)^{k-1}}{(10^{\lfloor\strut\log_{10}(p)\rfloor+1})^k}$$

It doesn't make much sense at first, but it explains arbitrary sequences, how they evolve from simple start of 1. For example fibonacci sequence on $1/89$ is actually:

$100-89^n / 100^n+1 => 11^0/100^1 + 11^1/100^2 + 11^2/100^3 + ... => 1/100 + 11/10000 + 121/1000000 + ... => 0.01123595505617977528089887640449438202247191$

Also sequence of $1/7$ would be ($10-7=3$):

$3^0 / 10^1 + 3^1 / 10^2 + 3^2 / 10^3 + ... => 1/10 + 3/100 + 9/1000 + ... => 0.142857...$

Or the arbitrary looking $1/17$ ($100-17=83$):

$83^0 / 100^1 + 83^1 / 100^2 + 83^2 / 100^3 + ... => 1/100 + 83/10000 + 6889/100000 + ... => 0.0588235294117647...$

Don't you think complicated summation on OT part d on StackExchange is still somewhat useful after this? I think it can argue arbitrary sequences of the decimal expansion and show how simple addition arithmetics causes some of the interesting sequences found like $1/7,  1/49,  1/89$ and such. And probably summation works for any base just by replacing 10 with b. And like you showed, not only reciprocals, but any $integer (x)/integer (y)$, if you prefix x before summation notation.

While "special" reciprocals in 10 base like x/2, x/5, x/3 have special decimal expansion, and they have more attracting summation method as you stated on your answer, they still fit to this, say more generic addition method.


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