Introduction to Infinity, Limits, and Why 0.999... Equals One

Date: 01/12/2008 at 11:42:34
From: Devan
Subject: 0.999...=1 and yes, I have read through the FAQ

I have read a good deal of the FAQs and other questions sent to you
concerning this problem, but I'm still not convinced that 0.999...
equals 1. 

The problem is that whenever you try to explain it, you tend to get
back to the point of 0.999 vs. 0.999... you say that since it is
constantly getting bigger, and gets infinitely close to 1, it must be
1.  But if 0.999... ever actually reached 1, that would bring an end
to infinity as quickly as 0.999 would.  In order for 0.999... to reach
1, there would have to be some point, somewhere along the number line,
where a finite quantity was added that would cause it to reach 1, and
then it would stop increasing.

Plus, whether it progresses infinitely or not, it still doesn't reach
one, because it adds a smaller value each time.  It's not like I'm
adding 1 a bunch of times, and saying that I'll never reach 100 
because I don't want to write out that many 1's.

Infinity louses up math pretty much anywhere you use it, and it seems 
to me that all you're saying is "Well, 0.999... is really really close 
to 1, and since it would be a lot harder to work with 0.999... than 1, 
we might as well just call it 1 and save everyone the trouble."  I 
have no problem with that, as I have as much trouble grasping the 
concept of infinity as everyone else on the planet, but when you're 
not working with it formally or towards a purpose, I see no reason not 
to admit that 0.999... is not actually 1.

Another thing I see wrong with this, is that if 0.999... is equal to 
one, what's to stop pi from being equal to 4, or, at the very least, 
3.2?  Pi is an infinitely long number, so it keeps adding up decimals.  
It should eventually get as close to 3.2 as 0.999... will ever get to 
1.

Date: 01/12/2008 at 16:17:14
From: Doctor Rick
Subject: Re: 0.999...=1 and yes, I have read through the FAQ

Hi Devan, 

Thanks for writing to Ask Dr. Math.  You wrote:

>the problem is that whenever you try to explain it, you tend to get
>back to the point of 0.999 vs. 0.999... you say that since it is
>constantly getting bigger, and gets infinitely close to 1, it must be
>1. 

We should be clear in our language.  The number 0.999... does NOT 
"constantly get bigger."  It is a NUMBER, and a number just is what it 
is; it doesn't change its value.

>but if 0.999... ever actually reached 1, that would bring an end to
>infinity as quickly as 0.999 would.  In order for 0.999... to reach
>1, there would have to be some point, somewhere along the number
>line, where a finite quantity was added that would cause it to reach
>1, and then it would stop increasing. 

What you're trying to describe is something that mathematicians 
formalize as a LIMIT.  You will learn all about this when you get to 
calculus.  The idea is that we have a sequence of finite decimals:

  0.9, 0.99, 0.999, 0.9999, 0.99999, ...

As we keep adding 9's at the end, the numbers get closer and closer 
to 1--but you're right, they never actually reach 1.

The problem with your arguments is that 0.999... is NOT this sequence 
(because the sequence is not one number, but many), nor is it any of 
the numbers in that sequence (because each of them has a finite number 
of 9's).

>Infinity louses up math pretty much anywhere you use it, and it 
>seems to me that all you're saying is "Well, 0.999... is really 
>really close to 1, and since it would be a lot harder to work with 
>0.999... than 1, we might as well just call it 1 and save everyone 
>the trouble."  I have no problem with that, as I have as much 
>trouble grasping the concept of infinity as everyone else on the
>planet, but when you're not working with it formally or towards a
>purpose, I see no reason not to admit that 0.999... is not actually >1.

Infinity is indeed a difficult concept to grasp. The theory of limits 
is the formalism that was developed to deal with infinite processes.  
It does so essentially by NOT dealing with infinity itself, but with
sequences of finite quantities, each of which we know how to work with.

If you try to think of 0.999... (or any other repeating decimal, such 
as 0.333...) in the same way we deal with finite decimals like 0.125, 
we run into a problem.  While we can work out something like 0.125 = 
1/10 + 2/100 + 5/1000, we can't do the same for 0.333... = 3/10 + 
3/100 + 3/1000 + ... --because we can't do an infinite number of 
additions!  As you suggest, this isn't the same as just having 100 
additions and getting tired before we're done; conceptually there is 
nothing different about doing 2 additions, or 100, or 1 billion.  But 
an infinite number of additions truly CAN'T be done.

Yet we NEED repeating decimals!  Do you recall what motivated their 
introduction?  When we do a division such as 1 divided by 3, the 
division process never ends; we never get a remainder of 0.  If we 
just cut off the process after some finite number of digits (no matter 
how great that number), the result would NOT be EXACTLY equal to the 
fraction 1/3.  If we're going to have a way to write any fraction as a 
decimal, we need a way to represent the idea of a division process 
that never ends.  And that way is repeating decimals.

