Love and Math by Edward Frenkel

The author is a professional mathematician and describes what it means to do mathematical research. There are some real gems in this book such as:

Which functions (Euler phi function) are used to encrypt credit card numbers.

Which is larger 2/3 or 3/5 – Most people know that 2 bottles of vodka for 3 people is better than 3 bottles for 5 people.

How we can obtain the Fibonacci sequence from its generating function.

What does a finite field mean.

There is also a good explanation of the Langlands Program.

Very few typos (the only two I picked up were both on page 85 - should be divisible not visible and penultimate paragraph should start with ‘As we’.)

It is really pleasing to see that the author does not shy away from the mathematics in his writing.

However this is not a book for the layman because the mathematics is totally inaccessible to the general audience. This is not a piece of writing for a popular audience as I mistakenly believed it was after reading a review before purchasing. To fully appreciate this book you should at least be an undergraduate in mathematics or physics as it is tough going in places.

Throughout the book the author has highlighted his personal struggles of being Jewish with the regime of the Soviet Union. This is really interesting as I was unaware of how the regime thought of all Jews as opponents, criminals, foes (these are my words). When depicting this the author mentions a number of locations in Russia which he should have illustrated with maps as most of us in the West will not be able to visualize the locations.

Additionally I have a few minor quibbles:

There are a number of terms omitted from the glossary such as invariant, winding number.

Should have explained the term monodromy through illustrations.

The author claims ‘It is customary to exclude 1 from this list” (of primes). I always thought of 1 as neither prime nor composite, just a unit.

I conclude that this is not a book for the layman.

Kuldeep Singh 15^th April 2015

The Education of a Mathematician by Philip Davies

This book is different from most other books for the mathematics layman. It is a very tangible and personal journey of the author from the 1930’s to end of the 20 century. In general I found it a very readable book on mathematics as well as mathematics education. There is also philosophy discussions in the book.

The author is a professional mathematician who gives his personal experience and what it means to be a mathematician with his travels to various countries.

The author discuss some really important questions such as the future role of mathematics and it applications. Also he reflects back at how mathematics has affected all of us in the past.

The last chapter describes the authors’ talk at the ‘The International Congress of Mathematicians’ in August 1998 given in Berlin. I really like this chapter with subheading such as Male vs Female Mathematics and Soft Mathematics vs Traditional Mathematics.

My only reservation is that it was hard to follow in places as the author is also a philosopher. 

Review of Mathematics in Victorian Britain by R. Flood, A. Rice and R. Wilson

This book is divided up into mathematics of various era and geographical locations. It gives a flavour of mathematics and mathematicians not only in Britain but also in the Commonwealth.

Since each chapter is written by a new author I found some of the chapters pretty technical and others very accessible where the emphasis is on the people involved. A glossary might have made the book more accessible on the technical side but it would have to be such a large glossary that the book loses its compactness.

There are some really interesting snippets about mathematicians and their social life. I found the last chapter by Jeremy Gray the most interesting and controversial and will let you read this to find out why.

However I do have one serious reservation – the size of the font is far too small and in general I struggled to read this book. Hence the three stars.

Any serious history of mathematics student should purchase this book as it provides good motivation to the methods and approaches that lie behind the mathematics.

The Simpsons and their Mathematical Secrets by Simon Singh

Until I read this book I was unaware of the sprinkling of mathematics in the Simpsons.

The book discusses the mathematicians on the writing teams of the Simpsons and Futurama. I must say that I was surprised to learn that most of the writers on these programmes are graduates or postgraduates of mathematics or physics.

It is an excellent book for a layman as all the mathematics is explained in detailed.

My reservation is that the book is a series of detached mathematical stories whose only common theme is that these stories appeared in the Simpsons or Futurama. Also these mathematical stories are so common that if you have read any mathematical popular book you would have come across them. So nothing new here.

However I do think any serious mathematics student should purchase this book as it provides good motivation to study mathematics at undergraduate level.

 

 

I am so proud that the front cover of my book is Mandela Bridge in South Africa.

Book of Proof by Richard Hammack

For any student who wants to read mathematics at undergraduate level should purchase this book. It is an introduction to mathematical proof and at its present price of approximately £8 is a good buy. I think you can download it free from the authors’ personal webpage.

