One of the world’s most influential math texts is getting a beautiful, minimalist edition – The Verge

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Little print-edition of Math Classics: Euclidean 《The Elements》(几何原理) – the second printing by volume after 《The Bible》.

The Euclidean Geometry was less emphasised in Secondary school from 1970s in the world (80s in Singapore) to be replaced by Vector Geometry (UK/USA) and Abstract Linear Algebra aka Vector Space / Affine Geometry (in France). It was a pedagogical mistake now these countries recognise – never “Throw the baby (Euclidean Geometry) together with the bath water (Old Maths )”.

IMO Competition usually has 2 out of 6 problems in Euclidean Geometry but no modern math or Calculus, reason being to cater for 100+ countries still with Classical Math. That’s why the Asian countries – South Korea (2017 IMO Champion), China (2nd IMO 2017), Vietnam (3rd IMO 2017) , HK, Taiwan, Thailand, – with old syllabus are top IMO but not the Advanced countries (France, UK, other European, Japan, USA, Singapore…).

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The Evolution of Software

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Zhihu 知乎: 环Ring, 域Field, (半Semi-)群Group, 幺半群 MonoId

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(I) 环和域
环 (Ring) 是德文, 这名词是David Hilbert发明的。
“环”最好的例子: 整数(Integer) 记作 Z (德文Zahl) 是个环。 他研究 Clock Arithmetic 时钟 是个 Modular Arithmetic (Mod 12)运算, 比如 15 = 3 mod 12

钟表上的整数 Z 数字 排成 一个钟的环状, 故称 此数是 “Zahl Ring” 整数 “环”。

Hilbert 的得意女弟子 Emile Noether 把 环论 发扬光大, 创造 “Noether Ring”。

如果 把 “12” 变成 任一素数 p , “环”就升级成”域” (Field) – 除了有环的( + – *) 运算, 还可以 (÷)。

Hilbert 发现 “环”有个重要性质是含有 “理想” (Ideal) – 理想 * 任何”外面”的东西 还回来”里面”。
例子: Z Ring 里有 “0”, 2Z, nZ…
0 * 任何Z数 => 回来 “0家族里面”
偶数 2 * 任何Z数 => 回来” 偶数 (2Z)家族里面”

大名鼎鼎的中国古代算术”韩信点兵”就是 含有 “环 和理想”的理论。

(II) 群 Group / 半群 Semi-Group / 幺半群MonoId
Group 群 有4个性质 C.A.N.I.
C: Close 封闭性
A: Associative 连续性
N: Neutral (or Identity = Id) 幺元
I: Inverse 逆元

Semi-Group 半群 只有2个性质: C.A.

Monoid 么半群 : C.A. + N (= Id) => Mono + Id

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2017 PSLE Math Question

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Answer: 10 (Yes!) but why ?

…(see explanation here )

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Garlic Butter Lobster

Cooking For Fun

1. Put lobster in boiling water for 4 mins.

2. Take lobster out and soak it into cold water.

3. In a bowl, add in minced garlic.

Then chopped chili, parsley and sprinkle salt. Mix well. Then add in butter. Stir them.

4. Cut lobster into halves.

5. Spread garlic butter onto lobster.

6. Heat a skillet on the stove. When it becomes smoke hot, put the lobster in with the butter side down.

7. Heat a bit then turn them over.

8. Place lobster in a plate. Pour garlic butter over it.

My husband loves it.

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Train delay: Students given full duration for paper (PSLE English Language and GCE O-level Music Performing)

Fortunately, students affected by the train disruption are given the full duration for their respective papers. It would have been quite a traumatic experience though.

Source: Straits Times

SINGAPORE – No exam candidates were affected by the train delay on the East-West Line on Thursday (Sept 28) morning, said the Singapore Examinations and Assessment Board (SEAB) in response to queries from The Straits Times.

As at 8.30am, no schools have informed SEAB that candidates sitting the PSLE English Language and GCE O-level Music Performing examinations were affected by the disruption, said the exam board.

Earlier in the morning, the Ministry of Education said that students who are affected by the train delay on EWL do not need to submit an excuse letter and will be given the full paper duration.

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Online Technology News Media Teknologi & Internet

The following is a new Online Tech News site:

Category: Media Berita Internet

Keywords related to the site:

Online Technology News Media Teknologi & Internet
Berita harian internet
Berita harian internet
Situs berita internet
situs berita teknologi
Berita online
teknologi online
berita teknologi
berita online
berita internet
situs berita teknologi
situs berita online
situs online

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Travel Advisory: Volcanic Activity in Bali

On 22 September 2017, Indonesian authorities raised the alert-level for Mount Agung on Bali island to level 4 (Awas or Warning) due to an increased possibility of a volcanic eruption within 24 hours. Indonesia’s National Disaster Mitigation Agency has warned residents and tourists to stay 9 kilometres away from the crater and up to 12 kilometres away to the north, northeast, southeast and south-southwest.

Given the possible eruption of Mount Agung, Singaporeans should defer non-essential travel to the affected areas at this juncture. Volcanic eruptions could result in ash clouds that could severely disrupt air travel. Singaporeans currently in Bali should monitor these developments closely and avoid Mount Agung and its vicinity. They are advised to take all necessary precautions for their personal safety, monitor the local news closely and heed the instructions of the local authorities, such as to be ready to evacuate at short notice. They should also purchase comprehensive travel and medical insurance and be familiar with the terms and coverage.

