1. I’m certain many of our readers would be interested in knowing how you got started. What led to your interest and love of the discipline?

Prof. du Sautoy: It was being exposed to real mathematics when I was at school. My teacher showed me some of the big stories of maths. We tend to focus too much on the vocabulary and grammar and forget to show kids the Shakespeare of maths. I was also lucky to attend the Royal Institution Christmas Lectures when they were first done on maths by Christopher Zeeman. I was very lucky to be given the chance to give the lectures myself in 2006 and this summer I am publishing a new book based on the lectures called The Num8er My5teries. The book is my manifesto for what we could be teaching kids.

2. Let’s talk about the perception of mathematics and science in today’s society. At a time where scientific awareness and reason are greatly needed to further progress the state of society and democracy, it appears to me that anti-intellectual, and in particular anti-scientific, stances are on the rise. It often seems that the majority of people believe they are not really expected to know much about mathematics and science, despite the far-reaching impact these disciplines have on our complex society and world.

Being “bad” at mathematics or failing to possess a fundamental grasp of other scientific areas is not only considered tolerable, but often flaunted without so much as a hint of shame. One would rightfully be considered ignorant for not knowing anything about Shakespeare, yet this is much less the case when dealing with the magnificent works of Euclid or Euler). At one extreme of the spectrum, there are also religious, political and economical pressures to invalidate evolution, global warming, and science as a whole. A recent reminder of the so-called war on science came through the high profile libel case against Dr. Singh. Thankfully that situation ended on a positive note when the judge concluded that, “Scientific controversies must be settled by the methods of science rather than by the methods of litigation […]”.

In your opinion, what is the best approach when it comes to fighting this trend and helping society to embrace reason and science? How do we increase trust in the scientific method and interest in the art of problem solving? How should we prepare the next generation to be more interested in mathematics and science?

Prof. du Sautoy: Increasingly our lives are being influenced by scientific developments which means that science impacts on everyone in society. By not engaging in the ideas of science you are effectively disenfranchising yourself from that debate. But it’s also the responsibility of scientists to be open to dialogue and share their science with society. That is why positions like the Simonyi chair for the public understanding of science are crucial in creating bridges.

I actually think there is an appetite for science among the public. TV, radio, newspapers and the bookshelves are full of interesting stories of science. I think we need to tap in more to the wonder and magic of science. Most scientists do science because they enjoy discovering how our universe works. I think most of us enjoy that feeling.

3. As a Simonyi Professor for the public understanding of science, you hold an important position that previously belonged to Richard Dawkins. As many know, his approach is very confrontational and direct. It places a great deal of emphasis on atheism and the damage that religion can – and often does – cause society. Despite being an atheist yourself (if we exclude your passion for Arsenal), it could be argued that you are a much less controversial figure and a diametrically opposite type of scientist. Could you describe how your strategy to promote the value of science and rational thought is different (from Dawkins) and how being a mathematician factors into this area for you?

Prof. du Sautoy: I think mathematics is a great choice for the chair. Mathematics is the language of science and bubbles underneath everything scientists do. It is probably at the other end of the scientific spectrum from evolutionary biology so it makes a good contrast. I see the role as an ambassadorial one, trying to create connections between society and the often foreign feeling land of science.

4. Your name is synonymous with the concept of symmetry. Can you briefly explain to our readers why symmetry is such an important topic?

Prof. du Sautoy: Symmetry is Nature’s language. Symmetry is present whenever there is structure or meaning. Our brains are evolutionarily programmed to be sensitive to symmetry. It is the key to our survival. It is also at the heart of many of the other sciences. Crystal structure and viruses depend on symmetry. Even the predictions we are making about what we might see in the LHC are thanks to a strange symmetrical object that exists in multidimensional space.

5. Can you tell our readers about your fund-raising project Symmetry4Charity, which helps provide kids from Guatemala with educational support? For those who haven’t read Finding Moonshine (published as Symmetry in the States), what led you to support a cause centered around Guatemala?

Prof. du Sautoy: I have two identical twin girls adopted from Guatemala (nothing to do with my obsession for symmetry). While living in Guatemala I encountered the charity Common Hope that gets kids in Guatemala off the streets and into education. Provided they stay in school their families receive health care and housing. It seemed a very empowering charity rather than one which made kids dependant. To help the charity I am naming new symmetrical objects that I have discovered in my research after people who donate to the charity. Just one more way that maths can help the world.

