ICM2014 — opening ceremony

First Female Mathematician to get Fields’ Medal: Maryam Mirzakhani!


Featured book:

Modern Mathematics in the Light of the Fields Medals

This small book demonstrates the evolution of certain areas of modern mathematics by examining the work of past winners of the Fields Medal, the “Nobel Prize” of mathematics. Foreword by Freeman Dyson.

Gowers's Weblog

I’d forgotten just how full the first day of an ICM is. First, you need to turn up early for the opening ceremony, so you end up sitting around for an hour and half or so before it even starts. Then there’s the ceremony itself, which lasts a couple of hours. Then in the afternoon you have talks about the four Fields Medallists and the Nevanlinna Prize winner, with virtually no breaks. Then after a massive ten minutes, the Nevanlinna Prize winner talks about his (in this case) own work, about which you have just heard, but in a bit more detail. That took us to 5:45pm. And just to round things off, Jim Simons is giving a public lecture at 8pm, which I suppose I could skip but I think I’m not going to. (The result is that most of this post will be written after it, but right…

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Private Candidate Tuition & Ad hoc O Level Maths Tuition

Private Candidate Tuition

Taking your Math exam as a Private Candidate?

Currently we are available to tutor private candidates on weekday mornings.

Contact our friendly tutor Mr Wu: email at mathtuition88@gmail.com

Ad hoc O Level Maths Tuition

Nowadays students are often so busy that regular weekly tuition may not be an option. (Due to stay back in school, extra lessons etc.)

We offer Ad hoc O Level Maths Tuition at our tuition centre at Bishan. Students can come for tuition when they need to ask questions or clarify concepts.

Fees will be charged per lesson; this will fit the students’ busy schedule perfectly.

Visit this page to learn more: https://mathtuition88.com/maths-tuition-centre/

Algebraic Topology Video by Professor N J Wildberger

This is the Introductory lecture to a beginner’s course in Algebraic Topology given by N J Wildberger of the School of Mathematics and Statistics at UNSW in 2010.

This first lecture introduces some of the topics of the course and three problems.

If you are curious about how to make the interesting flap of paper (Problem 1), the solution can be found here. 🙂

Featured book:

Divine Proportions: Rational Trigonometry to Universal Geometry

Author: N J Wildberger

This revolutionary book establishes new foundations for trigonometry and Euclidean geometry. It shows how to replace transcendental trig functions with high school arithmetic and algebra to dramatically simplify the subject, increase accuracy in practical problems, and allow metrical geometry to be systematically developed over a general field. This new theory brings together geometry, algebra and number theory and sets out new directions for algebraic geometry, combinatorics, special functions and computer graphics. The treatment is careful and precise, with over one hundred theorems and 170 diagrams, and is meant for a mathematically mature audience. Gifted high school students will find most of the material accessible, although a few chapters require calculus. Applications include surveying and engineering problems, Platonic solids, spherical and cylindrical coordinate systems, and selected physics problems, such as projectile motion and Snell’s law. Examples over finite fields are also included.

World’s Biggest Number that is actually used in a proof (Graham’s Number)

Check out this video on Graham’s Number — The World’s Biggest Number actually used in a mathematical proof! (Featuring Ron Graham himself!)

Featured book:

Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks

Magical Mathematics reveals the secrets of amazing, fun-to-perform card tricks–and the profound mathematical ideas behind them–that will astound even the most accomplished magician. Persi Diaconis and Ron Graham provide easy, step-by-step instructions for each trick, explaining how to set up the effect and offering tips on what to say and do while performing it. Each card trick introduces a new mathematical idea, and varying the tricks in turn takes readers to the very threshold of today’s mathematical knowledge. For example, the Gilbreath Principle–a fantastic effect where the cards remain in control despite being shuffled–is found to share an intimate connection with the Mandelbrot set. Other card tricks link to the mathematical secrets of combinatorics, graph theory, number theory, topology, the Riemann hypothesis, and even Fermat’s last theorem.

Diaconis and Graham are mathematicians as well as skilled performers with decades of professional experience between them. In this book they share a wealth of conjuring lore, including some closely guarded secrets of legendary magicians. Magical Mathematics covers the mathematics of juggling and shows how the I Ching connects to the history of probability and magic tricks both old and new. It tells the stories–and reveals the best tricks–of the eccentric and brilliant inventors of mathematical magic.Magical Mathematics exposes old gambling secrets through the mathematics of shuffling cards, explains the classic street-gambling scam of three-card monte, traces the history of mathematical magic back to the thirteenth century and the oldest mathematical trick–and much more.

Sum and Difference of Two Cubes

Students taking A Maths (Additional Maths) in 2014 should watch this: It may come out in the exam!