Singaporeans are advised to stay in touch with your family and friends so that they know you are safe. Singaporeans travelling overseas are also strongly encouraged to e-Register with the Ministry of Foreign Affairs at, so that we can contact you should the need arise. Those in need of urgent consular assistance may contact the Singapore Embassy in Jakarta or the Ministry of Foreign Affairs Duty Office at:

Embassy of the Republic of Singapore in Indonesia (Jakarta)
Jln H R Rasuna Said, Kuningan, Block X/4, KAV No 2,
Kuningan, Jakarta Selatan 12950
Tel: + 62 (21) 2995 0400 or +62 811 863 348 (24-hours)

Ministry of Foreign Affairs Duty Office (24-hours)
Tanglin, Singapore 248163
Tel: 6379 8800, 6379 8855

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PSLE Chinese Listening Exam (mrbrown) Very Funny!

Recently, there is a PSLE Chinese Listening Exam that does not make sense.

Question: Student A bought a new clothes. Student B asks Student A: “You bought new clothes?” Student A said: ” No, it is sewn by my mother, do you think it is beautiful?” Student B said: It is very beautiful, I didn’t knew your mother could sew?”.

What did Student A say next?

1) My mother will sew clothes for me whenever she is free.
2) My mother does not like to spend money to buy clothes.
3) My mother just started learning how to sew.

I am totally puzzled by this question. The three options seems equally plausible. How are we supposed to know which is the truth?

Answer is option 3 by the way.

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Recommended Books for Spectral Sequences

Best Spectral Sequence Book

So far the most comprehensive book looks like McCleary’s book: A User’s Guide to Spectral Sequences. It is also suitable for those interested in the algebraic viewpoint. W.S. Massey wrote a very positive review to this book.

A User’s Guide to Spectral Sequences (Cambridge Studies in Advanced Mathematics)

Another book is Rotman’s An Introduction to Homological Algebra (Universitext). This book is from a homological algebra viewpoint. Rotman has a nice easy-going style, that made his books very popular to read.

The classic book may be MacLane’s Homology (Classics in Mathematics). This may be harder to read (though to be honest all books on spectral sequences are hard).

***Update: I found another book that gives a very nice presentation of certain spectral sequences, for instance the Bockstein spectral sequence. The book is Algebraic Methods in Unstable Homotopy Theory (New Mathematical Monographs) by Joseph Neisendorfer.

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Interesting Facts about Green’s Theorem

Firstly, Green’s Theorem is named after the mathematician George Green (14 July 1793 – 31 May 1841). Something remarkable about George Green is that he is almost entirely self-taught. He only went to school for one year (when he was 8 years old). His father was a baker, and George helped out in the bakery. Later, at the age of 40 he went to Cambridge to get a formal degree, but even before that he had already discovered Green’s Theorem. It is a mystery where did George Green learn his mathematical knowledge from. (During his time there was clearly no such thing as internet.)

It is unclear to historians exactly where Green obtained information on current developments in mathematics, as Nottingham had little in the way of intellectual resources. What is even more mysterious is that Green had used “the Mathematical Analysis,” a form of calculus derived from Leibniz that was virtually unheard of, or even actively discouraged, in England at the time (due to Leibniz being a contemporary of Newton who had his own methods that were championed in England). This form of calculus, and the developments of mathematicians such as LaplaceLacroix and Poisson were not taught even at Cambridge, let alone Nottingham, and yet Green had not only heard of these developments, but also improved upon them.

One of the applications of Green’s Theorem that I find interesting is finding the area of the ellipse: (Scroll down to Example 16.4.3). I find the proof very neat, you may want to check it out.

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Pierre-Simon de Laplace; French Newton

To increase your interest in mathematics, let me introduce the French mathematician Pierre-Simon de Laplace, also known as the “French Newton” or “Newton of France”. He helped to calculate projectile motion for Napoleon’s artillery. Laplace was also the examiner for Napoleon when he entered military school. Laplace also invented “Laplace transform” and “Laplacian” which will be useful in advanced engineering calculations.

Some quotes:

In September 1785 Laplace subjected Napoleon to a rigorous examination in differential equations and algebra as well as the practical applications of mathematics.
Book on Napoleon

The French Revolution began in 1789. Laplace was fortunately situated for avoiding its dangers, in part because, like Lagrange, his talents were found useful in calculating artillery trajectories. Napoleon esteemed Laplace, and after the Revolution showered him with honors.

Napoleon himself was good at math, he proved a theorem called Napoleon’s Theorem. Napoleon was “close friends with several mathematicians and scientists, including Fourier, Monge, Laplace, Chaptal, Berthollet, and Lagrange.”

Napoleon also made the following quote:

The advancement and perfection of mathematics are intimately connected with the prosperity of the State. — Emperor Napoléon Bonaparte.

Hope the above interesting facts increase your interest in math.

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5 Skills Students Need To Cope With School Pressures

According to an article published by the American Psychological Association (APA), many teenagers in the USA say they experience stress in patterns comparable to what adults go through. Teenagers also report higher stress levels than adults during the school year.

Tutors from Leaps ‘n Bounds, a learning center in Dubai, also observe that teen stress is not just confined to adolescents in certain countries; it is slowly becoming a widespread issue.