6. I personally find the idea of drawing out people’s philanthropic side by naming a symmetric group after them to be both fun and brilliant, so I donated to the initiative myself. Some may argue though that you are selling (albeit for a very worthwhile and noble cause) your discoveries. How would you respond to that sort of criticism?

Prof. du Sautoy: I think if I was personally making money out of the project there may be some justification to criticise the project. But I think it is a great way to engage people with cutting edge mathematics while at the same time helping one of the poorest countries in the world.

7. As not all of our readers are familiar with group theory, could you please describe, in layman’s terms, the process of finding new symmetric groups? Do you use any mathematical software in your workflow?

Prof. du Sautoy: My brain, a pencil and yellow legal notepads! That’s the best equipment for exploring the mathematical landscape. Because the symmetrical objects live in multidimensional space you can’t draw pictures or build them. Instead the language of group theory provides an algebraic way to show how the symmetries of this object interact with each other. These objects are very special because they are connected with another important subject in mathematics: elliptic curves.

8. What mathematical literature would you recommend to those members of the general public who are interested in exploring symmetry further? As well, what titles would you promote to the budding undergraduate student?

Prof. du Sautoy: My book Finding Moonshine tells the story of our journey to understand and classify what symmetries are out there. It is also a very personal book talking about my own work as a mathematician. I tried to give readers a back stage pass to the mathematician’s lab. For a deeper introduction I would recommend Conway et al’s The Symmetry of Things.

We wish to thank Professor du Sautoy for his time. We also highly encourage you to donate to his very worthwhile cause. Common Hope needs to raise £1000 from 50 unique donors by the end of April. Let’s make this happen. If they’re able to do so, they’ll receive a permanent listing on the UK Global Giving website that would allow UK donors to benefit from Gift Aid where tax is added to the donation. Common Hope is an educational charity supporting and empowering children and their families in Guatemala. Donate £10 or more and you’ll be entitled to request a symmetric group to be named after you (you can even name it after someone you care about, as a gift).

If you haven’t seen it yet, you can watch Professor du Sautoy’s TED speech below:

Many parents face the challenge of helping their children with math homework, which for some stems in part to having developed a strong phobia or dislike of the subject themselves. Along with a psychological component, in many cases the challenge is augmented by a lack of basic skills (when it comes to knowing how to approach math problems and work their way through mathematical nomenclature). For some it’s like trying to help their child with French homework, when they don’t speak the language. Otherwise perfectly intelligent adults end up finding themselves worrying over problems that most math-savvy people would consider straightforward.

Math for Moms and Dads tries to solve this predicament by providing a vocabulary of essential terms, a very gentle introduction to problem solving and mathematical reasoning, fundamental concepts of elementary (primary) and middle school mathematics, and step-by-step solutions to basic exercises. It also stresses the importance of the parent-child and parent-teacher relationships when it comes to teaching and assisting with the learning of math. This book is very basic and relatively short, which means that it’s something most parents would be able to squeeze time into their schedule to read (which I feel is a positive element of this book). As someone with a passion for math, I’m biased and admit that I do not find this type of book terribly exciting myself, but I fully realize its usefulness for people who need a “less than scary” introduction (or refresher) to the subject.

The first chapter introduces the book and provides parents with a few pointers on how to use a calculator and when its usage is appropriate. The content on these pages will appear pretty obvious to a large number of readers, but this book tries not to make any assumptions, and as such it aims to cover concepts that many people might take for granted.

Chapter two details the mathematical vocabulary mentioned earlier in this article, and within this chapter parents will learn about fundamental math terminology, including terms such as absolute value, congruent, coordinate plane, diagonal, fraction, permutations and so on. The second part of the chapter provides the reader with more descriptive information about common, basic concepts like commutative and associative property, prime and composite numbers, rational and irrational numbers, union and intersection, linear and quadratic equations, etc.

Chapter three covers the basic rules necessary for resolving a variety of problems, including order of operations, exponents and their rules, properties of numbers, fraction and integer based arithmetic, expressions and equations, and so on.

Chapters four and five tackle the issue of solving homework exercises and preparing for math tests. Together these chapters help clarify how to approach mathematical problems, with examples that are solved in a step-by-step manner.

Chapter six is a pedagogical chapter about how to approach study, which covers topics such as how to create the right study conditions and find the ideal place in your house to turn into a homework area, as well as how to develop note taking and test preparation skills.

