Decimal Expansion of 1/7 -
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From: JEMebius on 15 May 2010 19:04 Hein wrote: > Most people look at the first four digits of 1/7 = .1428... and either > don't notice the appearance of multiples of 7 (2 times 7 and 4 times > 7) in the decimal representation or think that their existence in the > representation is a mere coincidence. It's not. Let's compute the > decimal expansion of 1/7 without doing long division, but using > instead the identity (***) > > x/(1-x) = x + x^2 + x^3 + ... > > (Note the identity holds if x is between -1 and 1.) > > > 1/7 > = 7/49 > = 7*1/(50-1) > = 7*1/50/(1-1/50) > = 7*.02/(1-.02) (now use *** with x=.02) > = 7*(.02 + .02^2 + .02^3 + ...) > = 7*(.02 + .0004 + .000008 + .00000016 + .0000000032 + ...) > = 7*(.02040816326...) > = 7*(.020408) + 7*(.00000016326...) > = .142856 + 7*(.00000016) + 7*(.00000000326...) > = .142856 + (.00000112) + 7*(.00000000326...) > > So, 2*7=14 and 4*7=28 are the first 4 digits because 7*.0204 = . > 1428. > > Also, notice that the first 6 digits of 1/7 are > > .142856 + .000001= .142857. > > For any positive integer j, the decimal representation of 1/j either > terminates or repeats with at most (j-1) repeating digits and we have > six (7-1) digits, so those six digits must repeat, so finally 1/7 must > be > > 1/7 = .142857142857142857142857142857142857.... > > > Cheers, > Hein Hundal Quotation: "For any positive integer j, the decimal representation of 1/j either terminates or repeats with at most (j-1) repeating digits (...)" Not true in general: make a careful study of http://en.wikipedia.org/wiki/Repeating_decimal or still better: spend several feet or yards of scrap paper in doing several different long divisions. For starters: 1/13, 1/27, 1/37, 1/41, 1/49, 1/53, 1/73, 1/81, 1/137, 1/239, 1/243. In my opinion the nicest are 1/487 and 1/487^2, both of which repeat after 486 digits. I myself discovered in my boyhood years that not only 1/7, but also 1/13, 1/21, 1/39, 1/63, and ultimately 1/999999 have repeating decimal expansions of period 6. A revelation was the factorization 999999 = 3^3 . 7 . 11 . 13 . 37 With this factorization in mind one easily finds all expansions of 1/Q of period 6. Enjoy! Johan E. Mebius
From: James Waldby on 15 May 2010 19:25
On Sun, 16 May 2010 00:04:11 +0100, JEMebius wrote: > Hein wrote: [snip nice derivation of 1/7 vs 0.(142857) ] [snip quotation shown below] > Quotation: > "For any positive integer j, the decimal representation of 1/j either > terminates or repeats with at most (j-1) repeating digits (...)" > > Not true in general: > make a careful study of http://en.wikipedia.org/wiki/Repeating_decimal > or still better: spend several feet or yards of scrap paper in doing > several different long divisions. > For starters: 1/13, 1/27, 1/37, 1/41, 1/49, 1/53, 1/73, 1/81, 1/137, > 1/239, 1/243. In my opinion the nicest are 1/487 and 1/487^2, both of > which repeat after 486 digits. Can you please state explicitly what you think is wrong with the quoted statement? Or perhaps provide a value of j that illustrates what you think is wrong? The numbers you gave are in agreement with Hein's statement. For example, at j=13, 1/13 repeats with cycle length 6, which is not more than j-1 = 12; at j=243, 1/243 repeats with cycle length 27, which is not more than j-1 = 242; and similarly for all the numbers you mention. [snip period-6-examples paragraph] -- jiw |