Teen stress can be caused by different factors including the pressure to perform well (or at least to pass) academically and in sports, and to have a great social life. In school, adolescents constantly face tough academic demands and responsibilities and experience social pressure.

Unfortunately, these challenges spill over even after the afternoon school bell rings, which can cause teenagers to feel even more stressed.

Dealing with Teen Stress

For teenagers to learn how to effectively deal with school pressures, they need to develop and rely on key personal skills. These include:

1.    Time management

All teenagers today always seem to be swamped with numerous activities: assignments, studying, extracurricular activities and sports. They need to find time for their friends, too.

Because teenagers need to have enough time to go through and complete these activities, they need to learn how to manage their time properly. Time management is an important skill they need to develop. This skill pertains to their ability to plan and control how they spend the hours in their day to complete their tasks and accomplish their goals.

 With proper time management, teens will be able to establish which tasks to prioritize and how to set their goals, and learn how to monitor where their time actually goes. As such, they will be able to avoid the stress of not having enough time on their hands to finish their assignments, complete their projects, meet their friends, and see their maths tutors in Dubai, if they have additional weekly tutorial or learning sessions.

2.    Setting realistic goals

Being number one in the class and, at the same time, for example, being the captain of the school football team are goals worth working hard for. However, overachieving teens tend to feel more pressure. When they fail or feel they didn’t perform up to expectations, they may develop low self-esteem and other negative feelings and attitudes.

Teenagers, therefore, are encouraged to lower their goals or set more realistic ones so that they can achieve more. By doing so, teens will also avoid pressure and boost academic success.

3.    Positive coping skills

 Coping skills are daily strategies and activities everyone uses or relies on to deal with, work through, or process emotions. Examples of positive coping skills include exercising, meditating, talking with friends or other family members, and having healthy hobbies such as reading and gardening.

Teenagers need to develop and practice positive coping skills instead of negative ones so that they will learn how to deal with stress through healthy ways. Positive coping strategies increase long-term resilience and well-being.

Negative coping techniques such as smoking and using drugs, on the other hand, may provide temporary relief from difficult emotions and pressure but lead to substance dependency and abuse.

For teenagers to effectively withstand adversity and deal confidently with daily stress and other challenges, they need to choose and apply positive coping strategies.

4.    Self-care

 For teens to better cope with pressure, they also need to have strong, healthy bodies. Teenagers, therefore, need to get enough sleep and rest, have a well-balanced diet, and get the right amount of exercise their bodies they need every day.

Adolescents need to take some time to pause from the relentless pace of everyday life and enjoy some creative activities that will help keep them from dwelling on or stressing over school pressures. This, in turn, will help them lower their stress levels.

5.    Optimism

Generally, stress is precipitated by stressful thinking. As such, teens can avoid stress and its negative effects by changing the way they think. When they have a positive mental attitude, they will have stronger coping strategies, better health, and a more stable, less stressful emotional life.

Adopting a positive way of thinking also helps teens complete their work and handle all their responsibilities. If they consistently think they won’t finish something or they don’t have enough time on their hands, they will lack the motivation to complete what they already started or even begin their task.

Teenagers only have a few more years before they enter another important phase in their lives: adulthood. But they can still enjoy all the experiences that come with adolescence and, at the same time, cope with all their school work and other activities without all the stress by simply developing the right skills.


Bushra Manna is one of the founders and Principal of Leaps and Bounds Education Centre – Motorcity. She has 20 years’ experience teaching the British and American curricula internationally at primary level – early middle school level, ages 4-12. Bushra believes in imparting deep learning to a child and not just rote learning, which is why she recommends the Magikats programme at her centre, to promote a genuine understanding with its multisensory, differentiated and interactive approach within a small group setting.



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Best Online Resources to Improve Your Math Skills

Best Online Resources to Improve Your Math Skills

Although Maths is a compulsory subject in all the educational institutions all over the world, many students consider it as a complete waste of time and skip this issue, claiming that they prefer not mathematically oriented disciplines but only social sciences. But will it really help you in your future or it is just something you have to learn because everybody does it? Math, as well as other exact sciences, is essential for the intellectual development of a person from early years, it helps to develop intelligence and better your critical, analytical, deductive and prognostic abilities together with improving your abstract thinking. Very often people understand the importance of logic and algebra when they are already adults and try to make up for lost time, but do not know where to find a qualified help quickly. We think that a man is never too old to study, that is why we have selected the best online sources to improve your knowledge of numerous subjects:

  • Online courses

There are plenty of courses available on the web, which can offer math courses online at low prices or even for free. Online learning is a trendy way of getting new knowledge and experience nowadays, but still, there are people, who prefer old and traditional methods and say it is useless and wastefulness. But, if you keep up with the times, you can visit such sites as Academic Earth, Edx, TED, University of the People, Coursera and others and get a unique experience in exploring new things from universities all over the world, just sitting on your sofa and holding a laptop.

  • Online libraries

For those, who still hesitate, exist online libraries, where you can download books you need and dive into learning by yourself. Of course, there is nothing better than a smell of a written word, but the theme of books varies depending on the site, and mostly they have a plethora of them available for free, so everybody could find the one he needs. The one drawback of using web libraries is that you have to organize an academic process yourself, thus have good time-management and organizational skills. Nevertheless, if you are a person, who is self-motivated and can move towards the goal on your own, this type of e-learning is exactly what you need. Among the most popular web libraries are The Online Library, Harvard Library, Wiley Library, Open Library and others.