This is the newest addition to the new syllabus.

Featured book:

Practical Algebra: A Self-Teaching Guide, Second Edition



Come join INSINC! Stand to win $$ for off-peak trips on MRT and LRT. More than $2,500,000 paid out so far!!

This is for Singaporean readers of my blog. Thanks for your reading and support. 🙂

Here is a good recommendation for those who take the train regularly, with chances to win cash prizes.

Take the train (MRT or LRT) regularly? You can win $$ for off-peak trips on MRT and LRT! More than $2,500,000 paid out so far!!

Sign up at: https://insinc.sg/r/NnRnLeCr/



The Mandelbrot Set

Check out this video on the very interesting Mandelbrot Set:

Famously beautiful, the Mandelbrot Set is all about complex numbers. Featuring Dr Holly Krieger from MIT.

Featured book:

The Fractal Geometry of Nature

Clouds are not spheres, mountains are not cones, and lightening does not travel in a straight line. The complexity of nature’s shapes differs in kind, not merely degree, from that of the shapes of ordinary geometry, the geometry of fractal shapes.

Now that the field has expanded greatly with many active researchers, Mandelbrot presents the definitive overview of the origins of his ideas and their new applications. The Fractal Geometry of Nature is based on his highly acclaimed earlier work, but has much broader and deeper coverage and more extensive illustrations.


[News] NUS makes it easier for students in three faculties to qualify for honours

More than 80 or 90 per cent of students on four-year direct honours programmes at publicly-funded universities here graduate with honours or the equivalent. But only 60 per cent of those in the three-year arts and social sciences, business and science degree courses at the National University of Singapore (NUS) qualify for the fourth year of study, which allows them to graduate with honours.

To close the gap, NUS is lowering the grade to qualify for the honours year in these three schools, which are among the larger faculties in the university and take in some 3,600 students a year. This means another 10 to 15 per cent – 400 to 500 students- from these three faculties can move on to the fourth year to study for their honours.

Previously, students in the three faculties require a Cumulative Average Point (CAP) of 3.5 and above to qualify for honours study. With the change, they need only 3.2. NUS, though, will stick to its policy of keeping the the three plus one structure. Students who fail to notch up a score of at least 3.2 will have to exit the course.

NUS Provost Tan Eng Chye said the university decided to lower the requirement as the quality of students has gone up over the years. Students need As and Bs to enter most of the courses now. Last year, for example, students needed a ABB to enter the arts and social sciences course and those entering business needed triple As.

Source: NUS makes it easier for students in three faculties to qualify for honours

Featured book:

Math Doesn’t Suck: How to Survive Middle School Math Without Losing Your Mind or Breaking a Nail



Infinity (BBC Video)

By our third year, most of us will have learned to count. Once we know how, it seems as if there would be nothing to stop us counting forever. But, while infinity might seem like an perfectly innocent idea, keep counting and you enter a paradoxical world where nothing is as it seems.

Mathematicians have discovered there are infinitely many infinities, each one infinitely bigger than the last. And if the universe goes on forever, the consequences are even more bizarre. In an infinite universe, there are infinitely many copies of the Earth and infinitely many copies of you. Older than time, bigger than the universe and stranger than fiction. This is the story of infinity.

See also: Georg Cantor

Georg Ferdinand Ludwig Philipp Cantor (/ˈkæntɔr/ KAN-tor; German: [ˈɡeɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfɪlɪp ˈkantɔʁ]; March 3 [O.S. February 19] 1845 – January 6, 1918[1]) was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are “more numerous” than the natural numbers. In fact, Cantor’s method of proof of this theorem implies the existence of an “infinity of infinities”. He defined the cardinal and ordinal numbers and their arithmetic. Cantor’s work is of great philosophical interest, a fact of which he was well aware. (Wikipedia)

Georg Cantor2.jpg

Recommended book:

The Princeton Companion to Mathematics

This is a one-of-a-kind reference for anyone with a serious interest in mathematics. Edited by Timothy Gowers, a recipient of the Fields Medal, it presents nearly two hundred entries, written especially for this book by some of the world’s leading mathematicians, that introduce basic mathematical tools and vocabulary; trace the development of modern mathematics; explain essential terms and concepts; examine core ideas in major areas of mathematics; describe the achievements of scores of famous mathematicians; explore the impact of mathematics on other disciplines such as biology, finance, and music–and much, much more.

Guest blog submission

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Welcome to the Teach100 community!

Recently, I added the Maths Blog to the Teach100 website. Glad to know that the blog has been approved!

“Thank you for submitting Singapore Maths Tuition to the Teach100! Your blog has been approved and is currently ranked at #427 of 601 blogs. Congratulations! We recently reached our 500th blog, and are excited to add your blog to our growing community!”



