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## Continued Fractions

We return to and . Note that

Notation 4

Note that , , and are just the 's in the Euclidean algorithm. The rational is written as a continued fraction

with all the , for and .

Lemma 6.4.1   Every rational with and has exactly one expression in this form.

Proof. Existance follows immediately from the Euclidean algorithm. As for uniqueness, suppose that

with the 's as before. Firstly as both are equal to . Since then

Thus and so on.

Now, suppose that given we wish to find equal to it. Then we work out the numbers and as in the Euclidean algorithm. Then by lemma (2.22). If we stop doing this after steps we get . The numbers are called the convergents'' to . Using lemma (2.23), we get that . Now the are strictly increasing, so the gaps are getting smaller and the signs alternate. We get

The approximations are getting better and better; in fact .

Next: Continued fractions for irrationals Up: Rational numbers Previous: Rational numbers   Contents
root 2002-06-10