Today is the 14th of March (3/14), Pi day. Happy Pi day everyone!
March 14, 2011
June 25, 2010
For which positive integers is rational? We present a simple solution to this problem using Galois Theory. We claim is rational if and only if . Indeed, first note that for any automorphism (where ), we have for some . But it is geometrically clear (from the unit-circle definition of cosine) that if and only if . Thus is fixed by at most two automorphisms. Now if is rational then it is fixed by every automorphism, so we must have . This is only satisfied for , and it is clear is rational for these .
May 23, 2009
According to an article in the Wall Street Journal, mathematicians have the best job in the U.S. “According to the study, mathematicians fared best in part because they typically work in favorable conditions — indoors and in places free of toxic fumes or noise — unlike those toward the bottom of the list like sewage-plant operator, painter and bricklayer. They also aren’t expected to do any heavy lifting, crawling or crouching — attributes associated with occupations such as firefighter, auto mechanic and plumber… The study also considers pay, which was determined by measuring each job’s median income and growth potential. Mathematicians’ annual income was pegged at $94,160…”