Chapter seven is entitled “When will I use this, anyway?”, and it attempts to convince both parents and their children that learning mathematics is an important and useful real world skill. I felt that this chapter (which is about a subject – the importance of math beyond the classroom – I believe strongly in) was on the weaker side, but it may still be useful to some.

Lastly chapter 8 deals with parent-teacher communication, a topic that I felt was important for this kind of book.

Should you feel that your own math skills are not your strongest suit or if you need a concise and easy to follow along with refresher course on numerous basic math topics, so that you can better assist your child with their studies, you will likely find this book right up your alley.

If you are a publisher and would like to have your books reviewed, please contact me at antonio@math-blog.com. As a policy, we will only publish reviews for book worth recommending, informing the publisher if a book doesn’t meet (in our opinion) the standard.

I’m hip-deep in the teaching of Mathematics 152, a discrete mathematics course titled “The Mathematics of Symmetry” designed by Paul Bamberg and taught at Harvard University. The course is seminar-style: the students take turns presenting the material in 5-to-15-minute assigned topics during class. The design puts an emphasis on learning to communicate mathematics, and so as I took over the course this semester I considered what I might do to further this goal. I wanted to encourage class participation, discussion and a sense of community, as well as tie the mathematics of the course to the wider experience of the students. So I tried an experiment: an assigned community math blog. The blog is open to and in fact aimed at the layperson public, but also serves as a community discussion board for the students. The 23 students in the class have been assigned 4 posting dates each, spread throughout the semester, which means the blog is updated at least once and sometimes twice a day.

I was inspired by my mother and father, who both assign “journals” to their students in psychology, english, classics and philosophy. The journals, updated regularly by students, are a sort of private diary of reactions to the course. They serve to draw connections with the sudents’ world outside the classroom, and encourage reflection on the material. Blogging, it occurred to me, is a sort of public journaling, and provide some of what my parents sought from the course journals, but this time in the form of a community project.

We’re now approaching the halfway mark for the semester, and I’ve been incredibly impressed with the students’ posts. They range from amusing to historical to musical to magical—even social commentary. There’s been no shortage of topic ideas, although I had hoped there would be more discussion via comments. I hope you’ll take a look at the blog and post some responses, so the students see that they are really reaching an audience out there on the internet: reaching beyond the course itself.

Site: The Math 152 Weblog.

1. General Math Cheat Sheet (iPaper and other formats)
2. Elementary Algebra Cheat Sheet (PDF)
3. Trigonometry Cheat Sheet (PDF)
4. Calculus Cheat Sheet (PDF)
5. Derivatives and Integrals Cheat Sheet (PDF)
6. Laplace Transforms Cheat Sheet (PDF)
7. Abstract Algebra Cheat Sheet (PDF)
8. Probability Theory Cheat Sheet (PDF)
9. Matlab Cheat Sheet (PDF)
10. Mathematica Cheat Sheet (PDF)
11. Maple Cheat Sheet (PDF)
12. Maxima Cheat Sheet (HTML web page)
13. LaTeX Cheat Sheet (several formats)

And since most of us like to show our math pride off when out and about as well, Amazon sells this awesome Math Cheat Sheet T-shirt with formulas on both sides (Also available for Science and Engineering). How awesome is this?

On the net I found an incredible lecture by the brilliant mathematician Timothy Gowers, entitled “The Importance of Mathematics”. In this keynote, Prof. Gowers makes a very strong case in favor of the value of math, of financing its relatively cheap research and its deep implications on human progress. You can watch the 8 parts that compose the whole video, in the following playlist:

For those who’d prefer it, a PDF transcript is also available.

It started as a personal blog and evolved into a site which openly welcomes contributions. This much is certainly true, yet I couldn’t help but ask myself, where is Math-Blog really going? There are plenty of mathematical resources out there on the web, if you know where (and how) to search for them. What’s the point then of this blog’s existence, I couldn’t help but ponder. Aside from my own desire to speak about my great passion for mathematics, I found in these letters all the motivation and reason I needed to make Math-Blog even more prominent amongst online math resources. For you see, there is, in fact, a little known world of people who secretly – or admittedly – have a deep fascination with mathematical subjects, though for one reason or another, didn’t actually end up becoming professional mathematicians. These people have all sorts of skill levels, but more often than not, they’re in need of some catching up (or refreshing) when it comes to the basics of mathematics.

