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Exercises for Diophantine Approximation
Last revised: September 16, 2004.
Please hand these in, on this sheet of paper, at the class after the last minilecture on the topic is presented.
1. Using the pigeonhole principle, find a rational approximation to e with denominator of 10 or less 
for which | e - | < . Start by using a calculator or Mathematica to calculate e, 2e, ... 10e, then proceed as was done for p.

2. Prove that the real number a defined by the infinite series
a = is irrational.
According to the theorems that are proved in Miller, all you need to do is to show that there are
infinitely many fractions for which | a -  | < . As with Liouville's proof, assume that there are only finitely many, and show that this assumption leads to a contradiction. The fact that you have an alternating series makes things easy.