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Converting a Fraction Into a Continued Fraction

Date: 11/29/2003 at 14:11:52
From: George
Subject: How do you convert a fractions into a continued fraction?

Dear Doctor Math,

I've searched math books and the Web but have still not found an
answer to my question.  I understand how to convert a continued
fraction into a fraction but do not understand how to do the opposite.
I have tried to work backwards to solve my problem but nothing has
worked.  How can I do it?

Date: 11/29/2003 at 16:04:30
From: Doctor Rob
Subject: Re: How do you convert a fractions into a continued fraction?

Thanks for writing to Ask Dr. Math, George!

Start with your fraction written in improper form, then express it as
a mixed number by dividing the denominator into the numerator and
writing the remainder as a fraction.  The whole number part of this
mixed number is the number to the left of the first plus sign.  

Invert the fractional remainder to get a new fraction.  Now express 
that new fraction as a mixed number.  The whole number part of this is 
the number to the left of the next plus sign.  Invert the fractional 
part to get another new fraction.  Repeat this until the fractional 
part turns out to be zero, at which time you stop.

For example, take 314/75.

   314/75 = 4 14/75.  Remember 4, and work with 75/14.
   75/14 = 5 5/14.    Remember 5, and work with 14/5.
   14/5 = 2 4/5.      Remember 2, and work with 5/4.
   5/4 = 1 1/4.       Remember 1, and work with 4/1.
   4/1 = 4.           Remember 4, and stop.

The continued fraction is

   315             1
   --- = 4 + -------------
   75                1
             5 + ---------
                 2 + -----
                     1 + -

Get the idea?

Feel free to write again if I can help further.

- Doctor Rob, The Math Forum 
Associated Topics:
High School Number Theory

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