Math can be a lifelong journey of discovery, even for those whose day job has nothing to do with mathematics or who may have fared poorly in an academic math setting. I know people who’ve spent the last 10 years studying Calculus, Real and Complex Analysis, Algebra and Number Theory on their own from advanced university textbooks, purely for the pleasure of learning, and now they’re highly skilled mathematicians – despite their lack of formal mathematical certification (or having never published a single math paper in a peer reviewed journal). This world of mathematicians who approach math out of love, not as a profession, like to be inspired, guided, and helped, especially at the beginning of their journey.

Thinking about all of these points helped me to realize that Math-Blog is capable of becoming an important reference point, a place that is essentially a world of “math for the rest of us”. This is the direction that I’d like to give to Math-Blog, and while I’m aware of the fact that it’ll require more frequent posting on my part and also the creation of introductory material (which I intend to prepare/do from now on), I’m happy to produce such a mathematical environment out of this site. And for those readers who may actually be studying to become, or currently are, a mathematician, or if you’re on the other end of the spectrum and are a high school student, don’t worry, just hang in there and I assure that there’ll be plenty of mathematical fun to be had by all.

2. Outside in (Turning a sphere inside out)

3. Flatland the film (Trailer)

The full movie is available on DVD, and of course, you can also get “Flatland: A Romance of Many Dimensions”, dirt cheap (a classic geek novel for less than 4 bucks). If you prefer, you could pick up the annotated hardcover version: “The Annotated Flatland: A Romance of Many Dimensions”. Highly recommended.

4. Math Education: An Inconvenient Truth

This short video shows what’s wrong with the current widely adopted methods of teaching mathematics (fortunately though, such practices have not caught on everywhere).

5. Math Education: A University View

You can consider this video a follow-up to the previous one. Clearly this education reform affects elementary school aged children, but the effects that it has on curricula at an early level also profoundly goes on to influences the education which is received by students at high school and even college levels.

Please note that we are now accepting authors and submissions for this website.

It’s not all bad though; in fact I was already planning to switch to a more serious provider with VPS or dedicated server plans. After endless research I picked Liquid Web which is fully managed, provides root access and support 24/7 on the phone. So far so good, they worked quite a bit on tweaking my server and I’ve actually called them at 3 and 5 AM and got a person on the phone in less than 30 seconds. I don’t know how well they will be able to cope with the “Digg effect” though, only time will tell.

With my “Refresh your High School Math” article on the front page of several social websites, the amount of feedback received has been terrific. It also allowed me to confirm a theory that I’ve always thought about: there is little consistency and standardization in the teaching of mathematics. I say this because the reactions to my basic math test were very highly varied. Many people said they were not able to solve the problems. That’s sad given the admittedly basic nature of the questions, but it wasn’t a big surprise. You could consider it the FizzBuzz of mathematics. What strikes me the most though, is how many people considered the test to be rightfully “middle school” material and far too basic (except for some parts of it), while others argued that it was way too advanced and too difficult for high school mathematics. This variety of reactions shows that the topics and depth of coverage in math classes in high school are quite different throughout the world.

So I’d like to explicitly ask a question to my readers, what topics did you cover while studying mathematics in high school and in what state/country were you? Please use the comment section to answer, thank you.

One person commented on the blog, and another wrote me privately, to express their concern that “harder books are not necessarily better books” and that teaching which is geared towards only the smartest kids is a mistake. I want to point out that I’m in no way advocating teaching for the brightest minds only. Wide access to mathematics is something that should be encouraged all over the world, and I’m pretty sure it will help take society in a better direction. In fact, education in general – and mathematics, technical and scientific education in particular – are key for the development of every country and ultimately for good of humankind.

However my point was that with wider access to higher education mathematics, we should not reduce the expected and established testing standards. In other words, there is a fair level of understanding that we should expect from people who major in math or from students who strongly depend on mathematics for their future careers. Furthermore, the textbooks adopted should be mathematically sound and provide the right intellectual stimulation for those who could use it. That said, there is nothing wrong with teachers trying to use different styles of teaching to reach a wider audience, or for students who struggle with the level of math presented in the textbook, to supplement it with simpler books in order to get an easier start. Hence, it’s perfectly OK for a student (who for example is taking an undergraduate class in programming in C) to read C For Dummies if The C Programming Language by K&R is too hard for them off the bat. But that does not imply that the class should adopt “C for Dummies” as their textbook nor that the examination should be based on such a book. So to summarize this point, feel free to study any number of introductory books, as long as you know that if you plan to be serious about mathematics, you should be able to eventually read and understand standard books and be able to solve most of the exercises put forward in them.