Know the Best Math Institutions around the World

This is a guest post by Maria Mcquire:

Know the Best Math Institutions around the World

There are many academic institutions in the world and rankings on certain faculties help determine the best ones in certain fields. Students who are extremely in mathematics will automatically want to enroll and join universities that are highly ranked as far as mathematics is concerned. Year by year, the rankings are released and can be accessed through the internet and various academic public publications. In addition to stellar performance, there are other factors that are considered during the rankings. This article will mention a few of the top mathematics institutions around the world.

The University of Cambridge

This is not only one of the oldest universities in the United Kingdom but also in the world. Its scholarly achievements have made it popular and famous the world over. Its high performance in mathematics and other fields is accredited to research and well trained and knowledgeable faculty tutors. External examiners and bodies such as the Quality Assurance Agency have all endorsed the performance of the university time and again. The University of Cambridge has vast studying resources such as museums, libraries and other collections. Teaching is done by lecturers who are experts and authorities in their field through seminars and lectures making this university the very top of this list.

The University of Toronto

The University of Toronto is one of the best in the world not only in mathematics but when it comes to research. Its research capabilities in all fields are rivaled only by the prestigious Harvard Universities. One thing that makes this institution great in mathematics is its system of graduate supervision and teaching strengths. The university attracts and enrolls professionals, graduates and undergraduates in the mathematical field and they are all taught by the best researchers. It has both on campus and off campus students. International students get to enjoy learning mathematics and visiting come of Canada’s most breathtaking scenic sites.

The Australian National University

This top ranked university was established back in 1946 and has risen over the years to be the top university in Australia and one of the best in the world. Mathematically speaking, the university’s teaching methods and perfectly teacher to student ratio ensure that the mathematical faculty is of international standards. Campus students do not even have to venture outside as the campus has all the social amenities needed. To make it even better, the university is well networked with some of the world’s leading academic institutions and this means that it has access to the latest information in research.

National University of Singapore

Asia is not left behind as far as mathematical prowess is concerned as the National university of Singapore puts it on the global map. One of the things that make it a top institution is its well-structured exchange programs with other universities that are mathematical giants. For this reason, the university’s best students apply for ESTA VISA in order to join other brilliant mathematical minds in other universities. The university also offers joint degree programs with other leading universities. Due to the university’s strength in research, it is affiliated with global research bodies such as International Alliance of Research Universities.

Author: Maria Mcquire


Math equation could help find missing MH370 plane

Math equation could help find missing Malaysian plane

Source: http://america.aljazeera.com/articles/2014/3/12/mathematical-equationcouldhelpfindmissingmalaysianplane.html

Bayes’ Theorem helped researchers locate Air France Flight 447’s black box in 2011

(Video: How Bayesian Search found the USS Scorpion)

Days after a Malaysian airliner with 239 people aboard went missing en route to Beijing, searchers are still struggling to find any confirmed sign of the plane. Authorities have acknowledged that they didn’t even know what direction it was heading when it disappeared.

As frustrations mount over the failures of the latest technology in the hunt for Malaysia Airlines Flight MH370, some scientists say an 18th-century mathematical equation – used in a previous search for an Air France jetliner’s black box recorder – could help pinpoint the location of the Malaysian plane.

Indonesian Air Force officers examine a map of the Malacca Strait during a briefing following a search operation for the missing Malaysia Airlines Boeing 777, at Suwondo air base in North Sumatra, Indonesia, on Wednesday.

In 2009, Air France Flight 447 en route to Paris from Rio de Janeiro vanished over the Atlantic Ocean, triggering the most expensive and exhaustive search effort ever conducted for a plane. After two years, officials could only narrow the location of the plane’s black box down to an area the size of Switzerland.

But Flight 447’s black box was found in just five days after authorities contacted scientific consultants who applied a centuries-old equation called Bayes’ Theorem.

Read more at: http://america.aljazeera.com/articles/2014/3/12/mathematical-equationcouldhelpfindmissingmalaysianplane.html

What is Bayes’ Theorem

Mathematically, Bayes’ theorem gives the relationship between the probabilities of A and B, P(A) and P(B), and the conditional probabilities of A given B and B given A, P(A|B) and P(B|A). In its most common form, it is: (Wikipedia)


(Check out this post on probability formulas to learn more about Probability)

Proof of Bayes’ theorem (Theorem useful for finding MH370 plane)

The proof of Bayes’ theorem is actually relatively simple, the only requirement is to know the formula for conditional probability (Learnt in H1/H2 Maths): \displaystyle \boxed{P(A|B)=\frac{P(A\cap B)}{P(B)}}

From this, we have \displaystyle \boxed{P(A\cap B)=P(A|B)P(B)}

Similarly, \displaystyle \boxed{P(B\cap A)=P(B|A)P(A)}

But since \displaystyle P(A\cap B)=P(B\cap A), we have P(A|B)P(B)=P(B|A)P(A). Dividing throughout by P(B) gives Bayes’ Formula: \displaystyle\boxed{P(A|B)=\frac{P(B|A)P(A)}{P(B)}}

Sincerely wishing that the MH370 plane will be found soon, and hopefully the passengers are still alive.

