## 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.

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.