Having clarified the first concern, I’d like to provide an answer for the second point, which actually interests me the most. A few readers wrote me emails about how they feel enthusiastic about the post and the opportunity to study mathematics again, but how those books are way too advanced for them, because they simply forgot all the mathematics taught at a high school level. So I’ve received a few “how can I get a refresher of high school math?” type of questions. The mathematics that you learned in high school is classified as pre-calculus, and as you can expect it is propaedeutic to learn math at an higher level. It is normal that you forgot quite a few formulas, but having a good grasp of the essentials of precalculus can make a big difference when trying to master calculus. You should have a decent knowledge of basic algebra, trigonometry, exponential, logarithmic, and analytic geometry. Calculus itself will provide you with a refresher of some of these topics and give you a deeper understanding not only of “how” but rather “why”. That said, Calculus without a decent precalculus base can be a big challenge for most people. Before proceeding to suggest a few resources, let’s try to establish if you actually need a refresher course or not. Here is a (simple and of course incomplete) list of some basic exercises. If you haven’t a clue or struggle to find a lot of the solutions for them, a refresher may be in order.

Simple Precalculus Questions:

1) Factor the following polynomials:

2) Solve for x:

3) Find the values of x for which:

4) Evaluate:

5) Solve for x:

6) Solve for x, where :

7) Write the equations of the following curves in the Cartesian plane:

1. Parabola
2. Hyperbola
3. Circle
4. Ellipse

8 ) Find the vertex, focus, and directrix of the parabolas given by the equations:

9) Find the center, vertices, foci, and eccentricity of the hyperbola given by the equation:

10) Find the equation of a circle whose center is at and radius .

11) Determine the center and radius of the circle with equation:

.

How did it go? Did you experience many struggles and the feeling that “I used to know this stuff”? If so, then it is a good idea to go for a refresher before attempting calculus right away. The following are two books that you may find useful to respectively learn and refresh basic math in a well organized manner:

• Precalculus by Michael Sullivan: a big book, which is quite extensive and thorough. If you want an all-in-one book that covers all you need to know about precalculus and more, in a clear but college oriented manner, than this is without doubt an excellent choice. It will likely make the step up to Calculus quite easy.
• Schaum’s Outline of Precalculus: it has a less prosaic approach but it’s still very clear and easy to read. If you were pretty good at math in high school and you just forgot a few things because you haven’t touched these topics in a while, then pick this book up. It is adequate for already mathematically inclined people who are in a rush to brush up the skills they once had.

If you feel entirely clueless and would like a “for dummies” type of book, the following two titles seem to have a good table of contents and excellent reviews:

• The Complete Idiot’s Guide to Precalculus
• Pre-Calculus Demystified

If you would like to use some free resources available online instead, here are some lessons:

• Precalculus Tutorial
• Topics in PRECALCULUS
• OJK’s Precalculus page
• Exploring Precalculus
• Precalculus problems
• Prof. Ward’s lecture notes (PDF, 23 pages)
• Dave’s Short Trig Course
• Precalculus Lessons
• Collection of links related to Precalculus
• Wikipedia entry on Precalculus

If you know of any other resources that are available for free, or if you successfully used other books for these purposes, please feel free to use the comment section to add to the discussion.

Widely used calculus books must be mediocre. — W. Rudin

The temptation to discard that statement as elitist may be strong, but it is worth noticing how there is so much truth to it. Education is embracing a dangerous downward spiral towards the oversimplification of mathematics in favor of letting every student pass, despite their actual understanding of the subject.

Dumbing down mathematics is a dangerous trend which affects students at all levels, from primary school where children are no longer taught how to perform division by following the standard algorithm to a complete emphasis on anti-racist mathematics and calculus courses where delta and epsilon are not mentioned while teaching limits, because they are considered to be “too complicated” or “too confusing” for most students. Of course, this is not the case everywhere, and bright students who study on their own will always exist (despite the fact that the social satire of Idiocracy may be more accurate that we like to think).

This problem exists because educational changes such as these reflect negatively on the competence level of the average student, meaning that a decent chunk of the next generation of mathematicians, engineers and scientists will be facing increasingly complex, unsolved problems and challenges with an inadequate amount of preparation. Most students have a tendency to adapt their study levels around the difficulty of the examinations they will be exposed to. In other words, no matter how easy a Calculus course is, there will always be a majority of C students. Making the courses dumber has only resulted in the creation of artificial A, B and C students who wouldn’t stand a chance against a C student of the 50′s, in terms of understanding and mastering the subject.

