Jump to content


Welcome to BrainDen.com - Brain Teasers Forum

Welcome to BrainDen.com - Brain Teasers Forum. Like most online communities you must register to post in our community, but don't worry this is a simple free process. To be a part of BrainDen Forums you may create a new account or sign in if you already have an account.
As a member you could start new topics, reply to others, subscribe to topics/forums to get automatic updates, get your own profile and make new friends.

Of course, you can also enjoy our collection of amazing optical illusions and cool math games.

If you like our site, you may support us by simply clicking Google "+1" or Facebook "Like" buttons at the top.
If you have a website, we would appreciate a little link to BrainDen.

Thanks and enjoy the Den :-)
Guest Message by DevFuse
 

Photo
- - - - -

Classic puzzle: .9999..... AND 1


Best Answer ThunderCloud, 20 October 2013 - 03:24 AM

Spoiler for

Go to the full post


  • Please log in to reply
11 replies to this topic

#1 BMAD

BMAD

    Senior Member

  • Members
  • PipPipPipPip
  • 1664 posts
  • Gender:Female

Posted 20 October 2013 - 12:32 AM

Here is a classic exercise:  Which of the following are true and why?

 

0.99999..... < 1

0.99999..... = 1

0.99999..... > 1


  • 0

#2 ThunderCloud

ThunderCloud

    Advanced Member

  • Members
  • PipPipPip
  • 102 posts
  • Gender:Male
  • Location:New England

Posted 20 October 2013 - 03:24 AM   Best Answer

Spoiler for


  • 0

#3 bonanova

bonanova

    bonanova

  • Moderator
  • PipPipPipPip
  • 5673 posts
  • Gender:Male
  • Location:New York

Posted 20 October 2013 - 11:00 AM

Spoiler for Simplest proof

  • 0
The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

#4 BMAD

BMAD

    Senior Member

  • Members
  • PipPipPipPip
  • 1664 posts
  • Gender:Female

Posted 20 October 2013 - 02:16 PM

So are we saying only one case is true?


  • 0

#5 ThunderCloud

ThunderCloud

    Advanced Member

  • Members
  • PipPipPip
  • 102 posts
  • Gender:Male
  • Location:New England

Posted 20 October 2013 - 04:28 PM

So are we saying only one case is true?

 

Yes. Both sides of the relations specify real numbers, which are well-ordered. So, exactly one of the relations must be true.


  • 0

#6 phil1882

phil1882

    Senior Member

  • Members
  • PipPipPipPip
  • 521 posts

Posted 04 November 2013 - 05:31 PM

hmm. there's a mathematician on youtube, norm wildberger, whom i kinda like who would disagree.

the problem being that we cant actually "know" infinity. try writing infinite 9's. you can't.

so at best we can say 0.9999999 or wherever you care to stop approximately equals 1.

to prove this point, accurately calculate pi +e +sqrt(2). if you're being honest, you would calculate it as pi + e +sqrt(2).


  • 0

#7 Prime

Prime

    Senior Member

  • Members
  • PipPipPipPip
  • 872 posts
  • Gender:Male
  • Location:Illinois, US

Posted 04 November 2013 - 06:06 PM

Spoiler for a simple proof


  • 0

Past prime, actually.


#8 bonanova

bonanova

    bonanova

  • Moderator
  • PipPipPipPip
  • 5673 posts
  • Gender:Male
  • Location:New York

Posted 05 November 2013 - 03:12 PM

hmm. there's a mathematician on youtube, norm wildberger, whom i kinda like who would disagree.
the problem being that we cant actually "know" infinity. try writing infinite 9's. you can't.
so at best we can say 0.9999999 or wherever you care to stop approximately equals 1.
to prove this point, accurately calculate pi +e +sqrt(2). if you're being honest, you would calculate it as pi + e +sqrt(2).


The notation convention 0.999 ... denotes an endless string of 9's.
You can't write the decimal representation of 1/3, but you can write 0.333 ...,
Thus, given the meaning of that notation, one can meaningfully write 1/3 = 0.333 ...

Similarly one can meaningfully write 1 = 0.999 ...
  • 0
The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

#9 phil1882

phil1882

    Senior Member

  • Members
  • PipPipPipPip
  • 521 posts

Posted 05 November 2013 - 06:38 PM

well, not entirely sure i agree. for example, how would you denote 1/13?

you need a consistent way of denoting any rational number, not just ones that repeat the last digit.

personally, i like the bar method, where the repeated values have a bar over the top over the length of the repeating part.

so

  _
0.9 = 1

would be more accurate (although more cumbersome.)


  • 0

#10 phil1882

phil1882

    Senior Member

  • Members
  • PipPipPipPip
  • 521 posts

Posted 05 November 2013 - 07:00 PM

here's my challenge for you.

accurately add 1/13 to 45/89 in decimal representation.


  • 0




0 user(s) are reading this topic

0 members, 0 guests, 0 anonymous users