It has a large number of exercises to reinforce your understanding and some good illustrations.

I do not know who published this book but it lacks a polished output and gives the impression that it has been printed on a local printer and bounded together; maybe this is why it can be sold at such a cheap price. Additionally it only gives solutions to odd numbered questions which can be off putting to a student who wants to use this book for self-learning.

How to study for a Mathematics Degree by Lara Alcock

2^nd April 2013

This is a book that every potential undergraduate student of mathematics should purchase. It has a very thorough description of what is involved with being an undergraduate in mathematics at university.

Alcock has succeeded in writing a relevant, interesting and beneficial account of what it is like to be a maths undergraduate. She highlights the differences between pure and applicable mathematics and what skills are needed to flourish in these. However the emphasis is on pure mathematics throughout the book.

Additionally the author does not shy away from doing some serious mathematics even though the book is written for the layman. The technical mathematics is described in simple terms that all sixth formers will understand.

The book is very well organised and has good quality writing style – well expressed, clarity, coherence etc. It is split into two – mathematics and study skills.

My only reservation is the lack of illustrations. Including these would have more of an impact but it is not a serious omission.

I would recommend this book to any student or teacher of mathematics.

Kuldeep Singh

Review of The Mathematical Universe by William Dunham

This is a popular mathematics book for the layman. The author has succeeded in writing interesting articles in mathematics which are laid out alphabetically. Dunham discusses the problems and personalities in the history of mathematics. If you want a student hooked into mathematics then this the book they should read.

The technical mathematics in the book are excellently described in simple terms that the average layman can understand. Dunham really does bring mathematics to live in this book.

Initially I was reluctant to purchase a book which is mapped out in alphabetic order. However Dunham managed to make the progression from one chapter to the next seamless. Chapter 1 is on arithmetic and the last chapter called Z is on complex numbers. 

The book is very well organised and has good quality writing style – well expressed, clarity, coherence etc. Dunham has struck the right balance between writing thoroughly and for the amateur or non-professional. Illustrations are sprinkled throughout the book and the author has used these with dramatic effect.

I would recommend this book to any student or teacher of mathematic as well as for non mathematicians.

Kuldeep Singh 

1st Feb 2012

The Story of Mathematics by Anne Rooney

27 December 2011

This book is full of interesting facts on the history of mathematics such as where our symbols + , – , = and square root originated from. There are also details of mathematics of the 20^th Century such as fractals and fuzzy logic. In places the book is fascinating reading such as ‘Pascal’s Triangle is called Khayyam’s Triangle in Iran.’ A student doing mathematics would find this book intriguing and learn some entertaining facts about the history of the subject.

The author has made good use of colour in diagrams but the diagrams are not referenced. There is also no caption for tables.

The layout of some details is rather peculiar. For example page 27 of the book claims that minus sign was first used by Johannes Widmann but does not mention who Widmann is until page 130.

In general the book is full of interesting facts but does lack detail in places. I think it would have been a better book with fewer facts but more details and mathematics about some of these facts.

I personally do not like the text layout in two columns per page. It just doesn’t flow as well as a traditional one column per page book. Additionally it is confusing in places with various diagrams and boxed information on the same page. However I can see the advantage of being a portable book of 208 pages, something that you can fit into your pocket.

Font size of the comprehensive index is rather small with three columns to the page.

Even with these reservations I would recommend this book to any student or layman who is interested in the history of mathematics.

Kuldeep Singh

Femininity, Mathematics and Science, 1880-1914

This is an excellent account of the history of women in the academic world of mathematics, science and engineering between 1880 and 1914.

Jones focuses on the barriers women faced at the turn of the 19^th century such as the scientific laboratory which was seen as a harsh environment for women. These labs were places of manliness and heroism. An example is Cavendish Laboratory at Cambridge which is named after Henry Cavendish who passed electric current through his own body. Even Darwin had theorised that women’s intellect was not on par with men and women were lower down the evolutionary scale, closer to animals. The author also provides an interesting history of the suffragette’s movement.

Jones has concentrated on two particular women, Hertha Ayrton and Grace Young, who both attended Girton College Cambridge to study mathematics. The author states that Hertha was the first Jewish woman to enter Cambridge and that she renounced Judaism so that she could assimilate into the middle class scientific society. Both women took the mathematical tripos examination at Cambridge with Hertha Ayrton opting for the applied mathematics route whilst Grace chose pure mathematics.