  • Online Tutors

There are different types of school teachers, some of them focus on the needs of each student, other – use discipline as a key to successful learning, but the thing that is common for all of them, is strict correspondence to the school program, established in the country. School program doesn’t care whether the student understood the material he had just learned, or he still needs time to master it. This is the time when online tutors come to stage. If you need to learn math or other subjects that you have missed at school or simply improve them, they can assist you as they are 100% students oriented and can present the material according to the students’ abilities and needs.

  • Online lessons

This type of lessons is a perfect variant of distance learning for those, who are limited on time and want to master something quickly and get great results. Individual classes can take place in the form of face-to-face conversation by means of various apps or programs such as Skype and other video calls. Among the advantages of this type of e-learning is easy access – you can learn from home and build up your own learning schedule, also there is wide choice of tutors available on the Internet, no matter how far away they are, different educational content – video and audio which you can find for free on the web and, of course, you can choose the subject or skills you want to level up – it could be mathematics assignment writing, organic chemistry, etc.

There are plenty of ways how to study maths online, and it depends on which directions of studying will you choose, the pros and cons of each method based on your goal and preferences. The one thing you should remember is that it is never late to gain new knowledge or skills. As Benjamin Franklin once said, “An investment in knowledge pays the best interest.”


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An ancient Babylonian tablet known as Plimpton 322

Source: NY Times

One of my favorite YouTube Math Professors, Norman Wildberger, has made a historical math discovery: that the ancient Babylonian tablet known as Plimpton 322 is actually a trigonometric table.

“It’s a trigonometric table, which is 3,000 years ahead of its time,” said Daniel F. Mansfield of the University of New South Wales. Dr. Mansfield and his colleague Norman J. Wildberger reported their findings last week in the journal Historia Mathematica.

Check out my other blog posts on Prof. Norman Wildberger:
1) Algebraic Topology Video by Professor N J Wildberger

2) Video on Simplices and Simplicial Complexes

3) Critique on the Modern Axiomatic Approach of Mathematics

4) AlgTop1: One-dimensional objects

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Garry Kasparov Masterclass Review

Garry Kasparov Teaches Chess is the latest Masterclass by World Champion Garry Kasparov. I am sure that even from the trailer, you can see that it is of exceptional quality.

According to the FAQ, “the class will test players with a US rating of 1300-1700 and include more advanced concepts and examples that will benefit 1800+ rated players”. Most amateur players will fall under this range.

The price may seem steep ($90), but it is lifetime access so one can watch it over and over again. Also, there are 20+ lessons.

Click on this link to purchase: Garry Kasparov Teaches Chess

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The Summer 2017 Mathematics A To Z: Morse Theory


Today’s A To Z entry is a change of pace. It dives deeper into analysis than this round has been. The term comes from Mr Wu, of the Singapore Maths Tuition blog, whom I thank for the request.

Summer 2017 Mathematics A to Z, featuring a coati (it's kind of the Latin American raccoon) looking over alphabet blocks, with a lot of equations in the background. Art courtesy of Thomas K Dye, creator of the web comic Newshounds. He has a Patreon for those able to support his work. He’s also open for commissions, starting from US$10.

Morse Theory.

An old joke, as most of my academia-related ones are. The young scholar says to his teacher how amazing it was in the old days, when people were foolish, and thought the Sun and the Stars moved around the Earth. How fortunate we are to know better. The elder says, ah yes, but what would it look like if it were the other way around?

There are many things to ponder packed into that joke. For one…

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Education and the Blockchain – Should We be Teaching Blockchain in Schools?


It goes without saying that tech progress is moving at a rapid pace. Futurists point to Moore’s law – the idea that tech capabilities double every two years – as evidence for tech’s expansion into nearly every facet of our lives.

Teaching Technology

Education has seen its own dramatic tech advances. Kids can learn math from gamified apps while riding in the backseat of the family minivan. Students can hire an online algebra tutor and learn from anywhere via Skype. Aspiring students can virtually attend free Ivy-league classes (Massive Open Online Course, or MOOC) with millions of other learners of all ages and backgrounds. And NASA now collaborates with high school students in inventive hardware and robotics projects.

The most significant advance in computer-based education isn’t AI or virtual-based learning or even big data – it’s the blockchain. Blockchain has its origins in cryptocurrency, i.e. Bitcoin. The blockchain is essentially a way of managing data transactions – and it’s considered a radical disruption of traditional banking.

Plus, its applications in education – both virtual and classroom-based – have the potential to change everything about schools, from instruction to student achievement.

Exposure Versus Creativity

In the US, three-quarters of children have access to a smartphone. But on its own, that’s not necessarily a good thing. Kids who simply learn to operate a phone, just downloading and playing games, become consumers. The future lies with creators.

US Department of Labor statistics tell us that 2020 will bring with it 1.4 million computer specialist job openings. But American universities produce woefully inadequate numbers of graduates in the right fields – enough to fill a mere 29% of the jobs.

So what’s wrong with the picture? Why the big gap? There are many societal reasons we could point to, but one thing seems to stand out. We’re teaching tech literacy the wrong way.

Textbook-style curriculum may have its place, but not in tech ed. When kids are taught to memorize coding sequences and churn out the same answers to the same textbook questions, there’s no creative spark. No outside-the-box thinking.

