Wednesday, April 15, 2009

Rotations and translations revisited

Bowling Green, April 3 - "Rotations and translations revisited" (contributed talk, 2009 Ohio MAA Spring Meeting). ABSTRACT: "By using a series of elementary plane geometry problems, we will illustrate the general result stating that the composition of a number of plane rotations of angles adding up to an integer multiple of 2*pi is a plane translation. We will discuss the case of collinear rotation centers, the case in which the rotation centers form a regular polygon, a general connection with discrete Fourier transforms, and also a limit case in which the number of rotation centers goes to infinity."

Seven Bridges of Königsberg...

Monday, April 6, 2009

An irrational logarithm

A short problem proposed by myself for a recent student team competition, held in Bowling Green:

Show that the logarithm in base 4+sqrt(5) of 6+sqrt(5) is irrational.
The proof proceeds by contradiction. Assume the logarithm in base 4+sqrt(5) of 6+sqrt(5) equals p/q with p,q positive integers. Then

(4+sqrt(5))^p=(6+sqrt(5))^q (*)
By taking conjugates,

(4-sqrt(5))^p=(6-sqrt(5))^q (**)
Multiplying (*) and (**) term by term we get

The fundamental theorem of arithmetic tells us this cannot happen.