The author describes how by the end of the 19^th century men were deserting the mathematics tripos for natural sciences tripos which consequently made this masculine in character. However the new women’s colleges (Girton and Newnham) retained their preferences for mathematics.

The author makes a really fascinating point on the use of language in mathematics (something I did not realise) . ‘The feminised language distinguished pure mathematics from the applied and helped women to feel comfortable within the discipline’. Her she is referring to proofs being elegant and theorems - beautiful. Jones gives some really interesting definitions of pure mathematics such as ‘Mathematics is absolute knowledge, permanent and unchanging over time’. The author goes on to say ‘Around 1900 Pure Mathematics prided itself on being uncontaminated by the real world’.

Jones highlights how mathematics was changing by the end of the 19^th century. It was moving away from algebraic geometry to a more abstract area such as set theory and Gottingen in Germany led the way. Gottingen was the leading centre for mathematics by the end of the 19^th century. Grace Young moved to Gottingen between 1900 and 1908 and became a member of the Gottingen Mathematics Club. By this time it was not unusual to see a small group of women in Gottingen mathematics lectures. However the author highlights some serious impediments on women because there were severe lack of academic openings for women in Germany as well as England.

Grace’s work at Gottingen had an enormous influence on the development of Cambridge mathematics according to the author. Her and her husband were at the forefront of the new mathematical analysis.

Hertha Aryton’s research at Central College London was in arc lights being used for search lights, street lights and other public lighting. When her husband died in 1908 she had no further dealing with Central College London.

Jones states that women encountered fewer obstacles in infiltrating mathematics as it did not necessarily require an institutional base. Hertha required a lab whilst Grace needed pencil, desk and access to a mathematical education.

In general this book is an excellent history of the barriers faced by women in academic circles around the 1900’s. I was not aware of such rich history and there are very few books in this particular field. I particularly liked the rare photographs in the book and would have preferred to have more of these. Also the author has made good use of graphs by highlighting the number of male and female students taking the mathematics and natural sciences triposes. My only gripe would be the cost of the book at £55. I think this book should be within the financial budgets of students.

I only found a couple of typos. On page 40 it should say 1890’s not 1990’s. On page 169 it should say 23 mathematical problems rather than 123 mathematical problems.

Kuldeep Singh

What’s Luck Got to Do with It? By Joseph Mazur

The book is divided into three parts – history, mathematics (probability and statistics) and the psychology of gambling.

Generally speaking this is a well written contemporary book including many interesting facts, such as the term ‘crook’ originates from the crooked dice used by gambling cheats. Another interesting section is the author’s exploration of the significance of the number 7 – the colours of a rainbow, the number of days in a week, the sum of the numbers on opposite sides of a cubical die.

Joseph Mazur gives a very tangible description of the law of large numbers, using the example of the probability of heads being approximately ½ for a large number of throws of a coin. Using the same theme the author highlights examples of the difference between the terms expected value and the mean. There is also an excellent account of the Normal Distribution.

The author gives a good description of the recent international financial crisis and reveals how closely linked Wall Street is with Las Vegas.

Mazur has made good use of illustrations, in particular when explaining possible outcomes of a game like throwing dice (Page 21).

My gripe is that sometimes numerical values are given but it is not made clear where these values have come from. For example on page 136 when referring to the lottery in New Hampshire, the book states ‘The total number of combinations of picking all six numbers is 5,245,786’. Whilst this is correct it would have been helpful to write this number as the combination formula C with superscript 42 and subscript 6 so that the reader knows that you are selecting 6 out of 42.

An issue which might cause problems for the reader is the flow of the text. The book is peppered with callouts which are mathematical justifications of particular statements. These are located at the back of the book with reference in the main text. Hence the reader is continually flicking back and forth. I think these callouts should be in the main text. You can of course choose to ignore these callouts and still understand the general concepts.

While not a serious omission I would like to have seen a callout defining standard deviation, as it is a term used throughout the mathematics section.

The book would have added substantial value if it included answers to popular statistical questions, such as: why car insurance is cheaper for women.

