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December 22nd, 2010, 08:42 PM  #1 
Member Joined: Dec 2010 From: Viet Nam and US Posts: 36 Thanks: 0  Is 0.9999... equal 1?
We all know that 9.0 is unequal 9, or 9.99 is different from 10. But there's something wrong with this problem below: 1) Simple prove Let 2) Prove using geometric sequence formula Let The geometric sequence above has Since , we have: So what is the problem here 
December 22nd, 2010, 09:10 PM  #2 
Senior Member Joined: Nov 2010 Posts: 502 Thanks: 0  Re: Is 0.9999... equal 1?
I don't quite see what you're getting at with the second half, but I have two comments. Firstly, 0.999.... is equal to 1. Secondly, 9.0 is equal to 9, unlike your first line. 
December 22nd, 2010, 09:27 PM  #3  
Member Joined: Dec 2010 From: Viet Nam and US Posts: 36 Thanks: 0  Re: Is 0.9999... equal 1? Quote:
I'm just curious about the accuracy of math. I've learned that, 9.0 is different from 9, exactly only 9=9. One more digit after dot makes a difference. And then I've never known that 0.999.. equal 1, mathematics says that 0.999 < 1. So I'm just concerned about my knowledge, there must be something wrong!  
December 22nd, 2010, 09:35 PM  #4 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: Is 0.9999... equal 1? The expression , but any truncation is less than 1. It is only when the nines go on infinitely that we have 1. 
December 22nd, 2010, 09:52 PM  #5  
Member Joined: Dec 2010 From: Viet Nam and US Posts: 36 Thanks: 0  Re: Is 0.9999... equal 1? Quote:
By saying 9.0 not equal 9, I mean that 9.0 is more accuracy than 9, 9.0 points out that after dot, we have a 0, so it is not a Round(9.1) or something. Easier to imagine, let's say 9.000000000 is not exactly equal 9, because it says after dot, there are 9 more zeroes! It's so complicated to imagine, I've learned that from school, I wish I could explain it better! And back to my problem, basically 0.999.. is not equal 1 , so according to you, it's only equal 1 when we have a lot of nines. So there's something inaccuracy in Math, isn't it? (According to my own opinion only!)  
December 22nd, 2010, 10:02 PM  #6 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: Is 0.9999... equal 1?
Not when we have a lot of nines, but only when we have an infinity of nines...not a thousand, or a million, or a centillion, or a googolplex, or any finite number.

December 22nd, 2010, 10:15 PM  #7  
Member Joined: Dec 2010 From: Viet Nam and US Posts: 36 Thanks: 0  Re: Is 0.9999... equal 1? Quote:
So we have to accept that 0.999...=1 if there are infinity number of 9 after dot? Math requires highest accuracy in all subjects, I think, and Math itself claims that 0.999....<1 because . It always exists a number which is the difference between 1 and 0.999... or 1 and In this case, that number is . Thus, there is no way that 0.999.. equal 1. This problem is interesting to me ^^!  
December 22nd, 2010, 10:19 PM  #8 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: Is 0.9999... equal 1?
What is ?

December 22nd, 2010, 10:25 PM  #9  
Member Joined: Dec 2010 From: Viet Nam and US Posts: 36 Thanks: 0  Re: Is 0.9999... equal 1? Quote:
So that's all the problem about, 0.999... = 1 it's about limit. I can go really close to 1 but not exactly 1, however, we consider it is 1, am I right?  
December 22nd, 2010, 10:32 PM  #10 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: Is 0.9999... equal 1?
If we allow n to grow unbounded, then the difference is zero, and we have exactly 1.


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