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### Dividing by 9, 99, 999....

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Date: 9/7/96 at 20:31:54
From: Joe Snyder
Subject: Dividing by 9, 99, 999....

Dear Dr. Math,

I was fooling around on my calculator.  Somehow, after a while, I
found a pattern in what I was keying in.

This is what I found:

x / 99... = .xx ...

(the number of 9s is the same as the number of digits x has)

Example No.1:
456 / 999 = .456456456...  (456 repeat)

Example No.2:
12,345,678 / 99,999,999 = .12345678 12345678 12345678...

Example No.3:
9,999 / 9,999 = 1  and  0 / 9 = 0

(*cannot use all-9 numbers or 0s*)

My question is:

Is there any property that states this or is similar to this?  What is
it and why does it work that way? (This isn't really a property, but
you know what I mean.)

Any answers or replies will be greatly appreciated!

Joe Snyder
```

```
Date: 9/10/96 at 14:37:38
From: Doctor Ana
Subject: Re: Dividing by 9, 99, 999....

My father, a math teacher, always talks about taking his calculator
with him everywhere, even to bed, to play with numbers. I'm glad he's
not the only one who likes math enough to do that! Here is one
explanation for the property that you found. (Sorry, you weren't the
first one to find it, but don't let that keep you from trying!)

456      456               456
--- =   ----      +      ------
999     1000             999000

Prove that to yourself first of all. Then we can see that we have

456             456
--- = .456 +  ------
999           999000

456        456        456
------ =  -------  + ---------
999000    1000000    999000000

So, now we have

456                         456
--- = .456 + .000456 +   ---------
999                      999000000

And the pattern keeps repeating forever.

This same method will work for any denominator that contains all 9s,
no matter how many of them there are.

Have you looked at the pattern for the sevenths and the elevenths yet?
They're almost as interesting as the ninths. Try 1/7, 2/7, 3/7  and
1/11,  2/11, 3/11, 4/11 on your calculator and figure out the pattern.

-Doctor Ana,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Number Theory

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