I only found one typo on page 121 where the Greek symbols mu and sigma are missing in the statement before the formula. Additionally this statement should follow after the formula.

Review of Linear Algebra and its Applications by D.C. Lay

The most obvious benefit of this particular book is its size. At fewer than five hundred pages it is a comparatively light book. Despite its compact nature, it does include some clear and helpful illustrations, and features some interesting applications
However, the layout is far from appealing, and at first glance can be confusing. The answers to the random exercises can be difficult to locate, and the solutions are often too brief to be of any use. In general the proofs are difficult to follow because they are in compact notation, and like many books of this type, it makes little provision for students struggling to remember previously mentioned results and definitions. In some cases a theorem is stated on page X and then the proof of this theorem is given on page X+2 without restating the theorem. This means the reader is constantly flicking between pages X and X+2. The inclusion of repeated theorems would not hamper the confident mathematician, but would be invaluable to a less self assured student.
There seems to be little attempt made to inspire the reader, and its approach is largely clinical and less personal.

Review of Elementary Linear Algebra by Larson and Edwards

This book has an attractive layout, with its theorems and definitions clearly and repetitively presented in highlighted boxes. It is easy to follow and could certainly prove useful to a weak student.
However, many results are stated without providing proof, leaving the student to produce the proof as part of the exercise. Underpinning this type of mathematics with proof is certainly a fundamental part of the education process, but in my experience many students find this an intimidating step. I believe it is vital that students have proofs demonstrated repeatedly, until they become confident enough to formulate their own. This book could undoubtedly benefit from the inclusion of more examples.
In addition, there is a noticeable lack of illustrations, giving the book a very dense feel.
Generally speaking, while this book might provide a valuable handbook for some students, I don’t believe its style engages the attention of the reader and it doesn’t attempt to provide a thorough explanation of linear algebra.

Review of Elementary Linear Algebra by Anton and Rorres

This is a polished, well written text and features a wide variety of interesting contemporary applications towards the end of the book
Its definitions and theorems are highlighted and boxed, and it makes excellent use of illustrations.
However, after their initial introduction, the definitions and results are not reprinted. This makes the book awkward to use and the reader is often forced to search back through the book to find a particular formula or theorem. It is my opinion that the average student is unlikely to absorb this information at their first attempt, and I think it is important to repeat important points until the student is completely familiar with them.
This book also fails to summarise each chapter. My experience of students is that they find a modular approach to mathematics easier to absorb, and I think it is imperative that at the conclusion of each section, new information is condensed and consolidated before the next module commences.
Its explanations are somewhat brief in places, and it uses a compact notation. A confident mathematician would find this method of presentation clear and concise, while the nervous undergraduate finds this kind of brevity difficult to interpret.

Journey through Genius by William Dunham

This book gives a thorough treatment of the history of important mathematical results. There are a number of interesting mathematical examples set in an historical context which makes the book very joyful to read.
The author has hit the right balance between the mathematics such as proofs of theorems and the history behind each theorem. It is good to see the author does not shy away from producing proofs of results which many popular writers tend to eschew so that they can increase their sales. This can be challenging at times for the reader but they can skip these without losing the flow.
Dunham has a fantastic writing style which keeps the reader hooked and intrigued.
Another great asset of the book is that it is portable and reasonably cheap at around £10. I manage to read most of it in Starbucks with pen and paper of course.
However I have following reservations:
Font size is too small and it is particularly difficult to read some of the fractions.
Two typographical errors are – On page 170 the fraction should be 5/128 and not 5/12. On page 238 the factorization should be over 2a and not just a.
On page 235 the first Fermat prime 3 is missing.
A less serious issue is that once the author has covered a particular concept he expects the reader to fully digest it. Dunham has tackled this by signposting his earlier results but I think it would have been more readable for the layman to see the statement of the result again.
This is a book for anybody interested in history of mathematics or mathematics in general. You do not need to be a mathematician to appreciate this book.
Overall I would say this is an excellent book and would recommend anybody interested in mathematics to purchase this.
Kuldeep Singh
24th August 2009