In the best way, blockchain is wildly unconventional. To advance the world-changing potential of anti-dogmatic thinking, we need to encourage kids’ inventiveness. If the educational focus is on robot-like achievements rather than innovation, where will we find our climate change-tackling problem solvers?

We’ve labeled a generation of kids “tech-savvy” without giving them the tools to move from consumption to creation. It’s a waste of their brain power to hook kids on the addictive side of tech without pulling back the curtains and showing them the remarkable inner workings. Children and teens want to know how things work.

One solution? Teach tech like art. Coding has more in common with drawing than accounting. Yes, there is a necessary foundation in understanding digital languages and principles – but without encouraging creativity, we’re creating a generation of the same brain. Even gamified learning, if done improperly, can be perilously bland.

Tackling the Education Gap

There are few key components of a sound approach to teaching creative thinking around technology.

  1. Let it be accessible. Kids will shy away from a big learning curve – learning and doing need an intimate relationship.
  2. Remove the achievement roof. Learning platforms and educational approaches which employ standardized tests as the litmus for success – and for what the content can achieve –inhibit creativity. Rather than saying “do this to produce this result,” what if we said, “here are your tools – now, what can you create?” Consider The Lego Movie’s message of the importance of imagination – for future tech innovation, we need makers, not managers.
  3. Embrace a shifting curriculum. In other subjects, things might stand as eternal truths; the Magna Carta will always have been signed in 1215. But in technology education, things move at a blistering pace. A particular tool or lesson may become quickly outdated, so the educational format needs flexibility, just like the subject it teaches.

Blockchain is set to change the world. But as we continue to encounter environmental and societal problems, we need amazing minds to solve them. Revolutionizing how we teach technology education might be the answer we didn’t know we needed.

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ZAPZAP : Math + Game

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Mathematical Functions vs Programming Functions

Math Online Tom Circle

Key Points:

  • Pure Function
  • Partial Function
  • Total Function

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Symmetry, Algebra and the “Monster”

Math Online Tom Circle

Very good introduction of Modern Math concept “Group” to secondary school math students by an American high school teacher.


  • Symmetry of a Square
  • Isometry (*) or Rigid Motion (刚体运动) = no change in shape and size after a transformation
  • What is a Group (群 “CAN I” ) ? = Closure Associative Neutral Inverse
  • Monster Group = God ?
  • String Theory: Higgs boson (玻色子) aka “God Particles”

Note (*): “保距映射” (Isometry),是指在度量空间 (metric space) 之中保持距离不变的”同构“关系 (Isomorphism) 。几何学中的对应概念是 “全等变换”

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Math Problem Solver – SymboLab

Math Online Tom Circle

A new alternative to the UK Mathematica’s “Wolfram Alpha” online tool:

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Son of taxi driver among this year’s President’s Scholars


Mr Lee Tat Wei lives in a four-room flat in Woodlands. He and his older brother went to neighbourhood schools. His father is a taxi driver and his mother works as a part- time sales assistant.

Despite his humble background, the 19-year-old said he has never felt shortchanged. “My parents gave me an environment that money couldn’t buy. They never pressured me to get straight As. They taught me to live in the moment,” said the Anglo-Chinese School (Independent) graduate who had a perfect score of 45 for his International Baccalaureate diploma exams.

Mr Lee, who is one of the five recipients of the President’s Scholarship this year, will be going to read liberal arts at Yale University.

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I’m a Rare Breed: An Elite Chess Player Who’s Open About His Faith

A nice interview by Wesley So, one of the top chess grandmasters from Philippines.

On the small planet where elite chess players dwell, very few people worship Jesus Christ. If anyone discovers that you’re one of those “superstitious,” “narrow-minded idiots,” you’re likely to see nasty comments accumulate on your Facebook fan page. On a regular basis, I receive emails from strangers lecturing me about the dangers of following Jesus. Out of pity or disgust, they wonder how I, the world’s second-ranked chess player, can be so “weak-minded.” I have been assured that identifying openly as a Christian will interfere with sponsorship, support, and invitations to events. I have been told that spending time reading my Bible, praying, and going to church will inevitably weaken my performance. People plead with me to at least keep quiet. They say thanking God publicly makes me look ridiculous. So why did I make such a risky move?

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From Wikipedia:

As a young player, So’s aggressive and tactical style of play caught the attention of a former Philippine chess champion, International MasterRodolfo Tan Cardoso. Cardoso said of So:

“The young lad…would sacrifice a queen or any other pieces in his arsenal to get a winning attack….He cannot afford decent training given by well known GM-coaches and has to rely on his pure talent…before competing.”

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UK Textbooks à la Chinese / Singapore Math Style

Math Online Tom Circle

[Original Financial Times Article] Google: UK maths books fail DfE test

$latex boxed {text {The Ideal Math = (Chinese + English) * French }}&fg=aa0000&s=3 $

UK, USA, France are copying Chinese Math (from which the “Singapore Modeling Math” derived) in Primary schools, this proves my above-mentioned “Ideal Math” formula is correct. The 2 Asian countries were top in 2015 PISA Math Test for 15-year-old students, while UK was ranked 27th.

It remains to see if China / Singapore reciprocate the French Math theoretical foundation rigor in High Schools (Junior Colleges) and in the first 2 years of undergraduates for STEM (Science, Technology, Engineering, Math) students.

