0.99... Repeating Can't Really Equal One, Can It?

Date: 08/14/2008 at 01:49:03
From: John
Subject: .999 repeating = 1 . . . . again

Doesn't .999 repeating simply get infinitely close to 1, but never
quite get there?  Kind of as if it had an asymptote at 1?  It always
gets closer and closer, but will never quite get there, because if it
did then all the 9s would turn to 0s, and then it would be 1.  Each 9
that is added simply makes the number more precise, closer to 1.  But
the number will ONLY become more PRECISE, it will never become, well,
precise, only MORE precise.  

Date: 08/14/2008 at 09:38:09
From: Doctor Peterson
Subject: Re: .999 repeating = 1 . . . . again

Hi, John.

Sure, if you ever STOP somewhere in writing out 0.999..., then it 
will not be equal to 1.  But that's not what a repeating decimal means.

Think about 0.333... . That equals 1/3, right? But something like

  0.33333333333333333333333333333333333333333333333333

does not equal 1/3, because I've left off all the digits after the 
50th.  The value of any non-terminating decimal like 0.333... is 
defined to be, not the value when you stop somewhere, but the "limit" 
that you approach closer and closer as you take more digits into 
account.  So in fact what you say is true of ANY such number, even 
something like pi = 3.14159...; each digit you add gets closer to the 
actual value, giving you better and better approximations without 
ever actually reaching it.  But that actual value is still out there, 
beyond all those approximations; it is not one of them.

So the value of 0.999... is not the value of

  0.99999999999999999999999999999999999999999999999999

but the value, namely 1, that you are approaching as you add more 
digits.  (In fact, if you could stop somewhere and get a value of 1, 
that would prove that 0.999... was NOT 1, since the next digit would 
make it greater!)

By the way, if 0.999... were not 1, how would you explain the fact 
that

  1 = 3 * 1/3 = 3 * 0.333... = 0.999... ?

If you have any further questions, feel free to write back.


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

Date: 08/14/2008 at 15:51:17
From: John
Subject: Thank you (.999 repeating = 1 . . . . again)

Touche, sir/madame . . . . touche!