Review of Lewis Carroll in Numberland by Robin Wilson
The book tells the story of Charles Dodgson who is better known as Lewis Carroll the author of various fictions such as Alice in Wonderland and Through the Looking Glass. The book is divided into eight fits and describes how Charles Dodgson was not just the writer of fictions but also a professional mathematician contributing to linear algebra, logic, mathematical puzzles, geometry etc. The book is essentially a biography of Charles Dodgson with a few opening quotes of Carroll’s work.
Charles Dodgson was born in 1832 in Cheshire and studied at Oxford graduating with a first class honours in 1854. One of his hints in studying mathematics was:
“Never leave an unsolved difficulty behind. It is bound to haunt you in some proof or solution later on”.
Wilson also describes in detail the great interest that Charles Dodgson took in photography. He claims that Charles become one of the most important photographers of the 19th Century. The book is sprinkled with some of the images that Charles photographed throughout his life.
It is good to see that the author does not shy away from putting some of the mathematics that interested Charles Dodgson. The mathematics in the book ranges from his defence of Euclid’s Elements to his book on Elementary Treatise on Determinants. However his main interest was in mathematical logic in which he wrote Symbolic Logic which was published in 1896. He also wrote various mathematical puzzles.
Over the last 100 years a lot has been written about Dodgson’s interest in children normally suggesting something disturbing but Wilson refutes all these claims. I do wonder how the political correct will accommodate this refutation with the book containing photographs of young children taken by Dodgson.
Wilson describes how not only is Charles Dodgson a mathematician and an author but also a deeply religious man and a keen walker.
There are some real engaging stories about Charles Dodgson such as when he put a case for a Mathematical Institute at the University of Oxford in 1868. However Oxford had to wait another 65 years before a Mathematical Institute was built.
Another fascinating story the author describes about Charles Dodgson is when his Oxford College (Christ Church) was in financial difficulty. Dodgson proposed that his salary be lowered from £300/year to £200/year. In present day circumstances this would be an unthinkable (or even stupid) act!
In 1881 Dodgson, aged nearly 50, resigned his mathematical lectureship so that he could devote more time to writing books.
Charles Dodgson passed away in 1898 aged nearly 66 in Guildford.
The book is really well written with both characters, Charles Dodgson and Lewis Carroll, being described as a mathematician and an author of fiction.
The book can be hard to follow in places if you are not familiar with A level mathematics but it is possible to skip these parts and maintain the flow of the book. It is a hard balance to strike between putting mathematics into a book like this which can lead to decreased sales and having no mathematics which would be a very serious omission. Robin Wilson has struck the right balance between these two conflicting notions.
Tuesday, October 28, 2008
Wednesday, July 30, 2008
Review of ‘The Pythagorean Theorem’ by Eli Maor
This is an excellent book on the history of the Pythagorean Theorem. I learnt a great deal of history of mathematics in relation to Pythagoras’s Theorem. This book is suitable to any student who has basic knowledge of calculus but the layperson will also find it interesting.
The book starts with the assertion that the Babylonians knew Pythagoras’s Theorem 1000 years before Pythagoras but it was the Greeks who proved the result.
There are a number of gems in the book which are not that well known in the mathematics community:
Hypotenuse is derived from the Greek words hypo meaning ‘under’ or ‘down’ and teinen meaning ‘to stretch’. Maor points out the reason for this is that the hypotenuse of a right triangle in Euclid’s Elements was always on the bottom. (I did not know this).
There are over 400 proofs of Pythagoras’s Theorem.
It was the French lawyer ‘Francois Viete’ who first converted verbal algebra into symbolic algebra.
Many more of these gems crop up throughout the book.
Maor does give a number of different proofs of Pythagoras’s Theorem.
More importantly the author does not shy away from producing mathematical expressions and symbols in a popular book like this. Here are a few examples:
1. Every even perfect number is of the form 2^(n-1)*(2^n-1).
2. Viete’s Identity product which expresses 2/π in terms of √2.
3. Shows how the area of one arch of the cycloid is 3 times the area of the circle generating it.
4. Gives an excellent brief description of Hilbert Spaces and non Euclidean geometry.
5. Explains why Pythagoras’s Theorem is not valid in non-Euclidean geometry.
There are many more fantastic mathematical examples. The more serious mathematics is left for the appendices.
Additionally Maor has provided an excellent general history of mathematics such as:
The first woman mathematician was Hypatia (370 to 415).
The University of Gottingen was world renown for mathematics up until the Second World War.
How Edmund Landau (1877 to 1938) shunned all references to geometry. Maor points out that Landau wrote a 372 page book ‘Differential and Integral Calculus’ and it does not contain a single illustration.
How Euler discovered differential geometry but its modern form is due to Riemann and Gauss.
There are also non-mathematical examples of history in the book such as the first European University was Bologna founded in 1088 and why the Christians burned the Library of Alexandria.
You will learn a lot from this book because it has been thoroughly researched and shows the different fields where Pythagoras’s Theorem is used.
The author has also made excellent use of illustrations so the layperson can understand without learning all the details.
Maor has an exceptional method of writing very technical mathematics in a seamlessly way.
The book starts with the assertion that the Babylonians knew Pythagoras’s Theorem 1000 years before Pythagoras but it was the Greeks who proved the result.
There are a number of gems in the book which are not that well known in the mathematics community:
Hypotenuse is derived from the Greek words hypo meaning ‘under’ or ‘down’ and teinen meaning ‘to stretch’. Maor points out the reason for this is that the hypotenuse of a right triangle in Euclid’s Elements was always on the bottom. (I did not know this).
There are over 400 proofs of Pythagoras’s Theorem.
It was the French lawyer ‘Francois Viete’ who first converted verbal algebra into symbolic algebra.
Many more of these gems crop up throughout the book.
Maor does give a number of different proofs of Pythagoras’s Theorem.
More importantly the author does not shy away from producing mathematical expressions and symbols in a popular book like this. Here are a few examples:
1. Every even perfect number is of the form 2^(n-1)*(2^n-1).
2. Viete’s Identity product which expresses 2/π in terms of √2.
3. Shows how the area of one arch of the cycloid is 3 times the area of the circle generating it.
4. Gives an excellent brief description of Hilbert Spaces and non Euclidean geometry.
5. Explains why Pythagoras’s Theorem is not valid in non-Euclidean geometry.
There are many more fantastic mathematical examples. The more serious mathematics is left for the appendices.
Additionally Maor has provided an excellent general history of mathematics such as:
The first woman mathematician was Hypatia (370 to 415).
The University of Gottingen was world renown for mathematics up until the Second World War.
How Edmund Landau (1877 to 1938) shunned all references to geometry. Maor points out that Landau wrote a 372 page book ‘Differential and Integral Calculus’ and it does not contain a single illustration.
How Euler discovered differential geometry but its modern form is due to Riemann and Gauss.
There are also non-mathematical examples of history in the book such as the first European University was Bologna founded in 1088 and why the Christians burned the Library of Alexandria.
You will learn a lot from this book because it has been thoroughly researched and shows the different fields where Pythagoras’s Theorem is used.
The author has also made excellent use of illustrations so the layperson can understand without learning all the details.
Maor has an exceptional method of writing very technical mathematics in a seamlessly way.
Tuesday, July 22, 2008
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.
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.
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.
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.
Thursday, June 12, 2008
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.
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.
Thursday, April 24, 2008
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.
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.
Monday, April 21, 2008
Sunday, December 09, 2007
Saturday, March 10, 2007
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.
Thursday, March 01, 2007
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?
It doesn't make any difference does it?
Thursday, September 07, 2006
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.
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.
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