Compared with Fields Medals (equivalent to Nobel Prize in Math), the picture is reversed – where USA, France and UK are top. The secret lies in the formula : the multiplying factor – (* French), ie the math theoretical foundation notably…

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How (not) to memorise mathematics

Math Online Tom Circle

Many excellent Math students after leaving universities more than 10 years forget 90% of math they learned, save some primary school arithmetics – few could do Singapore PSLE Modelling Math or solve quadratic equations.

The “Story-Telling” memory technique via “Signposts” can be used to reconstruct math from first principles:

Note: Lewis Carroll: the author of “Alice in wonderland”

Cambridge Professor Tim Gowers (Fields Medalist) suggested the similar pedagogy of “Memorise by First Principles”.

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Math drives Biology

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The Ring Z/nZ, Fermat Little Theorem, Chinese Theorem (French)

Math Online Tom Circle

Revision: Modulus Arithmetics

(1/2) Fermat Little Theorem

(1/2) Chinese Theorem

(Note: This is the “RING” foundation of “The Chinese Remainder Theorem” which deals with remainders )

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The scientist nuns: In pursuit of faith and reason

Source: Aleteia

Making a career out of science, just like joining a religious order, requires dedication and discipline. Some tireless souls have managed to do both.

In 1965, Mary Kenneth Keller became the first woman to obtain a PhD in Computer Science. She was also a nun.

Born in Cleveland, Ohio, in 1913, Keller entered the Sisters of Charity of the Blessed Virgin Mary in Dubuque, Iowa, in 1932. Eight years later, she professed her vows, before obtaining B.S. and M.S. degrees in mathematics from DePaul University in Chicago, where she became fascinated by the incipient field of computer science.

As a graduate student, she spent semesters at other schools, including New Hampshire’s Ivy League college Dartmouth, which at that time was not coeducational. For her, however, the school relaxed its policy on gender, and she worked in the computer center, where she contributed to the development of the BASIC programming language that became so instrumental to the early generation of programmers.

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Ideal Math Education

Math Online Tom Circle

$latex boxed { text {Ideal Math Education } = (C + E) * F}&fg=aa0000&s=3$

C = Chinese 中文 = Primary school Arithematics sans “Algebra” (= Singapore Modeling Math). Abacus-Algorithmic thinking.

E = English = Secondary School Math
=> Applied, Tricky, Math Olympiad-style
=> Engineering, Business, Applied Science.

F = Français (French) = High-school / Baccalaureate & University Math = Theoretical, Abstract
=> Rigorous Math Philosophy for Advanced Concepts
=> New frontier Scientific Research.

Why (C+E) ?
C + E = Basic Math Foundation.

Why * F ?
If multiply by Theoretical F, like flying with added wings (如虎添翅)。

if (C+E) -> 0 (less applied), or
if F -> 0 (lack theories),
then Total Math Education -> 0.

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Theorem of the Day

Just to recommend this excellent website: Theoremoftheday where they feature one mathematical theorem each day.

The nice thing is that each theorem is a one-page summary, good for getting acquainted with the theorem, and subsequently you may read it up in more detail.

The website does have a XML feed, though it would be nice if there were a email subscription (with weekly emails).

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How to win Sir Roger Penrose’s Chess puzzle (that computers can’t solve)


Despite chess computers being very highly rated and winning virtually all human grandmasters, there are still certain positions that the computers can’t solve.

Sir Roger Penrose has documented one of them here:

Chess engines will state that black is winning by a large margin, when in fact White can easily draw, or even win!

Drawing should be easy. Just move the king around (without moving the c6 pawn). The only black pieces that can move are the dark-squared bishops, which can’t checkmate your king.

Winning should be only possible if Black plays badly, e.g. Bishops all give up control of the c7 square. Then c7 followed by c8=B or c8=Q is checkmate!

Very nice study by Sir Penrose that illustrates the weakness of computers!

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Math Olympiad (Primary Schools) 一筐鸡蛋

Math Online Tom Circle

One basket of eggs.
Remove 1 by 1, nothing left in basket.

Remove 2 by 2, one left in basket.

Remove 3 by 3, nothing left in basket.

Remove 4 by 4, one left in basket.

Remove 5 by 5, short of one to complete.

Remove 6 by 6, 3 left in basket.

Remove 7 by 7, nothing left in basket.

Remove 8 by 8, one left in basket.

Remove 9 by 9, nothing left in basket.

请问筐里最少有几粒鸡蛋 ?
At least how many eggs are there in the basket?

[Hint] This is a Chinese Remainder Problem (韩信点兵“)

—— [Solution] —–

Let there be minimum X eggs in the basket.

Remove 1 by 1, nothing left in basket:
X = 0 mod (1) …[1]
=> trivial & useless !

Remove 2 by 2, one left in basket:
X =…

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Alternate Admission Route to NUS Computing


Good news to students who are interested to study Computer Science. There is now an alternative route for students who are short of the cut-off point (currently at least two A’s).

To win a place on the increasingly popular computer science degree course at the National University of Singapore (NUS), students need at least two As for their A levels. Next year though, students eyeing a computing degree can be admitted through another route.

They can take up a five-month-long computer programming course at NUS and if they do well, gain fast-track admission into the degree course, even though they may fall short of the required grades.

