Clear Kets

I don't know why I'm a fool for threads like this, but since everyone gets to put his or her two cents in... In one sense, one that may be clear only to me, Lou Reed said it best: So, from a life-centric point of view (the point of view that has a point of view)... well, you get my drift... Let's just say that the purpose of having a point of view is to express it.

Petra Bonfert-Taylor gives a lovely example in the first lecture of her introductory complex analysis MOOC at Coursera on how the study of certain cubic equations led to an understanding of the necessity of extending the number system into the complex plane. She mentions 15th/16th Century mathematicians Tartaglia, Del Ferro and Cardano and then discusses Bombelli's Problem, the solutions of \(x^3\ =\ 15x\ + 4\).

I remember an introduction I had to the SML programming language. SML does not even provide a loop construct in the basic out-of-the-box syntax. The classic recursion disaster, even given a termination condition, is computing the nth number in the Fibonacci sequence. There are 2n recursive calls. I forget whether tail recursion helps with this challenge. I thought I had down the MathML/LaTeX that worked for me in my first post, but the math-fu isn't there today, not even for a simple exponent 2 to the n.

\(\textbf{v}\ \in\ \mathbb{R^n}\) Hmm... guess not. Oh, well, season's greetings, all and sundry. OK, I've tried A. I've tried B. Tried C, which was reading the instructions, and that worked! Hooray!