Review of ‘The Prince of Mathematics Carl Friedrich Gauss’ by M. Tent

This is a fantastic biography of Carl F Gauss (1777 to 1855). It is a well written book and the author, M Tent, will achieve her goal of inspiring readers to explore the world of mathematics with this book. This is a book for the layman but will be a real inspiration for a sixth former or undergraduate student of mathematics.
The author highlights some well known stories about the young Gauss such as:
By the age of 10 he knew the formula for the difference of two squares and had a smart technique of adding up the first 100 consecutive natural numbers.
The author also claims that Gauss tried to prove the parallel postulate using the first 4 postulates in Euclid’s Elements. Towards the end of the book Margaret Trent shows how this lead to Gauss develop non-Euclidean geometry 25 years before the Russian mathematician Lobachevsky published his work on this topic.
By the age of 11 Gauss could prove the irrationality of the square root of 2.
By the age of 18 Gauss had constructed a regular polygon of 17 sides using unmarked straight edge and compass only.
Some of the material in the book is from Gauss’s diary which has results in shorthand notation such as
∆+∆+∆=N
This result says that any natural number N can be written as the sum of at most 3 triangular numbers. The author claims that this was Gauss’s Eureka moment and was a beautiful discovery.
For his PhD, Gauss proved the Fundamental Theorem of Algebra and also showed the mistakes made by Euler, Lagrange and Alembert in their proofs.
He also made a major contribution to number theory and proved the Fundamental Theorem of Arithmetic.
The author goes on to describe the political turmoil in Germany particularly in the Duchy of Braunschweig (Brunswick). The Duchy of Braunschweig had supported Gauss financially for 15 years but he was killed by Napoleon’s forces in 1806.
After the Duke’s death, Gauss was offered a post at the prestigious University of Gottingen which he took up. He remained there for the rest of his life. At Gottingen he had to teach as well as do research but in the beginning he did not enjoy teaching. He suggested to his wife “I would prefer simply to give my students a text and if they encounter any problems they can see me.” Amongst his students at Gottingen were Richard Dedekind and Mobius. At the University of Gottingen he had become the director of the observatory and published papers on infinite series, astronomy, optics, number theory and algebra.
Additionally between 1833 and 1855 Gauss worked very closely with Weber at Gottingen in the field of magnetism. They also produced the first telegraph system.
Whilst at Gottingen Gauss produced a map of entire kingdom of Hanover by using his triangulation and least squares methods.
Gauss had become an important figure in mathematics throughout Europe and beyond. Even the Universities of Berlin and Petersburg tried to lure Gauss to work for them but he refused each time.
The book is a biography of Gauss with his family and friends at the centre of his life. The only matter of concern is we don’t really know which of the stories are factual.
However this is a smashing book and definitely worth buying.

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.

Review of Letters to a Young Mathematician by Ian Stewart

'Review of Letters to a Young Mathematician' by Ian Stewart is a book which explains why a sixth former should study mathematics at undergraduate. Additionally it is an excellent book to motivate undergraduates to study mathematics at postgraduate level.
However the book contains no or very little mathematics so it is digestible for the general layman. I would have preferred more mathematics in the book because this letter approach could have been a novel way to put over some mathematical concepts.
All the 21 letters start with 'Dear Meg' who is the niece of a factitious mathematician writing the letters. It is set up with a question from Meg (you do not see the question) and the reply from the mathematician. The life span of the letters is about 15 to 20 years starting with the explanation of why Meg should read mathematics at university and ending with benefits of tenure and collaboration. Although the title of the last chapter ‘Is God a Mathematician’ brings in historical quotes such as ‘God is a geometer’ by Plato, ‘God is a mathematician’ by Paul Dirac and ‘God is a Pure Mathematician’ by Arthur Eddington. The book becomes a fantastic collection of letters into the life of a mathematician.
Stewart has quotes sprinkled in his book from the classic ‘A Mathematicians Apology’ by G Hardy. It seems like the book being reviewed is a supplement to the 20th Century Hardy’s classic. Whilst Hardy glorified in his non-applications of mathematics, Stewart shows why mathematics is universal used throughout our lives. He does not make a major distinction between pure and applied mathematics.
Humour is sprinkled throughout the text such as the Dean of a Faculty counting the number of lights in the ceiling of an auditorium. When the mathematician points out that there is no point counting them because there are 8 rows by 12 columns of lights so making it 96 altogether the Dean replies ‘I want the exact number’.
This is an excellent book and definitely worth buying.

What is mathematics?

Paul Erdos said mathematics is like a machine which converts coffee into theorems and proof.
Marcus in his book "Finding Moonshine" says mathematician is a pattern searcher.
Lord Kelvin asked the question, whom do you call a mathematician?
He answered a mathematician is a person who finds the integral of e^(-x^2) from plus infinity to minus infinity as easy as you find 2x2=4.

GCSE Web Site

GCSE Web site at http://homepages.feis.herts.ac.uk/~matqkks/

Sport

Rather disappointing performance by Arsenal today. They lost 2-1 against boro.

Diary

After a good six months I managed to get a lovely bike ride around the local county lanes. Hope to do the same tomorrow. The weather seems to be getting better and really looking forward to the summer.

Mathematical Jokes

Cos(x), sin(x) and e^(x) go to a party. Sin(x) and cos(x) are partying away but e^(x) is miserable and anti social. Sin(x) and cos(x) go up to e^(x) and say 'what's wrong, why don't you integrate?'
It doesn't make any difference does it?

Beautiful Mathematics Formulae

Mathematics
It is amazing how 2 divergent series and sequence converge to give Euler's constant:
As n goes to infinity we have (1+1/2+1/3+1/4+...+1/n)-log(n)=0.577.