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Topology application to Physics


The Strange Topology That Is Reshaping Physics

Topological effects might be hiding inside perfectly ordinary materials, waiting to reveal bizarre new particles or bolster quantum computing

Charles Kane never thought he would be cavorting with topologists. “I don’t think like a mathematician,” admits Kane, a theoretical physicist who has tended to focus on tangible problems about solid materials. He is not alone. Physicists have typically paid little attention to topology—the mathematical study of shapes and their arrangement in space. But now Kane and other physicists are flocking to the field.

In the past decade, they have found that topology provides unique insight into the physics of materials, such as how some insulators can sneakily conduct electricity along a single-atom layer on their surfaces.

Some of these topological effects were uncovered in the 1980s, but only in the past few years have researchers begun to realize that they could be much more prevalent and bizarre than anyone expected. Topological materials have been “sitting in plain sight, and people didn’t think to look for them”, says Kane, who is at the University of Pennsylvania in Philadelphia.

Now, topological physics is truly exploding: it seems increasingly rare to see a paper on solid-state physics that doesn’t have the word topology in the title. And experimentalists are about to get even busier. A study on page 298 of this week’s Nature unveils an atlas of materials that might host topological effects, giving physicists many more places to go looking for bizarre states of matter such as Weyl fermions or quantum-spin liquids.

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What Would You Like In The Summer 2017 Mathematics A To Z?


Summer 2017 Mathematics A to Z, featuring a coati (it's kind of the Latin American raccoon) looking over alphabet blocks, with a lot of equations in the background. Art courtesy of Thomas K Dye, creator of the web comic Newshounds. He has a Patreon for those able to support his work.

I would like to now announce exactly what everyone with the ability to draw conclusions expected after I listed the things covered in previous Mathematics A To Z summaries. I’m hoping to write essays about another 26 topics, one for each of the major letters of the alphabet. And, as ever, I’d like your requests. It’s great fun to be tossed out a subject and either know enough about it, or learn enough about it in a hurry, to write a couple hundred words about it.

So that’s what this is for. Please, in comments, list something you’d like to see explained.

For the most part, I’ll do a letter on a first-come, first-serve basis. I’ll try to keep this page updated so that people know…

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How to do Proof by Cases in LaTeX

If one searches online, one will find many different methods to do “proof by cases” in LaTeX. The most simple and convenient method in my opinion is to use the description environment.

Something like this:

\begin{proof} Proceed by cases.
\item[Case 1: This.] And so on.
\item[Case 2: That.] And more.

Source: Reddit

No additional package is needed. One drawback is there is no auto-numbering, but I am sure that is still ok, unless your proof has many many cases.

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What Is The Shape of the Universe (Video)

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(Important Changes) PSLE Math: Arrow -> vs Equal=

Source: Facebook

For those taking PSLE, please take note of this important update regarding the difference between arrow and equal sign. Forward this to your friends taking PSLE!

Basically, I think MOE is trying to instill students to be mathematically correct. (See update below: Marks will not be deducted in most cases but proper usage is highly encouraged.)

E.g. 100%=40 is wrong as 100%=100/100=1 technically. Similarly, 10 men = 40 hours is wrong as the units do not match (nor make sense).

Trying to enforce “units” instead of “u”, and banning “10 units -> 20” is a bit strict though, in my opinion.

Update: (

MOE responds

In response to queries, a Ministry of Education spokesperson clarified that the above information was not provided by the ministry.

The information above was originally sourced from the website of a private tuition centre, whose sources are currently unverified.

While the respective uses of the arrow and equal signs are accurate in the infographic, the MOE spokesperson said full credit will still be awarded to the student even when the signs are used interchangeably, as long as the student demonstrates a full understanding of the question.

Proper use of arrow and equal signs are, nonetheless, encouraged.

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PhD Comic: 大智若愚

The new PhD Comic is related to the Chinese proverb: 大智若愚, which is roughly translated as “a wise man looks stupid.; great wisdom takes the looks of folly [appearance of stupidity]”.

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SG Education News: More places for Medicine Students

Good news for those aspiring to be medical doctors.

Med school places to rise to 500 by next year

The National University of Singapore (NUS) has had more than 2,000 top students fight for the 300 spaces in its Yong Loo Lin School of Medicine …

Other top Education news:

Hong Kong, UAE, Singapore Priciest Places for Education

In Singapore, the government subsidizes schooling costs for locals and has doubled its education budget since 2005 as part of a plan to build the …

New NTU president has history of ties with Singapore

New NTU president has history of ties with Singapore … Council and the International Academic Advisory Panel of Singapore’s Ministry of Education.

Building a `Smart Nation’ in Singapore Doesn’t Come Cheap

Schooling isn’t cheap in Singapore: not just for citizens, but for the government too. The city state boasts one of the best education systems in the world …

How robots are teaching Singapore’s kids

In Singapore, admired globally for its education system, authorities are trialling the use of robotic aides to teachers in kindergartens. Two humanoid …
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Unique Cars and Parts

There are hundreds if not thousands of Automotive websites to be found on the net, so often it can be a little hit and miss as to finding the best of the best that is out there. Unique Cars and Parts is just one of those sites, one of the best that there is. Well researched and covering almost every automotive topic there is, the site also boasts a mammoth array of various media types, from old auto radio commercials, TV and cinema advertising, car brochures, press releases, car launches and biographies.