Review of ‘Recountings’ by Joel Segel

This is a fascinating history of mathematics at MIT covering the period of post WWII to the present. It gives a detailed account of how mathematics at MIT was converted from a service department for engineers to the high class research department over this period.
This is a book for the layman and also academics who work in such environments. Many mathematics departments throughout the world could do well be investing in this book and emulating a lot of the work done at MIT. The book highlights that the main reason for the transformation was by hiring some of best mathematicians in the world.
The book has a different style to mathematics books for the layman in the sense that it is based on a collection of interviews with 12 members of MIT and the widow of Norman Levison.
The author highlights some well known stories about the men (apart from the first interview which is with Fagi Levison all the other 12 interviews are with men) in mathematics at MIT such as how they were hooked into the institute from other organisations. Additionally the book highlights the arguments within the department between pure and applied mathematicians.
My main reservations regarding the book are:
• It contains no index. This is a serious omission.
• The book does not read well in places and it should have been more thoroughly reviewed.
Generally it was a good joy to read this book and definitely worth buying.
Kuldeep Singh

Review of ‘The Unfinished Game’ by Keith Devlin

I really enjoyed reading this book. It is an excellent account of the history of probability theory. It is essentially a correspondence between the two great French mathematicians, Fermat and Pascal.
The book has an interesting hook with the opening paragraph being a letter sent by Pascal to Fermat. The basis of this correspondence asks ‘how should we divide the stakes if a particular game is incomplete’. This lays the ground work for probability theory.
I liked the style of the author and the way he dipped into some straightforward mathematics in this book.
The history is particularly appealing with the explanation of how Graunt developed his mortality tables. It also goes on to state that Newton’s first great mathematical discovery, the binomial theorem, is based on Pascal’s triangle. Additionally the book explains with entertainment detail the personalities of Fermat and Pascal.
There are also some very fascinating applications of probability mentioned in the book such as how the repeated use of Bayes theorem predicted an attack on the pentagon and also the explanation of why DNA profiling is so reliable.
However the book has the following shortcomings:
It should have stated the dates of birth and death of all the mathematicians mentioned in the book.
On page 83 the author misses the first Fermat prime 3.
The last sentence in the second paragraph on page 102 should say ‘bet 24 to 40, that is 3 to 5, that a sixteen year old will die before the age thirty six’.
This is a book for anybody interested in history of mathematics or mathematics in general. You do not need to be a mathematician to appreciate this book.
Overall I would say this is a very successful book and would recommend anybody interested in mathematics or history of mathematics to purchase this book.

Review of ‘The Indian Clerk’ by David Leavitt

I really enjoyed reading this book. It is an excellent account of mathematicians and other academics at Cambridge between 1913 to 1919.
Initially it was difficult to know which parts of the book was fiction and which was fact. It is essentially based on historical facts but with entertaining fiction sprinkled throughout. However it is only when you read the ‘Sources and Acknowledgments’ section you realise the scattering of fiction.
I did like the style of the author and the way he dipped into some mathematics in this book. You do feel that you have understood some of background mathematics behind Ramanujan, Hardy and Littlewood.
However I did not see the need for the visual detail of gay sex in such a book. This should not put the reader off because it is only highlighted in rare instances.
This is a book for anybody interested in history of mathematics or mathematics in general. You do not need to be a mathematician to appreciate this book.
Overall I would say this is a very successful book and would recommend anybody interested in mathematics fiction or fact to purchase this book.

