*In which we start our study of Diophantine approximation and meet continued fractions.*

- We concluded the proof of Bertrand’s postulate from last time.
- Proposition 56 (Dirichlet):
*Let be a real number and let be a natural number. Then there is a rational with such that .*We saw that we could prove this quite easily using the pigeonhole principle. - Definition of
*continued fractions*(finite and infinite), and*partial quotients.* - Lemma 57:
*There is a one-to-one correspondence between finite continued fractions and rational numbers.*We saw the important point that the continued fraction of a rational number must terminate, and saw the link to Euclid’s algorithm.

#### Further reading

Davenport (*The Higher Arithmetic*), Baker (*A concise introduction to the theory of numbers*), and Hardy and Wright (*An Introduction to the Theory of Numbers*) all have good sections on continued fractions, including material beyond what we’ll cover in lectures.

#### Preparation for Lecture 19

Our next job will be to think about infinite continued fractions. Can you find a continued fraction for ? Is it unique? If you truncate the continued fraction at some point, you get a finite continued fraction and so a rational number. How does it compare with ? Investigating these truncated continued fractions before the next lecture would be very helpful.

March 13, 2012 at 7:21 pm

[…] see how continued fractions can fit into a number theory course, see these three lecture […]