ABSTRACT: Lego Bricks are very good at making rectangular structures: they have right angles built right into them. But have you ever tried to build diagonals at different angles? Pythagoras can help in a very obvious way, but we will extend classical results using something called a continued fraction. We`ll learn just what that is, and get very good at approximating irrational numbers. A warning to participants: During this talk, you will play with Lego bricks.
Level: I

ABSTRACT: We will present the motivation for a theorem which gives a necessary condition for the simple continued fraction of any real algebraic number. We point out that the contrapositive of the theorem can then be used as a sufficient condition for the transcendence of a number and we apply it to the number e in particular. We also consider the converse and provide a counter example, showing that it is not true in general. In conclusion we mention a problem originally posed by Jacobi, which the theorem does not come close to answering but which may be answered perhaps by applying Galois Theory to continued fractions.
Level: II