# Wilson Ong's Blog

## March 14, 2011

### Happy Pi Day!

Filed under: Miscellaneous — Wilson Ong @ 8:23 pm

Today is the 14th of March (3/14), Pi day. Happy Pi day everyone!

## June 25, 2010

### When is cos(2*pi/n) rational?

Filed under: Miscellaneous — Wilson Ong @ 5:05 pm

For which positive integers $n$ is $\cos\frac{2\pi}{n}$ rational? We present a simple solution to this problem using Galois Theory. We claim $\cos\frac{2\pi}{n}$ is rational if and only if $n=1,2,3,4,6$. Indeed, first note that for any automorphism $\sigma\in Gal(\mathbb{Q}(\zeta_n)/\mathbb{Q})$ (where $\zeta_n=e^\frac{2\pi i}{n}$), we have $\sigma(\cos\frac{2\pi}{n})=\sigma(\frac{\zeta_n+\zeta_n^{-1}}{2})=\frac{\zeta_n^j+\zeta_n^{-j}}{2}=\cos\frac{2\pi j}{n}$ for some $0\leq j\leq n-1$. But it is geometrically clear (from the unit-circle definition of cosine) that $\cos\frac{2\pi j}{n}=\cos\frac{2\pi}{n}$ if and only if $j=1,n-1$. Thus $\cos\frac{2\pi}{n}$ is fixed by at most two automorphisms. Now if $\cos\frac{2\pi}{n}$ is rational then it is fixed by every automorphism, so we must have $|Gal(\mathbb{Q}(\zeta_n)/\mathbb{Q})|=\phi(n)\leq 2$. This is only satisfied for $n=1,2,3,4,6$, and it is clear $\cos\frac{2\pi}{n}$ is rational for these $n$.

## May 23, 2009

### Mathematicians Land Top Spot in New Ranking of Best and Worst Occupations in the U.S.

Filed under: Miscellaneous — Wilson Ong @ 3:13 pm

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…”

## July 24, 2007

### Beginning of a New Blog

Filed under: Miscellaneous — Wilson Ong @ 9:05 am

The first post in this blog, two days after $\pi$ approximation day (22/7)… well it is only an approximation, 24/7 is still pretty close isn’t it? :s