Sunday, March 14, 2010
It is 3.14, the day we celebrate π. Laugh all you will, but without π circles would be impossible.
It is also Albert Einstein's birthday, so he is naturally our Wiki biography of the day. I admit, I never knew that he had been offered (and had refused) the position of president of Israel.
Princeton has a big celebration today, which reaches its crescendo at 3:14 pm. All the cool people will be there.
Well, actually not. Circles are circles. Pi arises only by defining a particular relationship involving circles. It is, in fact, "a derivative."
But what is worse about this post is the truncation of pi. Perhaps the most important mathematical fact about pi is that it is not only NOT rational (solves an equation of the form ax + b = 0, where a and b are in Z, but also is not algebraic, that is, satisfies no equation of the form a(n)x^n + a(n-1)x^(n-1) + ... + b = 0. It is "transcendental." Another famous transcendental number is e, which arises "naturally," no pun intended if you see the pun, in among other places, the calculation of compound interest where the compounding is done continuously.
But compound interest, calculated instanteously, would still exist whatever e, and vice versa.
Same is true of pi.
Now, note that sin(45 degrees) and cos(45 degrees) ARE algebraic. In fact, so are sin(60 degrees) and cos(30 degrees).
Can you explain why?