Turning a Repeating Decimal into a Fraction

Date: 10/18/2002 at 16:41:26
From: Danielle
Subject: Rational and irrational numbers

One of my math problems is .063490634906... and I need to know how to
figure out if that is rational or irrational. How do I make a decimal 
like that into a fraction?

Date: 10/18/2002 at 20:14:14
From: Doctor Achilles
Subject: Re: Rational and irrational numbers

Hi Danielle,

Thanks for writing to Dr. Math.

It is actually possible to turn any repeating decimal into a 
fraction. Here's how:

First, a bit of background:

The easiest repeating decimal I can think of is just 0.111111...  
That is equal to 1/9. Two times that is equal to 0.22222... or 2/9.  
Three times that is 0.33333... or 3/9 (or 1/3). Etc. (Note that 
0.99999... equals 9/9 or 1, for more on that you can see the Dr. 
Math FAQ: http://mathforum.org/dr.math/faq/faq.0.9999.html .)

If you have ONE repeating digit, just put the repeating digit over 9 
to get a fraction. So 0.555555... is a repeating 5 and it equals 5/9.

The next easiest repeating decimal I can think of is 
0.01010101010101...  This turns out to be equal to 1/99. Two times 
that is 0.02020202... or 2/99. 0.030303030303... equals 3/99 or 1/33.  
Etc. 0.0909090909... equals 9/99 or 1/11. Things get interesting when 
you get to 0.101010101010... That is equal to 10/99. 0.1111111... 
equals 11/99, which reduces to 1/9.  0.2525252525... equals 25/99.

If you have TWO repeating digits, such as 58 or 04, just put the 
repeating digits over 99 to get a fraction. So 0.0404040404... is a 
repeating 04 and it equals 04/99 or just 4/99.  And 
0.58585858585858... is just a repeating 58 and it equals 58/99.

If you have THREE repeating digits, just put them over 999 to get a 
fraction. So 0.102102102102... equals 102/999, and 0.038038038038 
equals 038/999 or 38/999.

For FOUR repeating digits, just put 9999 in the denominator.

In your problem, you have 06349 repeating. That's 5 digits. Can 
you figure out what to do with them to make a fraction?

If you're interested, there are a couple of other more difficult 
problems you can run into (note: the rest of this does NOT have to do 
with your problem). What happens, for example, if the decimal starts 
out NOT repeating, but then at some point starts repeating? An example 
of this is:

  0.000056565656565656...

[Where the 56 keeps repeating forever once it starts.]

Can you figure out how to deal with a problem like this?

Hint: 0.01111111... equals 1/90; 0.00111111... equals 1/900; 
0.00011111... equals 1/9000.

Even more difficult is a problem like this, where the misbehaving 
digits at the beginning are not just zeros:

  0.70643889889889889889889...

[Where the 889 keeps repeating forever once it starts.]

Can you figure out how to deal with a problem like this?

Hint: 0.80231111111111... equals 0.8023 plus 0.00001111111111...

All repeating decimals are rational numbers.  See the Dr. Math FAQ:

   Integers, Rational and Irrational Numbers   
   http://mathforum.org/dr.math/faq/faq.0.9999.html 

Hope this helps. If you have other questions about this or you're 
still stuck, please write back.

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