Review of ‘Taming The Infinite’ by Ian Stewart

This is a fantastic book on the history and applications of mathematics. It starts with Hindu Arabic numerals and ends with Chaos Theory. Of course every major mathematical theorem or topic during this period could not be stated in a book of this size and as the author states he had to be selective. My only complaint about the selection is the lack of ‘linear algebra’ because it is perhaps the second most (first being calculus) powerful mathematical tool ever invented.
To fully appreciate this book you must have a reasonable mathematical ability such as a good pass at Further Mathematics A level or equivalent.
Stewart does write in a way that will appeal to most readers and also you can dip into any chapter without digesting the previous chapters. The author has hit the right tone and progression.
A lot of research and time must have been invested into writing this book because of the coverage of applications, the history behind important mathematical developments, profiles of the leading mathematicians etc. I really do like the broad range of mathematical applications throughout the book. The author explains where differential equations are used in the field of physics and modern technology such as radio, tv and commercial jet aircraft and how important Navier Stokes Equation is in fluid mechanics. It goes on to explain where coordinate geometry and trigonometry are used in real life such as graphics, stock market fluctuations, navigation, surveying etc. This is an excellent resource for any A level mathematics teacher who wants to inspire his/her pupils.
The history of mathematics starts with the Hindu Arabic numerals and how they were brought to Europe by Fibonacci. It highlights major historical figures in the mathematics by placing a brief biography in a light shaded grey with an image of the mathematician. However I did not find this sort of feature for Leibniz which is a serious omission since he and Newton founded calculus.
I found the following minor typos:
1. Page 73 the result ‘sin(theta/2)=sqrt(1-cos(theta))/2’ should be ‘sin(theta/2)=sqrt(1-cos(theta)/2)’.
2. On page 156 the statement of Riemann Hypothesis should read ‘complex zeros lie on the line z=1/2 plus or minus it’ not ‘z=1/2 plus it’.
3. Page 260 the statement is written ‘x(t+3)’ should be ‘x(t+ epsilon)’.
This is an excellent book and would recommend that anybody interested in mathematics should purchase this book. The book is a fantastic resource for any college or university library.
Kulδεερ Siηgh
Sunday, 12 April 2009
Homepage at http://mathsforall.co.uk/

Yearning For The Impossible by John Stillwell

Review of Yearning For The Impossible by John Stillwell
This is a wonderful book to read around 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. There are some real gems in the book such as:
The word surd comes from the Latin surdus meaning deaf.
The product of 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.
Relationship between Gaussian primes and ordinary primes.
Hamilton’s quaternion 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 number.
There are many others like this.
However this book is not 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 the author has made excellent use of diagrams 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 which is 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 author starts by saying that the great Irish mathematician Hamilton (1805 to 1865) was the first to envisage complex numbers as pairs of real numbers. He goes on to describe the addition of pairs as component-wise addition, that is,
(a, b) + (c, d) = (a+c, b+d) and multiplication as (a,b).(c,d) = (ac-bd, ad+bc) . This system with addition and multiplication of ordered pairs defined in this manner satisfies all the rules of ordinary algebra (field) and the absolute value is multiplicative. The author claims that Hamilton tried for 13 years to extend this idea to triples (a, b, c) but failed. From the ashes of triples he revived a definition for quadruples (a, b, c, d) which ensured that the absolute value was multiplicative and all the rules of ordinary algebra was satisfied apart from multiplication being commutative (skew field). Hamilton called the system of quadruples with addition and multiplication the quaternion.
There is a wonderful chapter on the relationship between primes and Gaussian primes. The Gaussian prime is defined as a Gaussian integer which has an absolute value greater than 1 and cannot be factored into Gaussian integers of smaller absolute value. Numbers such as 1+i and 1-i are Gaussian primes but 2 is not because
2=(1+i)(1-i)
3 is Gaussian prime and an ordinary prime. This means that the ordinary primes are not a subset of the Gaussian primes. The author perhaps should have included that an ordinary prime of the form 1 mod 4 can be written as the sum of two squares but this can be factorised into Gaussian primes because a^2+b^2 = (a+bi)(a-bi). Therefore an ordinary prime of the form 1 mod 4 is not a Gaussian prime.
The last chapter is a nice description of the different kinds of infinity and countable sets. There is a really interesting quote in this chapter ‘ 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’.
‘Infinitesimal calculus brought almost the whole physical world within the scope of mathematics’. However 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 to mathematical reality such as ‘square root of a half a turn’ which means a rotation though a right angle. How are the two related?
Are they related by the polar or exponential form of a complex number?
There are a number of places where the book jumps from a simple introduction to the more advanced level within a few pages. An example of this is, we are introduced to the idea of a complex number and within 10 pages we are discussing Bezout’s Theorem.
In places I am not to sure what the author means. For example on page 45 the author says ‘complex analysis is known for its regularity and order.’ Clearly complex numbers don’t have order. For example if i>0 then i^2>0 which is false and if i<0>0 again this is false.
In addition there are a couple of typos such as on page 137 should read c1 not c2 in the formula of line 2. On page 169 it says substitute (7.6) in (7.1) but there is no (7.6).
Homepage at http://www.mathsforall.co.uk/