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Getting 0.99999...

Date: 04/15/98 at 07:40:13
From: Dusty
Subject: How do you get .9 repeating?

Is there any mathematical way to get .99999999999......  and so on?

Date: 04/15/98 at 09:35:29
From: Doctor Derrel
Subject: Re: How do you get .9 repeating?

Hi Dusty!

That's an interesting question that I've explored with my students, so 
I'll give you some hints. It may seem that I'm taking the long way 
around, but, trust me, we'll get there.

First, I'm going to assume that you know how to convert a fraction to 
a decimal. If you don't, write back and we'll help you with that one.

1) Write down the fraction 1/3 (one-third) on a piece of paper. Then 
add 1/3 to it. That gives you 2/3. Then add 1/3 to your answer. That 
gives you 3/3.

2) What is 3/3 as a decimal number?

3) I assume that you know how to do long division by hand. Convert the 
fraction 1/3 (one-third) to a decimal using long division. Keep going 
until ... well, you'll figure out when to stop. I'll give you a hint: 
if you have to sharpen your pencil a couple of times, you have 
probably gone a little further than necessary. :-)

4) By now, you have probably figured out that 1/3 is 0.33333..... 
Normally, this is written as 0.3 with a bar over the three to indicate 
that the "3" repeats forever, but since I can't easily do that here, 
I'll just write it as 0.3bar.  

5) Now, we'll do the same steps that we did in 1) above, but use the 
decimal number. So, write down 0.3bar, then add 0.3bar to it. (How 
is that the same as what we did in step one?) Can you figure out why 
0.3bar + 0.3bar = 0.6bar? Because these numbers go on forever, you 
will need to use a little logic to add them. (The algorithm that you 
learned for adding numbers doesn't work very well when you can't get 
to the rightmost number.) What is 0.6bar as a fraction?  Now, add 
0.3bar to your 0.6bar. What does that give you?

You should have figured out that 0.3bar + 0.6bar = 0.9bar, which is 
what you asked about. (If you had trouble, write back.) But don't stop 
there! The real fun is just starting! You know that 0.3bar = 1/3. From 
that, you know:

     0.3bar + 0.3bar + 0.3bar = 1/3 + 1/3 + 1/3

In step 5), you showed that the left side of the equation is 0.9bar. 
In steps 1) and 2), you showed that the right side of the equation is 
1. Therefore:

     0.9bar =  1

Think about that a while! :-) 

If you disbelieve that 0.9bar = 1, think about it for a while, talk to 
your teacher, and if you still aren't convinced, write back. Oh, and 
while you are at it, see if you can find a definition for the term 
"repetend." I hope we've helped! 
-Doctor Derrel, The Math Forum
Check out our web site!   

P.S. The Math Forum has a standard answer for "Why does 0.9999.... 
equal 1?" It gives you another way of thinking about the problem that 
is different from the way I gave you above. See:   
Associated Topics:
High School Number Theory
Middle School Number Sense/About Numbers

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