Also see: Bayesian search theory (Bayesian search theory is the application of Bayesian statistics to the search for lost objects. It has been used several times to find lost sea vessels, for example the USS Scorpion. It also played a key role in the recovery of the flight recorders in the Air France Flight 447 disaster of 2009.)

Fun Math Equals Better Student Participation

Fun Math Equals Better Student Participation

We are glad to have Mr Henry Thompson write a Math article on our blog. 🙂

Guest post by Henry Thompson of DegreeJungle.com:

One significant obstacle that students face when trying to understanding mathematics is that they devote a great deal of their energy to NOT enjoying themselves. Think about it; reading literature is satisfying, if the story is carefully chosen. Holding a conversation about up-to-date events in History, while studying critical analysis, is enjoyable. But, even for math teachers, working out a complex algebraic equation is simply not exciting.

Students rely on their professors to make mathematics convenient and more effortlessly appreciated. Thus, it makes good sense for educators to insert some sort of enjoyment into their math lessons as frequently as possible; particularly, if the diversion includes a little academic theory.

Today’s professors feel that great math education objectives should not only “address the program of study,” but should also present learners with new ways to discover life through the aperture of mathematics.

For this reason, groundbreaking educators around the globe have altered their approach to math education by leaving behind unimportant and boring learning objectives and implementing applicable and appealing math learning inside the classroom.

Yesteryear’s Math Programs Are Uninspiring

If teachers recall their pedagogic theories from college, they’ll remember that many lesson plans contained mathematical calculations at the hub of their programs.

Additionally, the framework in most old-school math textbooks contains terribly-fashioned word problems. It appears that a few textbook publishers hold fundamental challenges in developing math problems that are linked to real life.

Outdated textbooks only pay attention to computational formats as well, leaving out the reasoning that is produced behind the scenes, which is needed to solve math problems.

The folks at Degree Jungle recently talked to some math educators, who located their teaching credential programs through the infamous search engine, to find out what instructors in the twenty-first century should look for when analyzing conventionally-structured math programs.

A Brand-New Strategy for Teaching

A large number of math educators, today, recommend professors seek math learning-systems that guarantee relevancy, instead of those which put math calculations at the center of study; lessons that contain “real-life” relevance will most certainly motivate students to engage more.

The planet contains plenty of fascinating mathematic applications. A tree’s design is a consequence of fractional limb patterns. A tiny shellfish’s cask coils in an exquisite and attractive mathematical design. Profound mathematics dwells in the massive framework of the cosmos. Moreover, all things that folks explore throughout the day contain some sort of mathematical design.

Easy Tips for Applying Mathematics to the Real-World

Below are a handful of tips that educators can work with to help put real-world situations inside their educational math programs:

  • Instead of a worksheet that explains how to spend money, provide students with some real coins to count, or let them visit to the school store.
  • Cooking incorporates proportions and divisions.

Resources for Improving Engagement

Although adding real-world scenarios to math problems plays a vital part in ensuring an entertaining lesson, it is not the only unique educational approach for teaching math. There are countless mathematical strategies short of “real-life” applicability that are, nevertheless, exceptionally appealing.

  • Projecteuler.net delivers a collection of serious mathematical-CIS problems that will demand much more than just mathematical awareness to solve.
  • Fullerton IV Elementary School’s, Integers Across Disciplines, proposes another strategy. Educators there have developed tasks that force students to visit challenging mathematical problems and to discover that math demands practice and patience.
  • Euler’s graph theory using geography assists students in building mathematical tolerance and in discovering ways to conquer frustration. As an included reward, learners will understand that not all math problems have solutions.

A Mathematician’s Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form

A Mathematician’s Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form

A Mathematician’s Lament is a short book on the pedagogics and philosophy of mathematics by Paul Lockhart, originally a research mathematician but for many years a math teacher at a private school. Characterised as a strongly worded opinion piece arguing for an intuitive and heuristic approach to teaching and the importance of mathematics teaching reforms, the book frames learning mathematics as an artistic and imaginative pursuit which is not reflected at all in the way the subject is taught in the American educational system.

The book was developed from a 25-page essay that was written in 2002, originally circulated in typewritten manuscript copies, and subsequently on the Internet.