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### Point nine repeating equals one?

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

Dear Dr Math.

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)

Thanks,
Joe Redding
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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")
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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...?  .3333....is 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
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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!

Thanks,

Joe Redding
Web site: http://sasd.k12.pa.us/

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Associated Topics:
High School Analysis

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