From: ST102315 at brownvm.brown.edu (Jake)
Subject: A stupid proof that 1=.999...
Date: 18 Jul 1996
Message-ID: <4slq45$s3q@cocoa.brown.edu>
newsgroups: alt.fan.cecil-adams

If you want to convince someone that 0.9999... is equal to 1, it seems to me that there are two routes you can try. One is a rigorous mathematical proof, where you define 0.999... as an infinite series, define everything carefully, and show that the only possible conclusion is that 0.999... is equal to 1.

The only problem is that mathematical rigor is not convincing to many people. They distrust it, and may think that there's some trick happening that they're missing. They may not have the interest or patience to follow the argument to the depth required to understand and believe it. This is actually pretty reasonable--people will tend to be much less familiar with algebra (and certainly analysis) than with the processes of adding, subtracting, etc., so it only makes sense that most people will trust what they know--and any fool can see that 0.999... and 1 aren't the same number; they don't even look the same. (If this passage sounds in some ways insulting and condescending, it is not meant as such. There's no particular reason why people who haven't been trained to use mathematical rigor should understand it, any more than I understand how an architect designs houses. A math teacher might expect that a student would trust him or her to tell the truth when he or she says that 1=.99..., but on the other hand an unquestioning acceptance of the words handed down by the expert is hopefully not what the teacher is after anyways.)

The other way is to develop 'evidence' that what you're saying is true-- things which seem to indicate that if .999... equals anything, then it pretty much has to equal one. The very language I'm using should indicate that few if any of the pieces of evidence I'm discussing would count as a rigorous proof of anything--at least not without first proving the implicit assumptions that many of them are built on (assumptions like 3⋅.33333... = .99999.... or even 9 + .999... = 9.999....).

Having said that, I'll now give you a 'proof' in the second sense that shouldn't convince anybody due to its obvious stupidity. Maybe you'll enjoy it, though--I did. I'm going to start with a retread of how people are usually introduced to repeating decimals--feel free to skim or skip; the proof will be marked as such towards the end.

OK, first of all, the way most people are introduced to infinite decimals are through evaluating fractions like 1/3 as decimals. The long division looks like

  0.333
  ---------
3/1.000...
    9
    -
    10
     9
     -
     10
      9
      ...

Using this technique, one quickly comes up with 1/9=.11111...., 2/9=.22222.., and so on. Before long, one begins to start at the other end of the equals sign and wonder things like: What fraction is .27272727....? At this point, you're introduced to an algebraic way to determine the answer:

   x =  0.27...
100x = 27.27...
 - x = - .27...
 99x = 27
   x = 27/99 = 3/11

Do I believe the algebra? Yes, because now I can go back and check using the original method:

   0.272...
   ------
11/3.000...
   2 2
   ---
     80
     77
     --
      30
      22...

OK, so now what's .999...? The algebra says:

  x = .9....
10x = 9.9....
- x = -.9....
 9x = 9
 x = 9/9 = 1/1 = 1

But I don't know if I trust the algebra fully--this seems a bit suspect. Let's check the answer with the long division method:

PROOF
-----

  0.99...
  ------
1/1.00...
    9
    -
    10
     9
     -
     1...

My reasoning can be summarized as follows: 1 goes into 10 nine times, with one left over.

I know, too long a set-up, not a good enough punch line.

-jwgh