Shop Amazon – New Year, New You Event
Start Date: Dec 26, 2014
End Date: Jan 20, 2015
Type: Seasonal
Categories: Miscellaneous
Check out this New Year offer from Amazon, which ends Jan 20, 2015!
Buy a gift for yourself!
Shop Amazon – New Year, New You Event
Although the QZ8501 plane is now confirmed to be in the Java Sea, we still hope that there may be some survivors, who may have possibly swum to a nearby island. Hopefully, the rescuers may be able to rescue some of the passengers.
This is a followup on the previous post on the Probability of adding up to 8888.
Other than the QZ 8501 incident, there are many other incidents that have relation to “8888” or “8” in general. Some of them are the “8888” Uprising in Myanmar, the May 12 Sichuan earthquake, the Blizzard of January 25, and the Tibetan Riots of March 14.
To read more, visit: http://mathtuition88.blogspot.sg/2014/12/the-significance-of-8888-mh17mh370qz8501.html
One special fact about 8, is that it is the only natural perfect power that is one less than another perfect power. (2^3=3^2-1) This is known as Catalan’s conjecture or Mihăilescu’s theorem.
Lastly, you may want to watch this video on Chinese Lucky Numbers by Numberphile.
Featured Book:
Biblical Numerology: A Basic Study of the Use of Numbers in the Bible
A sane explanation of biblical numerology. Davis explains the conventional, rhetorical, symbolic, and mystical use of numbers in this fascinating study of the structure and syntax of biblical numbers.
Numbers are present throughout the Bible, and do have some meanings. Why did God create the world in seven days? (rest on the seventh day) Why did Jesus have 12 disciples, not more or not less? Read this book to find out!
The WordPress.com stats helper monkeys prepared a 2014 annual report for this blog.
Here’s an excerpt:
The Louvre Museum has 8.5 million visitors per year. This blog was viewed about 200,000 times in 2014. If it were an exhibit at the Louvre Museum, it would take about 9 days for that many people to see it.
女童记不住乘法表崩溃大哭”三五太難了”:
This 4-year-old Chinese girl cried when her mother ‘tortured’ her to recite Multiplication table by rote learning. She always got stuck at 3 x 5= ?
She complained it is too difficult.
Most parents are teaching the kids Mathematics the wrong way! No wonder they grow up with hate and fear for Maths subject throughout the entire life.
Same in schools and universities, Maths are taught the wrong ways by incompetent Math educators.
Source: http://mathtuition88.blogspot.sg/2014/12/math-of-qz8501-sum-of-3-numbers-that.html
We sincerely hope that Captain Iriyanto, the pilot of missing AirAsia flight QZ8501, and all other passengers survive and return safely. Hopefully the plane will be found soon.
Something mysterious about the recent missing airplanes is that their numbers add up to 8888, a mystical number in Chinese culture. MH17, MH370, QZ8501, 17+370+8501=8888.
We have calculated that it is not a common event at all, it is rarer than winning the top prize in 4D, a lottery in Singapore.
Probability of 3 random numbers (from 1 to 9999) adding to 8888: 0.0039497% , or around 1 in 25,000.
Probability of winning the First Prize in 4D: 0.01%, or 1 in 10,000.
Read more at: http://mathtuition88.blogspot.sg/2014/12/math-of-qz8501-sum-of-3-numbers-that.html
Featured Book:
Introduction to Probability, 2nd Edition
An intuitive, yet precise introduction to probability theory, stochastic processes, and probabilistic models used in science, engineering, economics, and related fields. The 2nd edition is a substantial revision of the 1st edition, involving a reorganization of old material and the addition of new material. The length of the book has increased by about 25 percent. The main new feature of the 2nd edition is thorough introduction to Bayesian and classical statistics.
Mind, Brain And Education
1. Spaced Repetition
2. Retrieval Practice
Tool: Test
Not to assess what students know, but to reinforce it.
Memory is like a storage tank, a test as a kind of dipstick that measure how much information we’ve put in there.
But that’s not how the brain works.
Every time we pull up a memory, we make it stronger and more lasting, so that testing doesn’t just measure, it changes learning.
Simply reading over materials to be learnt, or even taking notes and making outlines, as many homework assignments require, doesn’t have this effect.
Language learner: 80% retained.
Science: 50% retained.
Self-quizzing (focus less on input of knowledge by passive reading, focus more on output by calling out that same information from brain.)
Cognitive disfluency:
Tough topic, recall better.
Interleaved assignment: mix up different kinds of problems instead of grouping by type.
