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Point nine repeating equals one?

Date: 21 Dec 1994 13:16:05 -0500 
From: joe redding
Subject: Point Nine repeating equals one?

Dear Dr Math.

        I have had a question about this for years but I don't  
remember finding a satisfactory answer.

        When I learned how to convert repeating decimals to  
fractions, we were given the following example:
                  _                _
        Let n = .99      so  10n = 9.99

        Subtracting the first equation from the second yields:

        9n = 9 since the repeating decimals subtract out
        which gives us n = 1,  but we know that n = .99  so
        .99 = 1   

The problem I have is that I can't logically believe this is true,  
and I don't see an error with the math, so what am I missing or  
forgetting to resolve this?

(If memory serves, he also said that there are several other ways of  
proving that .9999...  = 1 but I don't remember them)

Joe Redding

Date: Wed, 21 Dec 1994 22:33:21 -0500 
From: Stephen Weimar
Subject: Re: Point Nine repeating equals one?

I think it is true and you did a beautiful job presenting it. 

> (If memory serves, he also said that there are several other ways of  
> proving that .9999...  = 1 but I don't remember them)

If it was not equal to one then there would be a number between it 
and 1. What number would that be?

-- steve ("chief of staff")

Date: 22 Dec 1994 14:26:44 -0500
From: Molly Foster
Subject: Re: Point Nine repeating equals one?

Dear Joe,

     Hello there!  Thanks for writing Dr. Math.  You asked an 
excellent question, and I liked Steve's first response to you, but I 
thought I might add two things.

     Another way to think about this is this:  Would you agree that 
1/3 = .33333...? the way to write 1/3 using decimals.  
If you multiply both sides of the above equation by three you get
1 = .99999...., right?

     I think the problem you are having, though, is BELIEVING it 
is true, right?  I admit, depending on how you look at it, it can seem 
false. After all, how can 2 different numbers be equal?  The thing is, 
these 2 numbers AREN'T different.  I think saying 1 = .9999... may 
seem contradictory to us because we aren't realizing that .999.... is 
a repeating decimal that really does go on forever.  Obviously saying 
1 = .9 is false, as is saying 1 = .99, 1 = .999, 1 = .9999, etc.  But we 
aren't dealing with finite decimals here.  So, you might think of 
.9999.... as another name for 1, just as .333... is another name for 1/3.  

     What Steve said really should clinch it for you, but I thought I'd
just add these thoughts anyway.  Hope it helps.

Sydney, Dr. "math rocks" Foster

Date: 22 Dec 1994 14:38:08 -0500
From: joe redding
Subject: Re: Point Nine repeating equals one?

Dear Steve and Molly,

        Thanks so much for your replies.  Both were what 
I was looking for.  Molly,  thanks for the second method of 
proving this to be true but does anyone know of the  
elusive third?

        By the way, I added Dr. Math to our Web page.  I think 
that it is a great service.   Keep up the good work!


Joe Redding
Web site:   

Associated Topics:
High School Analysis

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