Finding the Digit of a Decimal Expansion

Date: 11/14/98 at 14:06:06
From: breanna
Subject: Mathcounts

What digit will appear in the 534th place after the decimal point in 
the decimal representation of 5/13? 

I tried taking 534 and dividing them, but I just don't understand.

Breanna

Date: 11/16/98 at 19:15:02
From: Doctor Anderson
Subject: Re: Mathcounts

Hi, Breanna:

This is a very interesting question, and one that you really have to 
think about. When you see something that seems to ask you to perform a 
ridiculous task in math, usually you don't have to actually do it.  
Dividing 5 by 13 in long division, and going out 534 places, would 
take a long time, unless you were a computer. I will give a simpler 
example to help you along.  

Let's work on 2/27. You can punch this into a calculator if you like, 
or do long division. Either way you will find that:

    2/27 = 0.074074074074074074074...

If the question asked to find the 5th place after the decimal point, 
we would count off 5 numbers after the decimal and find the answer to 
be 7. 

Now to help you apply this to your own problem. There is a really nifty 
property of rational numbers (fractions): 

   either they terminate (meaning end, like 1.374)   or 
   they repeat (like 2/27) 

There is no other possibility.  

This means that when you divide out 5/13, there is going to be a 
pattern. How does this pattern help us? Well, look again at 2/27.  

Say I wanted the 301st digit. Notice that the 3rd digit is 4, and so 
is the 6th, and the 9th, and you can see that the 12th, 15th, etc, 
digits are all 4. So any digit where the place is divisible by 3, 
is a 4. Let's go for the 301st digit. First let's look at the 300th 
digit. 300 is divisible by 3, so the 300th is 4. What always follows 
a 4? Well, look at the expansion above. A 0 always follows a 4, and 
the 301st digit follows the 300th, so the 301st digit is a 0. 

See how this works? You will see that when you work on 5/13, it won't 
repeat every 3 places like this example, so it will be a bit different, 
but I think you can take this one from here. Good luck!

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