[math-fun] a fraction for .000100040009001600250036... (which?)
The question (from Pacher Christoph): What rational fraction is equal to .000100040009001600250036...? can be answered easily using (ordinary) generating functions*. First look up the sequence 1,4,9,16,25,36 on OEIS: http://oeis.org/A000290 In the "FORMULA" section we find the (odinary) generating function: G.f.: x * (1 + x) / (1 - x)^3. Since we want each term to be 4 digits of the decimal expansion, let x equal 0.0001. x = 0.0001 x*(1+x) / (1-x)^3 = 0.00010001 / 0.999700029999 Then turn it into a reduced fraction of integers: 0.00010001 / 0.999700029999 = 100010000/999700029999 = (2^4*5^4*73*137)/(3^6*11^3*101^3) The prime factorization of 100010000 and 999700029999 shows that the fraction is in reduced form. Verifying the result, to 300 digits using the Unx tool bc: scale=300 100010000/999700029999 .0001000400090016002500360049006400810100012101440169019602250256028\ 90324036104000441048405290576062506760729078408410900096110241089115\ 61225129613691444152116001681176418491936202521162209230424012500260\ 12704280929163025313632493364348136003721384439694096422543564489462\ 44761490050415184532954765625 We get a two-digit version by letting x=0.01, which gives 10100/970299=0.0104091625364964... what I could not find there but it also quite well-known:
The decimal expansions of the multiples of 1/7 (and of inverses of so-called full reptend primes) are cyclic shifted versions of each other. There exists a simple card trick (which I found in a book of Martin Gardner) based on this fact.
Christoph
Integers whose reciprocal has N-1 digits are "the primes with primitive root 10", http://oeis.org/A001913 The card trick is here: http://www.ams.org/samplings/feature-column/fcarc-mulcahy2 (found with Google "142857 card trick") - Robert * Finding the ordinary generating function for an integer sequence is the inverse of the process of finding the a Taylor/Maclaurin series expansion for a function. It is a bit surprising that this is how you would answer this type of decimal fraction question, but there it is. -- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com
On Sat, Jan 28, 2012 at 10:23 PM, Robert Munafo <mrob27@gmail.com> wrote:
What rational fraction is equal to .000100040009001600250036...?
You can also sum the series n^2 / 10000^n in other ways (for example: repeatedly multiply by 10000 and subtract), if you want to make it accessible to people who don't know about generating functions (like the 10th graders I work with). What I'm trying to understand is the length of the period of these things. I mean, it is a list of squares, but eventually carries start messing things up, which intuitively seems like it would make it not a repeating decimal, but I'm staring at the fractional form of it! How can you account for the carries and explain how many digits it takes for these things to repeat by analyzing the sequence of squares rather than the denominators? --Joshua Zucker
Joshua, That would be nice, because I am pretty averse to generating functions too. So can you show all the steps so it is clear what you mean? What do you multiply by 10000 and what do you subtract from what? And how do you end up with the numerator and denominator 100010000 / 999700029999? Does your method work for any integer sequence? How much do I need to know about the definition of the integer sequence or how to generate it? - Robert On Sun, Jan 29, 2012 at 03:58, Joshua Zucker <joshua.zucker@gmail.com>wrote:
On Sat, Jan 28, 2012 at 10:23 PM, Robert Munafo <mrob27@gmail.com> wrote:
What rational fraction is equal to .000100040009001600250036...?
You can also sum the series n^2 / 10000^n in other ways (for example: repeatedly multiply by 10000 and subtract), if you want to make it accessible to people who don't know about generating functions (like the 10th graders I work with).
What I'm trying to understand is the length of the period of these things. I mean, it is a list of squares, but eventually carries start messing things up, which intuitively seems like it would make it not a repeating decimal, but I'm staring at the fractional form of it! How can you account for the carries and explain how many digits it takes for these things to repeat by analyzing the sequence of squares rather than the denominators?
--Joshua Zucker
-- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com
participants (2)
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Joshua Zucker -
Robert Munafo