It is therefore very important to consider what are the great classical books available that can provide a solid basis for studying mathematics and that can guide, inspire and enlighten the student who wishes to learn mathematics the way it should be. To be factual there are still institutions which adopt valid textbooks and didn’t jeopardize their curricula too much, but that’s not the general rule unfortunately. What are then the best Calculus books? “Best” is very relative to the student, but in this context I will consider what is “best” for relatively bright students who have a genuine interest in calculus/mathematics. Books that I myself have found very appealing to me, and therefore my list is declaratively biased. All of the following books do a very good job of covering the material, explain the subject matter well, have mathematical rigor, proofs for the most important theorems, challenging exercises, and are able to really teach Calculus I at a sufficiently high level. Each of the books listed below is able to enlighten, guide and inspire the reader who is willing to put the time and effort into going through them:

1. Calculus by Michael Spivak (Updated in July 2008!): I can’t praise this book high enough, to me this is THE calculus book. It has a great selection of topics, careful and extremely rigorous proofs, and it goes well beyond the scope of calculus, so much so that a better title would be “Introduction to the beauty of Real Analysis”, because it really bridges between calculus and more advanced real analysis, showing how beautiful mathematics is. Spivak literally guides you in a enlightening experience, discovering Calculus, starting from the properties of the numbers and building on top of it. The examples are very meaningful, the explanations are clear, and the subject is so well presented and motived. It is, in my opinion, one of the most inspirational mathematics books I’ve ever read. The exercises proposed at the end of each chapter test different levels of understanding and don’t fail to challenge the reader on the subject. These are exercises which really make you feel like you are taking a second course on the topic. In fact, I’d say that the exercise sections of this book are especially valuable, and you’re highly encouraged to work through all of them. This book is great for anyone who is serious about mathematics and who wishes to have a very solid foundations upon which to face the challenges of more advanced courses down the road. Spivak’s book comes with a few selected solutions to odd numbered problems, but if you are self-studying or are disciplined enough, you may want to consider the associated answers book (which by the way is in print, despite what Amazon lists on their pages).
2. Introduction to Calculus and Analysis, Volume 1 by Richard Courant and Fritz John: a classical, well thought introduction to Calculus and Analysis in one variable, where explanations are very clear and the material is covered in a way which provides a good deal of motivation. The approach is more practical and less abstract than other Calculus books, while conserving a perfect balance between mathematical rigor and intuition. This book comes with plenty of exercises that will reinforce the knowledge of the student. The abundance of physics applications, make it ideal for physics majors and engineers as well. This is the first volume of a trilogy republished by Springer, if you are serious about Calculus, you may want to also consider the other two more advanced volumes: Volume II/1 and Volume II/2. An absolutely beautiful set of books.
3. Calculus, Vol. 1: by Tom M. Apostol: a very comprehensive book, methodical in the theorem/proof approach, adopted by many high-end universities as a first year textbook for courses in calculus or calculus and theory. Its coverage of the subject is impressive and provides a good selection of standard exercises. It is an excellent reference and textbook, albeit you may find it a bit dry and less inspirational than others at first, but you will eventually develop an appreciation of its teaching method. You may want to note that this is the first volume, and that the second volume is also worth getting: Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications.
4. A Course of Pure Mathematics by G.H. Hardy: this is the 10th edition of a book first published in 1908 by the great mathematician G.H. Hardy. It is a classic textbook that brought much needed rigor, and reformed the way math was taught in the UK in the twentieth century. This book is intentionally written to address the brightest students of the time, therefore it is a very inspirational and enthusiastic piece of work with plenty of elegant proofs and suggestions, and exercises that are definitely very challenging. For example, there are exercises coming from the Math Tripos examinations (at Cambridge) from early part of the last century, and they will definitely keep you busy for some time. Some notations are a bit outdated and it is mostly an introduction to real analysis that may be a bit too much as a first book. But this book is a masterpiece nevertheless, and it’s a classic that as a mathematician you will want to have in your library. I think it’s ideal as a reference and as a supplement to other textbooks (e.g., Spivak).

More than just calculus, these also serve as introductions to Analysis, and in general to mathematics at an undergraduate level. They are challenging, not for the faint heart, but ultimately a joy for math lovers.