Friday, July 18, 2008

Review of Impossible? By Julian Havil

This is not a book for the layman. I found this book heavy going and problems with lack of explanation in places.
However there are a number of gems which are definitely worth exploring. These are:
Simpson’s Paradox. This is an example where (a/b) > (c/d) and (p/q) > (r/s) but
(a+p)/(b+q) maybe less than (c+r)/(d+s).
Connection between the continued fraction of 1/e and the optimal number of r out of n.
Why the infinite sum 1/n is called the harmonic series.
Why order is lost in complex numbers.
Moreover there are some fantastic quotes by various mathematicians such as the following by De Morgan:
The ratio of log of -1 to square root of -1 is the same as circumference to diameter of a circle.
This book must be read in conjunction with the author’s other title ‘Nonplussed!’ because he refers to it in a number of places. I have not read ‘Nonplussed!’ but I do think a book like this should be totally independent of any other text.
A major problem with the book is progression is too fast. It is difficult to digest an idea and the author has moved on to higher dimension. A good example of this is the ‘Monty Hall’ problem. The author describes the Monty Hall problem and within a couple of pages he has moved onto various extensions and generalisations of the problem. It would have been better to progress at a slower rate so that the reader understands the initial problem and then is able to follow the extensions on this problem.
In general I found myself taking a lot of time to get through it, because I had to keep going back and looking at things again and again in a bid to understand.
There are a number of typos in particular the brief appendix at the end seems to be full of them. The infinite series for sin and cosine is wrong. It should have alternating signs. There is no fig 4 which relates to subintervals. The proof of log(2) is irrational is incorrect.
This sort of text should have a lot more diagrams.