Imaginary Erdős Number: What is that?

Students studying Mathematics at university will sooner or later hear of the famous eccentric Mathematician Paul Erdos, and the concept of Erdos Number. People who have written a paper with Erdos have a Erdos number of 1. People who have cowritten with the above people (with Erdos number 1), have Erdos number 2. Unfortunately, it is now impossible to get Erdos number 1, as Paul Erdos has passed away.

But what is an Imaginary Erdos Number?

This YouTube channel Numberphile has really succeeded in making Maths interesting! I watch almost every new episode that comes out, and will feature it on my website.

Featured Book:

MY BRAIN IS OPEN: The Mathematical Journeys of Paul Erdos

Paul Erdõs, one of the greatest mathematicians of the twentieth century, and certainly the most eccentric, was internationally recognized as a prodigy by age seventeen. Hungarian-born Erdõs believed that the meaning of life was to prove and conjecture. His work in the United States and all over the world has earned him the titles of the century’s leading number theorist and the most prolific mathematician who ever lived. Erdõs’s important work has proved pivotal to the development of computer science, and his unique personality makes him an unforgettable character in the world of mathematics. Incapable of the smallest of household tasks and having no permanent home or job, he was sustained by the generosity of colleagues and by his own belief in the beauty of numbers.
Witty and filled with the sort of mathematical puzzles that intrigued Erdõs and continue to fascinate mathematicians today, My Brain Is Open is the story of this strange genius and a journey in his footsteps through the world of mathematics, where universal truths await discovery like hidden treasures and where brilliant proofs are poetry.

What Maths do Engineers Learn: Singapore Engineering Maths (University) Tuition

Some of the Maths topics that Engineering Students need to learn are:

  1. Fourier Series
  2. Laplace Transform
  3. Total and Partial Differentiation
  4. Line Integral

All the above topics are rather challenging and deep. Fortunately, for most engineering students, application of the theorems would suffice, the deep proofs are not really necessary. It would be good to know them though.

Featured Book:

Schaum’s Outline of Advanced Mathematics for Engineers and Scientists (Schaum’s Outline Series)

This is the book you are looking for, if you are looking for a book to help ace Engineering Maths.

Happy Teacher’s Day!

Glad to receive some Teacher’s Day cards and presents from my students!

teacher's day cardWishing all teachers a happy Teacher’s Day this Friday, and also wishing students all the best for their upcoming exams.

Featured book:

A Mind For Numbers: How to Excel at Math and Science (Even If You Flunked Algebra)

Whether you are a student struggling to fulfill a math or science requirement, or you are embarking on a career change that requires a higher level of math competency, A Mind for Numbers offers the tools you need to get a better grasp of that intimidating but inescapable field. Engineering professor Barbara Oakley knows firsthand how it feels to struggle with math. She flunked her way through high school math and science courses, before enlisting in the army immediately after graduation. When she saw how her lack of mathematical and technical savvy severely limited her options—both to rise in the military and to explore other careers—she returned to school with a newfound determination to re-tool her brain to master the very subjects that had given her so much trouble throughout her entire life.

In A Mind for Numbers, Dr. Oakley lets us in on the secrets to effectively learning math and science—secrets that even dedicated and successful students wish they’d known earlier. Contrary to popular belief, math requires creative, as well as analytical, thinking. Most people think that there’s only one way to do a problem, when in actuality, there are often a number of different solutions—you just need the creativity to see them. For example, there are more than three hundred different known proofs of the Pythagorean Theorem. In short, studying a problem in a laser-focused way until you reach a solution is not an effective way to learn math. Rather, it involves taking the time to step away from a problem and allow the more relaxed and creative part of the brain to take over. A Mind for Numbers shows us that we all have what it takes to excel in math, and learning it is not as painful as some might think!

Book Review: Topology, James R. Munkres

Topology (2nd Edition)

This book is the best introductory book on Topology, an upper undergraduate/graduate course taken in university. I have written a short book review on it.


Book Review: Topology
Book’s Author: James R. Munkres
Title: Topology
Prentice Hall, Second Edition, 2000

It is often said that one must not judge a book by its cover. The book with a plain cover, simply titled “Topology”, is truly a rare gem and in a class of its own among Topology books.

One striking aspect of the book is that it is almost entirely self-contained. As stated in the preface, there are no formal subject matter prerequisites for studying most of the book. The author begins with a chapter on Set Theory and Logic which covers necessary concepts like DeMorgan’s laws, Countable and Uncountable Sets, and the Axiom of Choice.

