Welcome to the ToeQuest.
Page 1 of 2 12 LastLast
Results 1 to 10 of 12

Thread: 0.999...=1

  1. #1
    Master
    Join Date
    Oct 2007
    Location
    United Kingdom
    Posts
    785
    Thanks Given
    0
    Thanked 1 Time in 1 Post
    Rep Power
    26

    Awards Showcase

    0.999...=1

    There have been a few comments regarding this in the shoutbox recently, so I thought I'd start a thread to discuss the topic.

    The question of whether 0.999..=1 is an age old problem, which I'm sure everyone encounters sometime in their life/education. The simple answer to the question is that, yes, they are the same, as they are simply two different decimal expansions which represent the same number. One can think of a decimal expansion as an infinite sum. For example, take the sum 0.999..=9/10+9/100+9/1000+... . This is a geometric series whose sum is one. Further, consider the decimal expansion for 1: 1=1+0/10+0/100+0/1000+... . This is also a geometric series whose sum is one.

    I think that once we look at the problem in terms of infinite series, it is not surprising that two infinite series add up to the same finite value.

    Any questions or comments are welcome.
    ~neutralino

    If you haven't found something strange during the day, it hasn't been much of a day - John A. Wheeler.

  2. #2
    Grandmaster
    Join Date
    May 2007
    Location
    United States
    Posts
    6,657
    Thanks Given
    836
    Thanked 1,049x in 746 Posts
    Rep Power
    105

    Re: 0.999...=1

    Well I'm glad you started the thread Neutralino.

    So are you concluding that infinity equals something finite?

  3. #3
    Master
    Join Date
    Oct 2007
    Location
    United Kingdom
    Posts
    785
    Thanks Given
    0
    Thanked 1 Time in 1 Post
    Rep Power
    26

    Awards Showcase

    Re: 0.999...=1

    Quote Originally Posted by Profpat View Post
    Well I'm glad you started the thread Neutralino.

    So are you concluding that infinity equals something finite?
    No, I'm concluding that a sum of infinitely many terms can be equal to a finite number. Take the example I briefly mentioned in the shoutbox:
    [math]\sum_{k+1}^{\infty}\frac{2}{k(k+1)}=1+\frac{1}{3}+ \frac{1}{6}+\frac{1}{10}+\frac{1}{15}+\cdots [/math]
    [math]=2\left[\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+ \cdots \right][/math]
    [math]=2\left[\left(1-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+\cdots\right][/math]
    [math]=2.[/math]

    So we see that a sum of infinitely many terms can indeed equal a finite number.
    ~neutralino

    If you haven't found something strange during the day, it hasn't been much of a day - John A. Wheeler.

  4. #4
    Grandmaster
    Join Date
    May 2007
    Location
    United States
    Posts
    6,657
    Thanks Given
    836
    Thanked 1,049x in 746 Posts
    Rep Power
    105

    Re: 0.999...=1

    OK take the decimal .99999999999999999... and carry it out to a trillion trillion googleplex decimal points where does it equal 1 ? At infinity ?

  5. #5
    Master
    Join Date
    Oct 2007
    Location
    United Kingdom
    Posts
    785
    Thanks Given
    0
    Thanked 1 Time in 1 Post
    Rep Power
    26

    Awards Showcase

    Re: 0.999...=1

    Quote Originally Posted by Profpat View Post
    OK take the decimal .99999999999999999... and carry it out to a trillion trillion googleplex decimal points where does it equal 1 ? At infinity ?
    Yes, the decimal with infinitely many 9's (which is what the notation 0.99.. (recurring) means) is equal to one.
    ~neutralino

    If you haven't found something strange during the day, it hasn't been much of a day - John A. Wheeler.

  6. #6
    Grandmaster
    Join Date
    May 2007
    Location
    United States
    Posts
    6,657
    Thanks Given
    836
    Thanked 1,049x in 746 Posts
    Rep Power
    105

    Re: 0.999...=1

    But infinity never ends, you'll always have a .9 never ending.

  7. #7
    Master
    Join Date
    Oct 2007
    Location
    United Kingdom
    Posts
    785
    Thanks Given
    0
    Thanked 1 Time in 1 Post
    Rep Power
    26

    Awards Showcase

    Re: 0.999...=1

    Quote Originally Posted by Profpat View Post
    But infinity never ends, you'll always have a .9 never ending.
    Yes, that is what 0.999.. means. Think of it another way: the number 0.9999 is quite close to 1, but then the number 0.9999999999 is even closer, and 0.999999999999999999999999999 is even closer than that. Now consider the number 0.99... with infinitely many nines. This is the closest possible number to 1, but since we are dealing with real numbers I should be able to find another number closer to 1 still *. But, by definition I cannot, since there are infinitely many 9's. Thus 0.99.. must be equal to 1.

    * There exists a real number between any two real numbers. To show this consider r<s. Then r<(r+s)/2<s.
    ~neutralino

    If you haven't found something strange during the day, it hasn't been much of a day - John A. Wheeler.

  8. #8
    Grandmaster
    Join Date
    May 2007
    Location
    United States
    Posts
    6,657
    Thanks Given
    836
    Thanked 1,049x in 746 Posts
    Rep Power
    105

    Re: 0.999...=1

    OK Lets use Nobody's fraction 1/9.

    Now 1/9~.1111and if you keep carrying this out never stoping you'll always have another.1 to get it a bit closer, but being infinite it's never going to end and at no point because there is no end point to infinity will it ever =.111... It will always be a very very close approximation.

  9. #9
    Grandmaster
    Join Date
    May 2007
    Location
    United States
    Posts
    6,657
    Thanks Given
    836
    Thanked 1,049x in 746 Posts
    Rep Power
    105

    Re: 0.999...=1

    I'm sorry Neutralino, I may be dense and I know I am but 1 will never equal, be the same as, or be identical to .999...

    But the very best to you sir,

    Pat

  10. #10
    9th degree Black Belt
    Join Date
    Jan 2007
    Location
    United States
    Posts
    1,941
    Thanks Given
    0
    Thanked 2x in 2 Posts
    Rep Power
    40

    Re: 0.999...=1

    I had a convo with Lloyd way back when about this, and we put it into particulate terms if I remember right. Would it be fair to infer that .999... can be representative of 1 composite particle consisting of an infinite number of point particles; and 1 representative of a 1 point particle? This way both would equal 1, but of a different type: the composite, infinite; and the point, absolute.

    I've always considered there to be an extremely fine line between infinite and absolute, and for this reason I feel there will always remain debateable points as to whether or not the larger is larger than the largest. It might be the cause for the quantum fluctuation in there somewhere.

    "To show this consider r<s. Then r<(r+s)/2<s."

    If r is 1 and s is 2, the result is 3/2, but I don't see how that implies 4/2. The infinite set taken as one set I understand though.

    Special thanks to you, Neutralino, for your help before.

 

 
Page 1 of 2 12 LastLast

Thread Information

Users Browsing this Thread

There are currently 1 users browsing this thread. (0 members and 1 guests)

Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •  
Back to top