1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

1 is by definition 0.999999999 9?

  1. Jun 22, 2012 #1
    Hi there,

    I have a question regarding this statement:

    35jvtvt.png

    My question is whether we can say so...

    Thank you very much!
     
  2. jcsd
  3. Jun 22, 2012 #2
    OP, the answer is that .999... = 1. It's an equality. They're two expressions that represent the same number.

    The reason this is so is that .999... is defined as the infinite sum

    9/10 + 9/100 + 9/1000 + ...

    This is a geometric series whose sum is 1. This is proven in first-year calculus.

    Another way to see it is that there's no distance between the number denoted by .999... and the number denoted by 1. That is, suppose you say, well, .999... is 1/zillion away from1. But I'll just point out that if you take enough 9's, you'll eventually get WITHIN 1/zillion of 1.

    So if there's no conceivable positive difference between .999... and 1, then they must represent the same number.

    Possible conceptual objections to this reasoning are things like:

    * "But how can you have two different expressions for the same number?" Easy. 4 and 2 + 2 are two different expressions for the same number. It happens all the time.

    * There must be an "infinitesimal" difference between 1 and .999..." In the standard real number system, there are no infinitesimals. A distance is either zero or positive. Since there's no positive distance between .999... and 1, the distance between them is zero and they're the same number.

    Hope this helps. There are discussions of this topic all over the net.
     
    Last edited: Jun 22, 2012
  4. Jun 22, 2012 #3
    There is already a thread about this. Please visit the Frequently Asked Questions subforum.
     
  5. Jun 22, 2012 #4

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Do you realize the question you ask in your post and the question you ask in the title are quite different?
     
  6. Jun 22, 2012 #5

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Please read this: https://www.physicsforums.com/showthread.php?t=507001 [Broken]
     
    Last edited by a moderator: May 6, 2017
  7. Jun 22, 2012 #6
    Many thanks to all of you for reply. Everything is clear now.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: 1 is by definition 0.999999999 9?
  1. F(x) = 1/(x+1) - 2/9 (Replies: 3)

  2. 6÷2(1+2) = 1 or 9? (Replies: 4)

Loading...