Other Ways to See That 0.999... = 1

Date: 05/29/2000 at 18:42:29
From: Mick Rissling
Subject: 0.999... = 1? 

Hi. My friends and I were debating whether 0.999... = 1 (with me on 
the wrong side, apparently) the other night.

I have looked through he FAQ and found the answer, but I am not quite 
satisfied. I guess that I believe it, the proofs are quite simple, but 
I am having trouble getting my mind around a few things.

My problem arises from functions. Say you have an asymptote at x = 1 
on a graph and a function that approaches it (but never reaches it), 
my problem is how can a number (0.999...) equal another number (1) if 
it never reaches that value?

Another problem I have is that I was always taught that .333... is 
just a decimal approximation of the exact value 1/3. Have all my 
teachers been mistaken?

I guess that my main thought-obstacle is that you can keep adding 9's 
on the end of 0.999... to get it closer to 1, but if you have to keep 
getting closer out to infinity how can you equal it?

Any help would be appreciated.

Thanks

Date: 05/29/2000 at 18:55:42
From: Doctor Schwa
Subject: Re: 0.999... = 1? 

Hi Mick,

>My problem arises from functions. Say you have an asymptote at x = 1 
>on a graph and a function that approaches it (but never reaches it), 
>my problem is how can a number (0.999...) equal another number (1) if 
>it never reaches that value?

I think part of the question to answer is: What is a number? Some 
people would solve this whole problem by saying that 0.999 is a 
number, 0.999999999999999999999999 is a number, but the whole idea of 
having infinitely many 9s doesn't even make sense. See below for more 
on this idea.

>Another problem I have is that I was always taught that .333... is 
>just a decimal approximation of the exact value 1/3. Have all my 
>teachers been mistaken?

If you stop after finitely many 3s, it's an approximation. If there 
are infinitely many 3s, then it's exact.

>I guess that my main thought-obstacle is that you can keep adding 9's 
>on the end of 0.999... to get it closer to 1, but if you have to keep 
>getting closer out to infinity how can you equal it?

This is exactly the issue. What does it mean to say "there are 
infinitely many 9s"? We certainly don't mean to write them all down, 
one after the other. What mathematicians use it to mean is "you name 
any number of 9s you want, my number has even more 9s than that," or 
"take a look in the _____th decimal place, where you fill in the blank 
with any positive integer you want, and that decimal place will have a 
9 in it."

Maybe looking at the question the other way would help. What does it 
mean to equal 1? The idea of "equal" that we use here is that there's 
no distance between your number and 1. To prove there's no distance, 
think of any distance you want. Certainly, after enough 9s, it's even 
closer than that. So, the only possible distance is zero; any other 
distance is too big.

Another way to approach the question is to subtract.

      1.0000000....
     - .9999999....
     --------------
      0.0000000....

Sure looks equal to me. What about the "1" at the end, I hear you ask? 
Well, I'll write it as soon as I finish writing infinitely many 0s. 
Any decimal place you name (say, the four billion three hundred 
twenty-eight million two hundred seven thousand four hundred 
ninety-fifth) has a 0 in it. A number with a 0 in every decimal place 
is certainly 0.

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