## Number Theory: Lecture 18

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 $\theta$ be a real number and let $N$ be a natural number.  Then there is a rational $\frac{a}{q}$ with $1 \leq q \leq N$ such that $| \theta - \frac{a}{q} | < \frac{1}{qN}$.  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.

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 $\sqrt{2}$?  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 $\sqrt{2}$?  Investigating these truncated continued fractions before the next lecture would be very helpful.