Weekly Homework (hours)
A recent survey in 44 OECD countries reveals for 15-year-old students an average of 5 hours / week of homework.
| Country | Homework Hrs | Extra Hrs | PISA |
| Shanghai | 14 | Tuitions | 1 |
| Singapore | 9.4 | Tuitions | 2 |
| Hong Kong | 6 | Tuitions | 3 |
| Korea | 2.7 | Tuitions | 4 |
| Japan | 3.8 | Tuitions | 5 |
| Taipei | 5.9 | Tuitions | 6 |
| Finland | 2.8 | 7 | |
| England | 4.9 | 26 |
PISA 2012:
PISA is like the Army IPPT Test on physical fitness. A fit soldier and a weak soldier go to war, whether he can fight with courage under duress to win the battle, has nothing to do with his IPPT scores.
Same for PISA scores…that explains why Americans are poor in PISA but produce many entrepreneurs, Nobel prize / Fields scientists, whereas China, Singapore, Korea, HK have only few.
With the Star Wars Episode 7 coming up, all Star Wars fans are really excited. The trailer alone has reached 50 million views, barely a month after it was released.
Wait, can Star Wars be related to Math? Yes it can! Check out The Math of Star Wars which describes a Math related question related to Star Wars! As a Math Tutor, I try my best to relate anything and everything to Math! 😛
Christmas is ending soon, and hope everyone had a nice day, and happy new Year!
Featured Book:
Star Wars Workbook: 1st Grade Math
Does your child love Star Wars but hate math? Well, this might just work! A highly rated Workbook based on Star Wars.
One of the pros of Blogger is that JavaScript is allowed, whereas for WordPress, JavaScript is not allowed for security reasons. JavaScript is a pretty cool add on to a website, it can do simple calculations.
I have written a JavaScript Applet to Guess Birthday Month based on First 4 NRIC Digits, at http://mathtuition88.blogspot.sg/2014/12/javascript-app-to-calculate-birthdate.html.
Do check it out if you are interested! (For fun only, it may not be 100% accurate).
The math theory is based on my earlier post at: Formula to guess Month of Birthday from Singapore NRIC.
Featured Post:
Mathematics for 3D Game Programming and Computer Graphics, Third Edition
Refering to How to write Math Formulas on Blogspot / Blogger, Blogspot is also capable of rendering beautiful Math Equations based on LaTeX.
Blogger uses MathJax for rendering the Math Formulas, which look like the ones shown above.
Let’s try the WordPress version and readers can judge for themselves which is better.
My personal opinion is: they look the same to be honest. However, the Blogger way of typing is much more convenient, just using $ and $$, as opposed to WordPress requiring “$latex” and “$latex\displaystyle”, which is more cumbersome especially for long texts. However, experts like Terence Tao have opted for WordPress, which shows that WordPress does probably have some advantages.
Featured Book:
The LaTeX Companion (Tools and Techniques for Computer Typesetting)
In O Level E Maths, students are required to memorize the prefixes: kilo-, mega-, and giga-. For today’s computer literate generation, that should be no problem since kilobytes, megabytes, and gigabytes are used so commonly.
A practical usage of this is Computer Memory, and how to solve the dreaded Blue Screen of Death.
I have recently discovered a truly remarkable method on how to solve the Blue Screen of Death, using a free tool called Glary Utilities. Read more about it here and hope it helps!
Site: http://mathtuition88.blogspot.sg/2014/12/the-math-of-computer-memory-and-how-to.html
Featured Book:
Computer Math: Problem Solving for Information Technology with Student Solutions Manual (2nd Edition)
The following is a follow up video on my earlier post on Logarithm Question (Challenging).
The video is posted on: http://mathtuition88.blogspot.sg/2014/12/singapore-math-high-school-logarithm.html
Thanks for watching!
Featured Book:
John Napier: Life, Logarithms, and Legacy
Featured Posts:
Is Santa real? Can we prove it using Math that Santa exists / does not exist?
Check out this website at: http://mathtuition88.blogspot.sg/2014/12/the-math-of-santa.html
(Actually it is a sister website of Mathtuition88.com, I have decided to post more often there to build up the site.)
Featured Book:
Everyday Calculus: Discovering the Hidden Math All Around Us
Calculus. For some of us, the word conjures up memories of ten-pound textbooks and visions of tedious abstract equations. And yet, in reality, calculus is fun, accessible, and surrounds us everywhere we go. In Everyday Calculus, Oscar Fernandez shows us how to see the math in our coffee, on the highway, and even in the night sky.
