Page123Sentence6to8Page 123, Sentences 6 to 82 October 2006 Ordinarily I thumb my nose at blog memes, but someone I can't refuse has tagged me with the "sentences 6-8 from page 123 of your nearest book" one, so here goes: I'm also going to cheat by reaching for the books on my bedside table rather than those under my monitor since I'm not sure telephone directories have sentences per-se and I'm certainly not going to quote anything from Sommerville's Software Engineering—there's a reason it's under my monitor. From the complex perspective we see that 1/z is indeed a single function. The one place where the function 'goes wrong' in the complex plane is the origin, z = 0. If we remove this one point from the complex plane, we still get a connected region. It's from The Road to Reality by Roger Penrose. I'm not a big fan of Penrose's Mind series ([1], [2], [3]) but his new book appeared much more closely focused on what he's famous for (mathematical physics) and so I bought myself a copy with some of the book vouchers I received on my birthday. In the quoted sentences he's examining how 1/x (where x is a real number) which appears to be disconnected at x = 0 (the graph seems to be two separate curves) becomes a connected graph when we extend it into the complex plane as the function 1/z (where z is a complex number). I'm about a sixth of the way through and so far I've been getting exactly what I wanted out of it - insights into which pieces of mathematics Penrose considers important for physics and why. We'll see how the rest goes. :) |