0.999... and Infinity

Date: 09/17/2003 at 17:23:33
From: Shane
Subject: paradox?

What's wrong with this?

    x = 0.99999999...forever

so

  10x = 9.99999999...forever

so

  10x - x = 9.9999999 - 0.9999999 

       9x = 9.0

        x = 1

How does x = 0.9999...forever turn into x = 1?

Date: 09/18/2003 at 01:04:25
From: Doctor Mike
Subject: Re: paradox?

Shane, I'll tell you what's wrong with what you have written.

Nothing.  Nothing is wrong with it.  It's fine.
  
I can see, though, why you might be surprised about what you have
found.  Infinity is a pretty powerful concept.  

Why am I talking about infinity?  Because whenever you use a "forever"
repeating decimal, it doesn't just go on for a "really long time", it
goes on forever.  And whenever you have an infinite number of parts of
something (an infinite number of places in a number, in this case) you
might well expect something amazing to happen.
  
OK, let's get beyond the philosophy and on to exactly why this
happens.  An infinite decimal "officially" is considered to be a
sequence of numbers, by mathematicians who think about the origins and
foundations of our number systems.  For instance:

   0.9
   0.99
   0.999
   0.9999
   0.99999
   0.999999
   0.9999999
    and so on forever.

Another familiar example you may have encountered is

   3.
   3.1
   3.14
   3.141
   3.1415
   3.14159
   3.141592
   3.1415926
    and so on with the rest of the digits of pi, forever.
 
In each case, the final and official value of the number is the limit
of the sequence of these numbers that get "closer and closer" to that
limiting value.  
  
In the case of 0.99999... the limiting value is 1.  If you go out far
enough in the terms of the sequence, you can get "arbitrarily close"
to one.  
  
You might ask, "Can you go out far enough so that the terms in the
sequence are closer to 1.0 than one trillionth = 1/1000000000000 ?" 
Yes, anything past the term 0.9999999999999 is closer to one than one
trillionth.   

"Can you go out far enough so that the terms in the sequence are
closer to 1.0 that one octillionth?"  Yes. 

"What about 1/google?"  Sure, no problem.  

As close as you want to specify, you can go out far enough in the
sequence so that every term in the sequence, from there on, is closer
to 1.0 than the degree of closeness you specified.  That's what limits
are all about.  
   
Now, the bottom line is:  If this number 0.99999999... is closer to
one than anything that can be measured... if it is closer to one than
anything that you can even think about measuring... if it is closer to
one that any possible number you could think of... if it is closer to
one than any number that anybody could think of, no matter how
small... THEN essentially 0.9999999... is indistinguishable from one.  
 
Said another way, if your number 

  x = 0.999999999...

is such that the difference "x-1" is smaller than any positive number
(however small), then we consider x - 1 = 0, so that x = 1. 
  
I hope this helps you to understand your paradox.

Thanks for writing to Dr. Math.

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

Date: 09/18/2003 at 03:02:03
From: Shane
Subject: Thank you (paradox?)

Dr. Math is the most impressive thing I've found on the internet to
this day - and I've been on it a very long time.  I'm amazed at the
quality and promptness your answers, and you do it without asking for
anything in return.  You are a true credit to society, thank you!