Free Career Analysis:
Are you a Careerist, Entrepreneur, Harmoniser, Idealist, Hunter, Internationalist, or Leader?
Find out by doing this Free Career Analysis by renowned survey company Universum:
http://uledge.co/ambWWCY
Thanks a lot for your help! You will also benefit by finding out your Career Personality, which is released at the end of the survey!
Website: http://uledge.co/ambWWCY
This is a YouTube Video, based on my earlier post on Fibonacci Numbers and the Mysterious Golden Ratio!
Thanks for watching! Remember to speed up the video when necessary!
Featured Book:
The Fabulous Fibonacci Numbers
The most ubiquitous, and perhaps the most intriguing, number pattern in mathematics is the Fibonacci sequence. In this simple pattern beginning with two ones, each succeeding number is the sum of the two numbers immediately preceding it (1, 1, 2, 3, 5, 8, 13, 21, ad infinitum). Far from being just a curiosity, this sequence recurs in structures found throughout nature – from the arrangement of whorls on a pinecone to the branches of certain plant stems. All of which is astounding evidence for the deep mathematical basis of the natural world.
With admirable clarity, two veteran math educators take us on a fascinating tour of the many ramifications of the Fibonacci numbers. They begin with a brief history of a distinguished Italian discoverer, who, among other accomplishments, was responsible for popularizing the use of Arabic numerals in the West. Turning to botany, the authors demonstrate, through illustrative diagrams, the unbelievable connections between Fibonacci numbers and natural forms (pineapples, sunflowers, and daisies are just a few examples).
This is an example of Intro Video made using Flixpress!
Pretty easy to make, just take a few minutes. I will be adding more videos to my YouTube Channel soon.
Featured Page:
Source: http://blog.nus.edu.sg/provost/2014/12/23/su1101-the-science-and-art-of-the-su-option/
Check out this link by NUS Provost, Prof. Tan, who was also a Math Professor.
Now, which grades should you keep?
If you obtained A or A+, well done and keep the grade!
If you obtained B+ or A-, I would generally encourage you to keep the grade as well. For those who may be thinking of exercising S/U on a B+ to qualify for the Dean’s List, do note that there will NOT be Dean’s Lists for the first two semesters.
If you obtained Bs and Cs, it is a little tricky. In theory, you should exercise S/U on your worst grades. However, the challenge is to do so without foresight of the grades that you will get for subsequent semesters. You should base your decisions on your academic goals and your self-assessment of your expected academic performance for the rest of your candidature. If you do not have a goal right now, your first semester CAP (before any S/U options are exercised) may be a good guide.
The S/U option is really a lifesaver for those at the borderlines, for example Borderline First Class Honours (CAP around 4.5), or Borderline Second Upper (Cap around 4.0). It makes a difference to your final grades.
The S/U option will apply to all Level 1000 modules (with or without pre-requisites) and Level 2000 modules without other NUS modules as pre-requisites, unless otherwise stipulated by the Faculties/Departments.
Taken from: http://www.nus.edu.sg/registrar/edu/UG/graduation.html#SU
This is a great improvement from the earlier system, where students could only S/U non-core modules. New NUS students have it good!
Featured Book:
The Calculus Lifesaver: All the Tools You Need to Excel at Calculus (Princeton Lifesaver Study Guide)
This is my second ever YouTube Video, and it is about a rarely seen proof (Constructive) that Square Root of Two is irrational.
The video is based on my earlier post on: Constructive Proof that Square Root of Two is irrational
Thanks for watching! Please speed up the video according to your preference! You can speed up either 2x or 1.5x for best effect.
Featured Book:
The Irrationals: A Story of the Numbers You Can’t Count On
The ancient Greeks discovered them, but it wasn’t until the nineteenth century that irrational numbers were properly understood and rigorously defined, and even today not all their mysteries have been revealed. In The Irrationals, the first popular and comprehensive book on the subject, Julian Havil tells the story of irrational numbers and the mathematicians who have tackled their challenges, from antiquity to the twenty-first century. Along the way, he explains why irrational numbers are surprisingly difficult to define–and why so many questions still surround them. Fascinating and illuminating, this is a book for everyone who loves math and the history behind it.
The holiday season is here, and check out the deals on Groupon to save while buying presents or delicious food!
https://www.youtube.com/watch?v=buhghuawde0
This is my first video on YouTube! 🙂
Please subscribe to my channel for more Math Videos coming in the future!
