1. Joined
    25 Aug '06
    Moves
    0
    05 Mar '08 22:51
    Prove:

    0/2 + 0/4 + 0/8 + 1/16 + 1/32 + 0/64 + 1/128 + ... = 1/9

    Where the denominators are the powers of 2, and each numerator equals the number of odd digits in the denominator.
  2. Joined
    23 Oct '07
    Moves
    2831
    06 Mar '08 15:57
    i take your word for itπŸ˜‰
  3. Standard memberadam warlock
    Baby Gauss
    Ceres
    Joined
    14 Oct '06
    Moves
    18375
    06 Mar '08 16:33
    Originally posted by David113
    Prove:

    0/2 + 0/4 + 0/8 + 1/16 + 1/32 + 0/64 + 1/128 + ... = 1/9

    Where the denominators are the powers of 2, and each numerator equals the number of odd digits in the denominator.
    I really want to see how this one is done. I tried for a little bit but couldn't get much done and for the moment I'm not seeing how it can be done either.
  4. Sigulda, Latvia
    Joined
    30 Aug '06
    Moves
    4048
    06 Mar '08 18:221 edit
    I think it has something to do with number divisibility with 9. But I can't remember what properties were required for the number to be divisible with 9. Eh, that was just a wild guess.

    P.s. And, yeah, I'd also like to see this done.
  5. Joined
    11 Nov '05
    Moves
    43938
    06 Mar '08 18:592 edits
    Originally posted by David113
    Prove:

    0/2 + 0/4 + 0/8 + 1/16 + 1/32 + 0/64 + 1/128 + ... = 1/9

    Where the denominators are the powers of 2, and each numerator equals the number of odd digits in the denominator.
    1/9, you say it is 0/2 + 0/4 + 0/8 + 1/16 + 1/32 + 0/64 + 1/128 + ... = 1/9
    it is actually not so.
    The correct is 0/2 + 0/4 + 0/8 + 1/16 + 1/32 + 1/64 + 0/128 + ... = 1/9
    or if we show only the indices = 000 111 000 111 ...

    Go to the binary system, 1/9 (10) = 1/1001 (2), do the division 1/1001 and you get the result .000111000111... (a pattern of 3 zeroes followed by 3 ones and again and again).

    Just simple arithemtics.

    If you want a proof that .000111000111... is really 1/1001 binary = 1/9 decimal, here it comes, again in binary arithemtics:

    say that x = .000111000111...
    then 1000000x = 111.000111...
    and 1000000x-x = 111 exactly.

    Now we safely can go back to decimal arithemtics again.
    64x-x = 63x = 7
    x = 7/63 = 1/9

    And voilà we've shown that 1/9 is really .000111000111...
  6. Standard memberadam warlock
    Baby Gauss
    Ceres
    Joined
    14 Oct '06
    Moves
    18375
    06 Mar '08 19:03
    Originally posted by FabianFnas
    1/9, you say it is 0/2 + 0/4 + 0/8 + 1/16 + 1/32 + 0/64 + 1/128 + ... = 1/9
    it is actually not so.
    The correct is 0/2 + 0/4 + 0/8 + 1/16 + 1/32 + 1/64 + 0/128 + ... = 1/9
    or if we show only the indices = 000 111 000 111 ...

    Go to the binary system, 1/9 (10) = 1/1001 (2), do the division 1/1001 and you get the result .000111000111... (a pattern of 3 ...[text shortened]... arithemtics again.
    64x-x = 63x = 7
    x = 7/63 = 1/9

    And voilà We've shown that 1/9 is realy
    πŸ˜•
  7. Subscriberjoe shmo
    Strange Egg
    podunk, PA
    Joined
    10 Dec '06
    Moves
    7733
    06 Mar '08 19:11
    Originally posted by FabianFnas
    1/9, you say it is 0/2 + 0/4 + 0/8 + 1/16 + 1/32 + 0/64 + 1/128 + ... = 1/9
    it is actually not so.
    The correct is 0/2 + 0/4 + 0/8 + 1/16 + 1/32 + 1/64 + 0/128 + ... = 1/9
    or if we show only the indices = 000 111 000 111 ...