So what value do we assign to a repeating decimal such as 0.333..., 
if we can't treat it the same as a terminating decimal?  As I have 
said, it has to be a NUMBER--in this case, it has to equal 1/3 
exactly.  The only number available is the LIMIT of the sequence 0.3, 
0.33, 0.333, 0.3333, ...; that is, the number that this sequence 
APPROACHES as the terms go on without end.  As we have said, none of 
those terms is exactly equal to the limit (1/3); that's not the 
point.   The point is that the terms get closer and closer to 1/3.  
Furthermore, there is no other number that the terms get closer and 
closer to.  Therefore the limit of the sequence is 1/3; it is the 
only number uniquely associated with the sequence, so (by 
definition) we say that it is the value of the repeating decimal 
0.333...

>another thing I see wrong with this, is that if 0.999... is equal to 
>one, what's to stop pi from being equal to 4, or, at the very least, 
>3.2?  Pi is an infinitely long number, so it keeps adding up 
>decimals.  It should eventually get as close to 3.2 as 0.999... will 
>ever get to 1.

Here's what is wrong with your thinking this time: Once we know that 
the tenths digit of the decimal expansion of pi is 1, we know that 
pi can't be greater than 3.2.  And once we have found that the 
hundredths digit is 4, we know that pi can't be greater than 3.15. 
Each digit we learn puts limits on the range of possible values of 
pi--precisely because of one of your observations above: that we're 
adding a smaller value each time in the sequence

  pi = 3 + 1/10 + 4/100 + ...

I appreciate your thinking about this topic.  You are asking the right 
kind of questions, questions that lead on into the field of calculus.  
Please don't stop!

If you'd like to continue discussion of this topic, may I ask you to 
tell me your thoughts on repeating decimals in general.  Do you think 
that 0.333..., for example, is exactly equal to 1/3, and the only 
problem is with 0.999...; or do you have a problem with ALL repeating 
decimals?  I don't recall anyone broadening out the argument this way; 
everyone seems to be concerned about 0.999... = 1 specifically.  How 
about you?

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

Date: 01/15/2008 at 18:32:58
From: Devan
Subject: 0.999...=1 and yes, I have read through the FAQ

Okay, I see where my flaw is, and I think I get the limit thing, but
I'm going to make one more desperate attempt to save face before I
give up: where does this leave Zeno's runner, who had to cross an
infinite number of halfway points before he could reach the finish
line?  Of course, we all know that that would never stop a real 
runner, but if Zeno really did take, say, two minutes to cross each 
new halfway point, he never would reach the finish line.  It really 
is the same problems as repeating decimals; which, by the way, I do 
(or did) have a problem in general with.

Date: 01/15/2008 at 19:36:13
From: Doctor Rick
Subject: Re: 0.999...=1 and yes, I have read through the FAQ

Hi, Devan.

You wrote:

>where does this leave Zeno's runner, who had to cross an infinite
>number of halfway points before he could reach the finish line?  Of
>course, we all know that that would never stop a real runner, but if
>Zeno really did take, say, two minutes to cross each new halfway
>point, he never would reach the finish line. It really is the same
>problem as repeating decimals ...

A real runner would NOT take the same time to cross each halfway 
point. If he did, then yes, he would never reach the finish line.  He 
would get infinitely close, because the limit of the sequence

  1/2 + 1/4 + 1/8 + ... + 1/2^n + ...

is 1.  But there is no FINITE time (corresponding to a finite number 
of terms in the series) at which the series would equal 1.  That's a 
clear correspondence with the 0.999... problem, where the infinite 
series is

  9/10 + 9/100 + 9/1000 + ... + 9/10^n + ...

Another way to express the correspondence is that the first series 
represents the repeating binary fraction 0.1111... (base 2), which 
also equals 1.

As I said, a real runner does not take equal times to reach points 
that get closer and closer together.  Suppose, for simplicity, that 
the runner runs (walks?) at 1 kilometer/minute, and the course is 1 
km long.  Then it takes 1/2 minute to reach the halfway point, then
another 1/4 minute to reach the 3/4 point, and another 1/8 minute to
reach the 7/8 point, and so on.  The time it takes to reach the finish
line is

  1/2 + 1/4 + 1/8 + ... = 1 minute

just as we would expect.  In the last moment of the race, an infinite 
number of halfway points are passed.  Thus the entire infinite series 
for the distance is completed in a finite time, and since the infinite
sum equals 1 km, the runner does finish the race as expected.

>... repeating decimals; which, by the way, I do (or did) have a
>problem in general with.

I'm glad to hear it!  Many of those who write to us about this topic 
(and there are a lot of them) have no problem with 0.333... = 1/3. 
This tells me that their problem with 0.999... = 1 has more to do with 
an unfounded feeling that there can't be two decimal representations 
of the same number, and less to do with the real mathematical issue of 
what an infinite sequence of digits can mean.  You have been thinking 
well, and there is no need to save face.

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

Date: 01/17/2008 at 15:04:51
From: Devan
Subject: Thank you (0.999...=1 and yes, I have read through the FAQ)

Thanks very much for clearing up a point for me, and replying to my
update.