Questions About .999...

Date: 11/13/95 at 10:26:46
From: Anonymous
Subject: rational numbers

Is it not true that a rational number has to be able to be put in the form
A over B?  Ex: 0.3 = 3/10  

Also, is it not true that an infinite number's fraction form is that number 
over 9? Ex : 0.555.... = 5/9.  So, does that mean that 0.999... put in the 
fraction form is 9/9,  which is equal to 1?  Does that mean that 0.999... 
is not a rational number?

Can you help me with my problem?

(Robin Ogilvie - 13 years old) 

Date: 11/13/95 at 17:12:17
From: Doctor Ethan
Subject: Re: rational numbers

Hey Robin,

   You are exactly right.  And the answer to your last question is NO.

Let's look at why.

As you said, a number is rational if can be written as

 A
---
 B

So as you pointed out .3 = 3/10 and .55555..... = 5/9 are both rational.

Now look at .99999999.....

As you said, this is equal to 9/9 = 1.

Now why would this make you think that .99999 isn't rational?

You have just written down 1 and .9999999 in the form A/B where A and B
are both 9,  so 1 and .9999999 are both rational numbers.

In fact all repeating decimals [like .575757575757..], all integers 
[like 46], and all finite decimals [ like .472] are rational.

I hope that this helps.  Please write back with any more questions 
that you have.

-Doctor Ethan,  The Geometry Forum

Date: 11/16/95 at 17:17:46
From: Jay and Rob Ogilvie
Subject: Other question on rational numbers

        It's me again!  I'm the one that was wondering if .999.... can be a 
rational number?  Anyway, I have a few more questions I would like to ask, 
so here it is:
        In theory, a periodical .999..... will never reach the
whole # standing right?  If this is true, how can .999..... ever be equal to 
one?  I understand that 7/9= .777..... and that 8/9= .888..... so logically 
0.999..... should be 9/9, but we know that 9/9=1 and .999..... can never fully 
reach one.

Can you explain?

Date: 11/19/95 at 15:17:48
From: Doctor Ethan
Subject: Re: Other question on rational numbers

Well
  You are right in a way.

.99999...

When considered as the decimal point followed by a whole bunch of 9 nines, 
it will never reach 1.

However that is not what we mean when we right .999....

We mean .9 repeating all the way to infinity right there in that little symbol.

And yes if you had an infinite number of nines there all at once, it wouldn't 
be just close one it would be exactly one.  This a little hard to understand 
but it is the way that mathematicians think about that notation.

(Actually they would prefer different notation but they get by.)

I hope this has helped.

-Doctor Ethan,  The Geometry Forum