Featured Book:
Algebra Survival Guide Workbook: Thousands of Problems To Sharpen Skills and Enhance Understanding
$latex (a-b)^3=a^3-3a^2b+3ab^2-b^3$
So,
$latex
begin{array}{rcl}
a^3-b^3&=&(a-b)^3+3a^2b-3ab^2
&=&(a-b)(a-b)^2+3a^2b-3ab^2
&=&(a-b)(a^2-2ab+b^2)+(a-b)(3ab)
&=&(a-b)(a^2+ab+b^2)
end{array}
$
Hi Readers,
Are you looking for a printable 2015 Calendar, specially tailored for Singapore?
Check out this PDF printable 2015 calendar: Calendar
(Generated by http://www.calendarlabs.com/pdf-calendar.php)
Happy new year!
Featured book:
Give a child gifts that will last a lifetime – self-esteem, tolerance, values, and inner strength. This book is filled with inspirational stories for children and their families to share, all about kids making good choices and doing the right thing.
The values that children learn today will stay with them for the rest of their lives. This collection gives kids positive role models to follow in its 101 stories about doing the right thing and making healthy choices. You and your child will enjoy discussing the stories, making it a family event. Great for teachers to share with students too.
从来没有状元的老师, 却能培养出状元学生。中国历史的状元, 都是在科举失败的穷秀才教导出来的。看今日的Fields Medalists / Nobel Prize winners, 他们的指导教授多数没拿过这大奖。
法国有两位世界级的数学大师Evariste Galois, Charles Hermite, 相隔15年,同是一位中学数学老师培养。Mr. Richard 是巴黎路易大帝中学(Lycée Louis Le Grand) 的数学老师, 他能创新课材, 加上他本身是”数学迷”, 对学生因材施教, 慧眼视英雄于微时。Galois13 岁前是由妈妈家教(home-schooling), 中一才上学校。Richard看出他有数学天才, 个别教他读当代数学大师的书, 其他科目的老师却视他为功课低劣学生。
结果Galois发现群论 (Group Theory), 开创”抽象代数” (Abstract Algebra)。Galois不被世人接受, 因为他的理论太玄奥 (至今还是数学硕士班的难题), Cauchy, Fourier, Poissons, Gauss… 这些世界数学泰斗也看不懂! 他的悲剧是法国大革命, 因枪斗而死, 才21岁的生命。可是他对人类的贡献是”Larger than Life”.
Richard 收集Galois的作业, 留给15年后同一班另一学生Charles Hermite, 也是个数学天才。他比Galois的命运好一点, 考进Ecole Polytechnique (X),是排最后一名及格, 好过Galois重考2年都进不了。可是他的脚有问题, X第一年学生是军官训练, 他被X踢出校门。讽刺的是, 很多年后他回校被聘为教授。
Charles Hermite 证明 e是超函数 (Transcendental), 他的学生德国人Lindermann如法炮制, 证明pi也是transcendental.
Lindemann 从Hermite接过法国数学火种, 回去德国成为一代宗师, 培养了很多大师级的学生 (Félix Klein, Dirichlet, Jacobi, Gauss, …), 20世纪德国数学Gottingen University取代巴黎成为数学王国, 直至二战德国犹太数学家(Émile Noether, Artin, …)逃去美国, 才轮到Princeton University.
一位默默无名的中学/高中数学老师Richard, 百年树人, 改变了世界数学史!
My previous posts on Clash of Clans Math (on Mortar damage, and Gold mine) were fairly popular, so I have decided to write one more post! This shows that Math can be applied to almost everything, even games!
Vs 
Recently, players of COC will know that the Level 7 Giant has been released. For fans of the Giant (I am one of them), this is great news. The Giant is a cheap substitute for tanking vs the Golem, and can be used in many strategies for instance Giwipe (Golem, Wizard, Pekka), Garch (Giant archer), among many others.
In this post, I will use Math to concretely compare the Level 7 Giant and the Level 5 Golem. For fairness sake, we will compare 6 Giants with 1 Golem (since they take up 30 spaces). Sources are taken from http://clashofclans.wikia.com/.
Conclusion: Golem is around 30% better than Giants in term of HP.
Calculation: (8820-6600)/6600 x 100%=33.6%
For pure tanking, nothing beats a max level Golem.
Conclusion: Giants are 450% better than Golems in terms of DPS!
This can be quite significant, for example, when using Giants in Giwipe, often one does not even need to use wallbreakers, since the giants can break through the walls on their own. This frees up more spaces for wizards/other troops.
Spring Traps are the ultimate nemesis of Giants, since each Spring Trap can bounce 15 Housing spaces, or 3 Giants.
To avoid Spring Traps, place a few barbarians before sending out your giants. Hopefully the barbarians will activate (and waste) a few spring traps.
