Yearning for the Impossible: The Surprising Truths of Mathematics

This is a wonderful book about the subject of mathematics. I learned a lot from this book which can be used in general class discussion to motivate students in a particular field of mathematics. It contains some real gems, such as:

• The word surd comes from the Latin surdus meaning deaf.
• The product of a sum of two squares is itself the sum of two squares.
• A simple example of an elliptic function is the height of a point on an ellipse given in terms of the arc length.
• The relationship between Gaussian primes and ordinary primes.
• Hamilton's quaternions, which is another number system.
• 3-D symmetry is rare because only 5 regular polyhedra are fully symmetrical.
• If 1 was prime then we would not have unique prime factorisation.
• 27 is the only cube which is 2 more than a square.

There are many others like this.

However this book is not really for the layman, because, as the author says in his preface, "many of the ideas are hard and there is no way to soften them'. The author has made extensive and fantastic use of illustrations to soften the blow and give an intuitive view of the mathematics, in particular in the chapter on curved space.

I really do like the historical context the author has placed his mathematics. I did not know that the Chinese had approximated π by the fraction 355/113, accurate to 6 decimal places. Nor did I know that the Leibniz series was known to Indian mathematician(s) well before it was discovered by Leibniz. (However, the author does not give the name(s) of the Indian mathematician(s) who had discovered this series.)

The chapter on the fourth dimension has an excellent historical context where the author describes Hamilton's search for the arithmetic of triples, quadruples etc. The chapter on the relationship between primes and Gaussian primes is also wonderful. The last chapter is a nice description of the different kinds of infinity and countable sets, including the interesting tidbit that "Many people believe the continuum hypothesis put Cantor in a mental hospital,'

In general there are some really good statements in the book which can be used to hook students to study mathematics, such as "Calculus as we know it today, is perhaps, the most powerful mathematical tool ever developed' and "Infinitesimal calculus brought almost the whole physical world within the scope of mathematics.'

I do have a number of minor quibbles. Sometimes a proposition or theorem is not clearly stated and it is only when you are half way through a proof you realise what is happening. There are statements which do not relate clearly to mathematical reality, such as "the square root of a half a turn,' which turns out to mean a rotation though a right angle.

There are a number of places where the book jumps from a simple introduction to the more advanced level within a few pages. For example, we are introduced to the idea of a complex number and within ten pages we are discussing Bézout's Theorem. In places I am not to sure what the author means. For example, on page 45 he says "complex analysis is known for its regularity and order.' Clearly complex numbers don't have order, so what does he mean?

Kuldeep Singh's homepage is at http://www.mathsforall.co.uk/