    Go to the binary system, 1/9 (10) = 1/1001 (2), do the division 1/1001 and you get the result .000111000111... (a pattern of 3 ...[text shortened]... arithemtics again.
    64x-x = 63x = 7
    x = 7/63 = 1/9

    And voilà We've shown that 1/9 is realy
    The pattern that he is asked for is different than the one you have shown, the series for the numerators according to his stipulations are

    0,0,0,1,1,0,1,1,2,1,0,1,2,2,2,3,4,1,1,3,4,3,1,5,5,2,5,2 for the first 28

    that doesnt look like 0,0,0,1,1,1,0,0,0,1,1,1..........
  8. Joined
    25 Aug '06
    Moves
    0
    06 Mar '08 19:231 edit
    Originally posted by FabianFnas
    1/9, you say it is 0/2 + 0/4 + 0/8 + 1/16 + 1/32 + 0/64 + 1/128 + ... = 1/9
    it is actually not so.
    The correct is 0/2 + 0/4 + 0/8 + 1/16 + 1/32 + 1/64 + 0/128 + ... = 1/9
    or if we show only the indices = 000 111 000 111 ...

    Go to the binary system, 1/9 (10) = 1/1001 (2), do the division 1/1001 and you get the result .000111000111... (a pattern of 3 ...[text shortened]... .
    64x-x = 63x = 7
    x = 7/63 = 1/9

    And voilà we've shown that 1/9 is really .000111000111...
    The fact that there is another series whose sum is 1/9 does not mean that the sum of "my" series can't also be 1/9.

    Your series is the only one in which the denominators are 2,4,8,... and whose sum is 1/9 IF EACH NUMERATOR IS 0 OR 1.

    This doesn't mean that if the numerators are not restricted to being 0 or 1 there isn't another series whose sum is 1/9.
  9. Standard memberadam warlock
    Baby Gauss
    Ceres
    Joined
    14 Oct '06
    Moves
    18375
    06 Mar '08 19:29
    Originally posted by David113
    The fact that there is another series whose sum is 1/9 does not mean that the sum of "my" series can't also be 1/9.

    Your series is the only one in which the denominators are 2,4,8,... and whose sum is 1/9 IF EACH NUMERATOR IS 0 OR 1.

    This doesn't mean that if the numerators are not restricted to being 0 or 1 there isn't another series whose sum is 1/9.
    A hint on how it's done. Puhlease. Nothing too giving just a little hint! I love solving this type of things.
  10. Joined
    25 Aug '06
    Moves
    0
    06 Mar '08 19:41
    Originally posted by adam warlock
    A hint on how it's done. Puhlease. Nothing too giving just a little hint! I love solving this type of things.
    I'll wait a few days before giving a hint... maybe someone will solve it?
  11. Standard memberadam warlock
    Baby Gauss
    Ceres
    Joined
    14 Oct '06
    Moves
    18375
    06 Mar '08 19:44
    Originally posted by David113
    I'll wait a few days before giving a hint... maybe someone will solve it?
    how much number theory notion is needed? Can't you just tell what branch of math is more used? 😡
  12. Joined
    03 Mar '07
    Moves
    591
    07 Mar '08 01:532 edits
    Originally posted by David113
    Prove:

    0/2 + 0/4 + 0/8 + 1/16 + 1/32 + 0/64 + 1/128 + ... = 1/9

    Where the denominators are the powers of 2, and each numerator equals the number of odd digits in the denominator.
    wait. I'm confused. If you add anything positive to 1/2, it goes up. Since 1/9 is less than 1/2, it's mathematically impossible, isn't it?
  13. Subscriberjoe shmo
    Strange Egg
    podunk, PA
    Joined
    10 Dec '06
    Moves
    7733
    07 Mar '08 04:361 edit
    Originally posted by kayakninja
    wait. I'm confused. If you add anything positive to 1/2, it goes up. Since 1/9 is less than 1/2, it's mathematically impossible, isn't it?
    where do you see 1/2 in that series? what you should be seeing is 0/2 which = 0 ,
  14. Subscriberjoe shmo
    Strange Egg
    podunk, PA
    Joined
    10 Dec '06
    Moves
    7733
    07 Mar '08 04:40
    Are there any 0's in the numerator after 2^11?
  15. In Christ
    Joined
    30 Apr '07
    Moves
    172
    07 Mar '08 05:44
    I don't know if this is a good way to go about it, but I'd like to present a different viewing angle to the problem.

    In any doubling series:

    A digit is even if the digit immediately after it in the previous number is 0, 1, 2, 3, or 4. [Since the digit is doubled, and doesn't carry a one, the digit must become even].

    A digit is odd if the digit immediately after it in the previous number is 5, 6, 7, 8, or 9. [The digit is doubled, and it carries a one, so it must become odd].

    Therefore the number of odd digits in a power of 2 is equal to the number of digits greater than 4 in the previous number. Maybe that can help ('?'πŸ˜‰
Back to Top