Even better is the ability to sell your old car and or parts for free, or even list your auto business if you are in the trade. No cost really does mean no cost – not sure how they do it – but what they have created is brilliant. If you are a car aficionado or even have a simple passing interest in the history of automobiles, from the earliest contraptions found on the roads at the beginning of last century, the the more modern vehicles we drive today – pay this site a visit and we would recommend you bookmark it for future reference.

Click here to go to the Unique Cars and Parts Classic Car Website

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Les Categories Pour Les Nuls

Math Online Tom Circle

“Categories for Dummies”

Example 1:

Paris (P) -> Rome (R) -> Amsterdam (A)
Objects: cities {P, R , A}
Morphism (Arrow): railway

  • Identity: railway within the city
  • Associative: (P -> R) -> A = P -> (R ->A)

=> Category

Example 2:
A, B are categories

functor f : A -> B

f (B) has the “information” on A, with some loss of information since f may not be a MONOMORPHISM.

Example 3: Natural Transformation

A = 0 1 2 3 4
f : A -> B

B = Ladder steps:

g : A -> B

B = Staircase steps :

Natural Transformation: =>α

α : f (i) => g (i)

f(0)| =>g(0)||

α transforms naturally the Ladder to the Staircase.

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Debate (French): Set Theory vs Category Theory

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An Introduction to N-Categories

Math Online Tom Circle

Tom Leinster

N = 0 : 0-Cat

  • => Set, 0-morphism = function

N= 1: 1-Cat

  • => Cat, 1-morphism = functor

N= 2: 2-Cat

  • => 2-morphism = Natural Transformation

$latex text {f, g : 1-morphism }$

$latex alpha :: beta text { Natural Transformations : 2-morphism }$

Definition of n-Category:


0-Cat : Set
1-Cat : Cat

Examples of n-Categories:

  • Manifold
  • Top (Topological Space) : 2-morphism = homotopy


Best Technical Category Theory Book (2016) by Tom Leinster (Cambridge Press): “Basic Category Theory”

(Free downloadfrom arxiv)

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Category Theory : “How to Make Pi”

Math Online Tom Circle

Dr. Eugenia Cheng – Professor of Category Theory (Chicago University)

Author of the Best Selling Category Book Book : (for readers from 7-year-old to high school and undergraduate students)

How to Bake Pi ?- an Edible Exploration to the Mathematics of Mathematics”

[Loan from NLB(eg. AMK Branch)]


  • Factors of 30 = {2, 3, 5, 6, 10, 15, 30} which form a Cube with these factors as the vertices.
  • Knots
  • Bach music
  • Associativity: (sugar + milk + egg )
  • Button cake = order 2 group (0,1)
  • Bed mattress = rotate, flip, flop
  • Icosahedral virus


Best Technical Category Theory Book (2016) by Tom Leinster (Cambridge Press): “Basic Category Theory”

(Free downloadfrom arxiv)

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Program = Category 

Math Online Tom Circle



  • Category
  • Monad = Monoid + Endofunctor

Category Theory is replacing Set Theory as the foundation of Math. Nowadays, few Advanced Math papers are written without using Category to explain, and this trend is spreading to IT through Functional Programming languages (Google’s Kotlin, Haskell, Clojure…) - the latest paradigm to replace Object-Oriented languages like Java, C++, etc, as a safer “Strong Typed” languages for AI, BIG DATA…

$latex boxed {text {Type = Category }}&fg=aa0000&s=3$

Examples of “Types” in IT:

  • Integers
  • Real
  • Double
  • Boolean
  • String
  • etc

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Free tutoring app for peer-tutoring

Do try out the new app AsknTeach. Quite a good idea, the app creator reportedly invested more than $100,000 to $200,000 into it.

Only problem I can foresee is that overall quality of answers may not be that good, since it is by peer students, also the engagement rate may not be high. The same problem may occur: more people asking questions than answering.

That being said, if the top echelon of students can be motivated to answer it, the quality of answers will be great. E.g., the top secondary school students are easily at JC level or beyond, they can easily answer questions at their level.


One of the founders of free tutoring app EduSnap is back with a new peer tutoring app – AsknTeach.

Back in 2013, Mr Chia Luck Yong set up EduSnap as a social enterprise with two Singapore Management University schoolmates, Mr Anders Tan and Mr Shaun Tan, and launched the app the next year.

At the time, the free mobile platform drew attention as it positioned itself as the first of its kind helping Singapore students, and was reported about in major media outlets.

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Integral Domains 整环 (Abstract Algebra)

Math Online Tom Circle

Remember when you cancel a common factor at both sides of an equation, you must check if the factor is non-zero, otherwise you would miss some answers.

This is about Cancellation Law, related tofewNumber Theory Properties :

  • Zero Divisors,
  • Integral Domain.

Origin of “Integral” => Integers

Definition of Integral Domian:

Property:Cancellation Law



  1. Every Field is an Integral Domain.
  2. Every finite Integral Domain is a Field.

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South Korean Young Mathematician June Huh

Math Online Tom Circle

A weak South Korean Math studentJune Huhbecomes a top Mathematician in the Princeton Institute for Advanced Study by proving the Read’s Conjecture and Rota Conjecture.

“A path less taken to the peak of the Math World” :

This important idea is called Isomorphism 同构

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