How can .999999.... equal 1?

Date: 03/21/2001 at 15:07:26
From: Emily F. and Jenny B.
Subject: .999999..... I still don't get it

Dr. Math,

In my math class in school, my math teacher always talks about how 
whenever she has a problem she goes to your site and finds it or 
writes to you. I have a problem.

I know .999999.... is supposed to equal 1. My teacher demonstrated 
the subtracted thing and the other stuff you have on your site. I 
still don't get it. If .99999999.... goes on forever, wouldn't it be 
just a little below one? There would be just a tiny gap between it and 
one. Please explain this to me. 

Thanks,
Emily and Jenny

Date: 03/21/2001 at 16:22:11
From: Doctor Ian
Subject: Re: .999999..... I still don't get it

Hi Emily and Jenny,

There's no doubt that this equality is one of the weirder things in
mathematics, and it _is_ intuitive to think: No matter how many 9's 
you add, you'll never get all the way to 1. 

But that's how it seems if you think about moving _toward_ 1.  What if 
you think about moving _away_ from 1?  

That is, if you start at 1, and try to move away from 1 and toward 
0.99999..., how far do you have to go to get to 0.99999... ?  Any step 
you try to take will be too far, so you can't really move at all -  
which means that to move from 1 to 0.99999..., you have to stay at 1.  

Which means they must be the same thing!

Here's another way to think about it. When you write something like

  0.35

that's really the same as 35/100,

  0.35 = 35 / 100

right? Well, you can turn that into a repeating decimal by dividing by 
99 instead of 100:
                    __
  0.35353535... = 0.35 = 35 / 99

Play around with some other fractions, like 2/9, 415/999, and so on, 
to convince yourself that this is true.  (A calculator would be 
helpful.)

In general, when we have N repeating digits, the corresponding 
fraction is 

  (the digits) / (10^N - 1)

Again, some examples can help make this clear:
         _   
       0.1 = 1/9
        __
      0.12 = 12/99
       ___
     0.123 = 123/999

and so on.  

So, here's something to consider:  What fraction corresponds to
    _
  0.9 = ?

It has to be something over 9, right?
    _
  0.9 = ? / 9

The _only_ thing it could possibly be is 
    _
  0.9 = 9 / 9

right?  But that's the same as 1.  

Ultimately, though, this probably won't _really_ make sense until you 
come to grips with what it means for a decimal to repeat _forever_, 
instead of just for a  r-e-a-l-l-y  l-o-n-g  t-i-m-e.  

When you think of 0.999... as being 'a little below 1', it's because 
in your mind, you've stopped expanding it; that is, instead of 

  0.999999...

you're _really_ thinking of

  0.999...999

which is not the same thing. You're absolutely right that 0.999...999 
is a little below 1, but 0.999999... doesn't fall short of 1 _until_ 
you stop expanding it. But you never stop expanding it, so it never 
falls short of 1.  

Suppose someone gives you $1000, but says: "Now, don't spend it all,
because I'm going to go off and find the largest integer, and after I 
find it I'm going to want you to give me $1 back." How much money has 
he really given you?  

On the one hand, you might say: "He's given me $999, because he's 
going to come back later and get $1." 

But on the other hand, you might say: "He's given me $1000, because 
he's _never_ going to come back!"  

It's only when you realize that in this instance, 'later' is the same 
as 'never', that you can see that you get to keep the whole $1000. In 
the same way, it's only when you really understand that the expansion 
of 0.999999... _never_ ends that you realize that it's not really 'a 
little below 1' at all. 

I hope this helps.  Let me know if you'd like to talk about this some 
more, or if you have any other questions. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/