The first part of the book is on General Topology. The second part of the book is on Algebraic Topology. The book covers Topological Spaces and Continuous Functions, Connectedness and Compactness, and Separation Axioms. Some other material in the book include the Tychonoff Theorem, Metrization Theorems and Paracompactness, Complete Metric Spaces and Function Spaces, and Baire Spaces and Dimension Theory.

The book defines connectedness as follows: The space X is said to be connected if there does not exist a separation of X. (A separation of X is defined to be a pair U, V of disjoint nonempty open subsets of X whose union is X.) Other sources may define connectedness by, X is connected if \nexists continuous f:X\twoheadrightarrow \mathbf{2}.

Also, the proof of Urysohn’s Lemma in the book was presented slightly differently from other books as they did not use dyadic rationals to index the family of open sets. Rather, the book lets P be the set of all rational numbers in the interval [0,1], and since P is countable, one can use induction to define the open sets U_p. In hindsight, the dyadic rationals approach in other sources may be more explicit and clearer.

An interesting new concept mentioned in the book is that of  locally connectedness (not to be confused with locally path connectedness). A space X is said to be locally connected at x if for every neighborhood U of x, there is a connected neighborhood V of x contained in U. If X is locally connected at each of its points, it is said simply to be locally connected. For example, the subspace [-1,0)\cup (0,1] of \mathbb{R} is not connected, but it is locally connected. The topologists’ sine curve is connected but not locally connected.

In general, the content of the book is comprehensive. The other book, “Essential Topology”, did not cover some topics like the Urysohn Lemma, regular spaces and normal spaces.

The author’s approach is generally to give a short motivation of the concept, followed by definitions and then theorems and proofs. Examples are interspersed in between the text. The motivation tends to be a little bit too short though. For instance, in other books there is some motivation of how balls can determine the metric in a metric space, leading to the concepts of “candidate balls” \mathcal{C}=\{C_\epsilon (x)\}_{\epsilon >0, x\in X}. This useful concept is not found in the book Topology, nor the other book Essential Topology.

One interesting explanation of the terminology “finer” and “coarser” is found in the book. The idea is that a topological space is like “a truckload full of gravel”‘ — the pebbles and all unions of collections of pebbles being the open sets. If now we smash the pebbles into smaller ones, the collection of open sets has been enlarged, and the topology, like the gravel, is said to have been made finer by the operation.

Another point to note is that the book does not use Category Theory. Personally, I would prefer the Category approach, since it can make proofs neater, and it provides additional insight to the nature of the theorem. We also note that the other book “Essential Topology”, also does not explicitly use Category Theory. But upon closer examination, the book has expressed commutative diagrams in words, which is not as clear as in diagram form.

The organization of the book is similar to most other books, except that it covers Connectedness and Compactness before the Separation Axioms. The concept of Hausdorff spaces, however, is covered way earlier, immediately after the discussion of closure and interior of a set. This enables theorems like “Every compact subspace of a Hausdorff space is closed” to be proved in the Compactness chapter.

The author’s style is to combine rigor in proofs and definitions, with intuitive ideas in the examples and commentary. This makes it both a good textbook to learn from, and a good reference for proofs too.

This informal style in the commentary makes for a especially good read. For instance, a mathematical riddle is mentioned: “How is a set different from a door?” (For interested readers, the answer can be found on page 93.)

Also, there are many figures in the book, 84 sets of figures to be precise. This is rather good for a math book, and I would recommend the book to visual learners.

However, to learn Topology from this book alone may be difficult. Even though there are exercises to practice, there are no solutions and very few hints. Also, the book uses the terminology “limit point”, which can be confusing.

The book has surprisingly few typographical errors. While reading through the book, I only spotted a trivial one on page 107, where a function written as “F” should be “f” instead. Upon consulting an errata list, there was only one page of errors.

In conclusion, despite some shortcomings of the book, Topology is a great book, and if there was one Topology book that I could bring to a desert island, it would be this one.

Chapter Headings

Part I: General Topology

  • Set Theory and Logic
  • Topological Spaces and Continuous Functions
  • Connectedness and Compactness
  • Countability and Separation Axioms
  • The Tychonoff Theorem
  • Metrization Theorems and Paracompactness
  • Complete Metric Spaces and Function Spaces
  • Baire Spaces and Dimension Theory

Part II: Algebraic Topology

  • The Fundamental Group
  • Separation Theorems in the Plane
  • The Seifert-van Kampen Theorem
  • Classification of Surfaces
  • Classification of Covering Spaces
  • Applications to Group Theory

For more undergraduate Math book recommendations, check out Undergraduate Level Math Book Recommendations.