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A Maths Puzzle...

Discussion in 'Off Topic [BG]' started by Tituscrow, Jul 19, 2012.


  1. Tituscrow

    Tituscrow Banned

    Feb 14, 2011
    NW England
    ..well, not so much a puzzle, more something that seems counter-intuitive yet logically bombproof. A prize for anyone who shoots a hole in it.

    THREE WAYS TO PROVE 0.999(rec) = 1

    1. Switching between fractions and decimals,
    1/3 = 0.333(rec). Multiply both sides by 3 and you get 1 = 0.999(rec)

    2. Take 1 - 0.999(rec) = 0.000(rec). Therefore the distance between 1 and 0.999(rec) is nothing, so they must be the same.

    3. Let x = 0.999(rec). Multiply both sides by 10 to get 10x = 9.999(rec). Next, subtract the first equation from the second equation to get 9x = 9.
    Therefore x = 1.

    What am I missing (other than something better to do at this time of night) ?
     
  2. MatticusMania

    MatticusMania LANA! HE REMEMBERS ME!

    Sep 10, 2008
    Pomona, SoCal
    Beer.
     
  3. fdeck

    fdeck Supporting Member Commercial User

    Mar 20, 2004
    Madison WI
    HPF Technology LLC
    If somebody really wants to believe that 0.999.. != 1, then no number of proofs will help.
     
  4. Tituscrow

    Tituscrow Banned

    Feb 14, 2011
    NW England
    Yep. Fridge empty. Just half a bottle of Mrs Crows cider, and I have to be up early :)
     
  5. MatticusMania

    MatticusMania LANA! HE REMEMBERS ME!

    Sep 10, 2008
    Pomona, SoCal
    Im stocking up tonight for my weekend at the river. I, too, must be up early tomorrow.
     
  6. rr5025

    rr5025

    Nov 12, 2008
    Wouldn't that just equal 1 x 10^-whatever?
     
  7. Tituscrow

    Tituscrow Banned

    Feb 14, 2011
    NW England
    No. Its 1 subtract a number almost as big as 1, which must equal almost nothing.

    I think.

    Matticus, have a swig on my behalf my friend :)
     
  8. FretlessMainly

    FretlessMainly

    Nov 17, 2010
    I use math all day; what the heck does (rec) mean?

    Looked at another way, when buying a pint, they're equal; when building an airplane (aeroplane for you lot), they're not.
     
  9. Tituscrow

    Tituscrow Banned

    Feb 14, 2011
    NW England
    (rec) means recurring. Can't work out how to put the little dot above the third digit.
     
  10. FretlessMainly

    FretlessMainly

    Nov 17, 2010
    1/3 does not equal 0.333(rec), so your premise is incorrect. 0.333(rec) approaches 1/3 in the same way that 0.999(rec) approaches 1.
     
  11. seventhson

    seventhson Supporting Member

    Aug 12, 2005
    Seattle, WA
    IIRC, the proof involves a premise imagining that there exists some x that is smaller than 1 - .999...

    And that such an premise leads to a conclusion that is know to be false. Therefore the premise that such an x exists must also be false. Or something like that.
     
  12. FretlessMainly

    FretlessMainly

    Nov 17, 2010
    Makes sense. An asymptotic philosophy as it were. It's not really a math problem, it's more of a philosophical debate in some respects (although mathematical proofs do venture more than a little bit into the realm of philosophy).

    Basically, is an infinite number of 9's after the decimal is the same thing as 1? Literally, the answer is no, but practially, the answer is yes.
     
  13. seventhson

    seventhson Supporting Member

    Aug 12, 2005
    Seattle, WA
    If it's not exactly 1, then there would exist an X != 0 such that .999... + X = 1. Since such an X does not exist, .999... = 1 exactly.

    edit: i'm dumb, here's a smarty pants with the answer...

    http://polymathematics.typepad.com/polymath/2006/06/no_im_sorry_it_.html
     
  14. 0.999... is a repeating decimal number or an "irrational" number. To convert this to a "rational" number, you do the third problem.

    3. Let x = 0.999(rec). Multiply both sides by 10 to get 10x = 9.999(rec). Next, subtract the first equation from the second equation to get 9x = 9.
    Therefore x = 1.


    A rational number can be put in the form a/b. So, any "irrational" number can be converted into a rational to be put in the a/b form.

    eq1: x=0.666...
    eq2: 10x=6.666...

    subtract eq1 from eq2 gives 9x=6
    solve for x x=6/9
    reduce fraction x=2/3
     
  15. How about: none of this actually 'exists' anyway, except in your mind.
    That help?

    : D
     
  16. BassyBill

    BassyBill The smooth moderator... Gold Supporting Member

    Mar 12, 2005
    West Midlands UK
    It's amazing how we higher primates are fixated on the number 10 and the decimal system. This conumdrum is a great example. Try converting everything to base 9 and the problem goes away. ;)
     

  17. a repeating decimal is a RATIONAL number.

    it is rational because it can be written as the ratio of two integers...

    in this case,eg, 999/1000= .999 or 99,999/100,000=.99999

    davesignatureII-1.
     
  18. BassyBill

    BassyBill The smooth moderator... Gold Supporting Member

    Mar 12, 2005
    West Midlands UK
    Correct.

    Or, +1. :D
     
  19. Evil Undead

    Evil Undead

    Oct 31, 2009
    +1

    0.9999rec will never be equal to 1. Otherwise it would just be called 1.
     
  20. BassyBill

    BassyBill The smooth moderator... Gold Supporting Member

    Mar 12, 2005
    West Midlands UK
    Hmmmm... did you see the exclamation mark? ;)
     

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