Conclusion: Golems are more resistant to Spring Traps.
Conclusion: Superficially, it seems good that Golems take 80% less damage than giants from Multi-mode Infernos. However, a bit of thinking reveals that the inferno in multi-mode will be attacking your other troops (for example wizards) instead, together with the 1 Golem. Hence, in other words, Golems also tank 80% less damage than giants from Multi-mode Inferno towers.
This calculation gets a little complicated. The inferno (Level 3) has 36 DPS initially, then 140 DPS after 2 seconds, then a whopping 1400 DPS after 5 seconds.
Conclusion: Giants survive 260% longer than Golems under Inferno (Single Target) Fire!
Other last points to note are that everytime a giant dies, there is a switch of targets, potentially attacking weak but crucial units like wizards or witches. This is a downside of giants.
Featured Product:
Fans Game Design Men T Shirts – O Collar T-Shirt
An interesting news to share:
Billionaire Ronnie Chan Chi-chung seems to have it all figured out. Were the Hang Lung Properties chairman to live his life over again, it would not be as a businessman — he’d be a mathematician or scientist instead.
Chan, who offered this little gem during a speech at the Hang Lung Mathematics Awards ceremony, said he may have more material wealth than famed mathematician Yau Shing-tung but much less intellectual wealth.
As co-founder of the awards, which were set to encourage secondary school students to pursue maths and sciences, Chan urged youngsters to go the extra mile and become mathematicians or scientists as they can contribute more to society than what a businessman can.
Read more at: http://www.thestandard.com.hk/news_detail.asp?we_cat=21&art_id=152297&sid=43527569&con_type=1&d_str=20141212&fc=8
Hopefully this can encourage students currently studying Maths, be it O Level Maths, JC H1 or H2 Maths, or even University Maths!
Featured Book:
The Shape of Inner Space: String Theory and the Geometry of the Universe’s Hidden Dimensions
New Scientist
“It is a testimony to [Yau’s] careful prose (and no doubt to the skills of co-author Steve Nadis) that this book so compellingly captures the essence of what pushes string theorists forward in the face of formidable obstacles. It gives us a rare glimpse into a world as alien as the moons of Jupiter, and just as fascinating…. Yau and Nadis have produced a strangely mesmerizing account of geometry’s role in the universe.”
Nature
“Physicists investigate one cosmos, but mathematicians can explore all possible worlds. So marvels Fields medalist Shing-Tung Yau…. Relating how he solved a major theoretical problem in string theory in the 1970s, Yau explains how the geometries of the vibrating multidimensional strings that may characterize the Universe have implications across physics.”
Tuition for toddlers – necessary or over the top?
Is tuition for toddlers necessary? Some food for thought.
As a tutor, I always try to value add and teach students something that is not taught in schools, for example tips or tricks in Math, or methods to check for careless mistakes.
Also certain key concepts may not be taught in school, even in elite schools. Many students, including students from the elite Nanyang Girls’ High School or ACS(I), have no idea initially that the discriminant
has something to do with the quadratic formula
. After my explanation though, they are enlightened, and finally can understand why
means that the quadratic has no real roots! 🙂
A good tutor can help find out what the student does not know, and teach to fill in the gaps of knowledge. Without a tutor, often a student does not know what he/she does not know! (until the exam comes)
A Maths exam is not an IQ test! It is a test of knowledge and preparation, whoever is more prepared (whether through tuition or self-studying) will get more marks.
Tuition, in the right format and spirit, is actually something good. Aristotle, the philosopher, was a tutor to Alexander the Great, one of the greatest kings that ever lived.
Featured Books:
Recommended Books for GEP
AFP News/AFP File Photo – File Photo of Children attending preschool/kindergarten
As his tutor flashed one flashcard after another to him, little Gabriel Tan glanced longingly towards the door.
A sharp look from his mother put him back in focus, and he obediently repeated the words on the flashcards.
“Can I go to the playground now, mummy?” asked the little boy hopefully.
“No, you have to do your memory exercises next,” said his tutor, pre-empting the mother’s reply, and the boy’s face fell again.
Gabriel is only three years old, but he has been receiving hour-long tuition lessons three times a week after his nursery classes so that he can “keep up” when his mother finally enrolls him at the coveted primary school she is an alumnus of – Nanyang Primary.
“I don’t think he is very smart, so to make up for that, he has extra tuition…
View original post 52 more words
The beauty I see in algebra: Margot Gerritsen at TEDxSTANFORD:
From equations to matrices… to Google search, MRI body scan, …
Watch this very inspirational video about learning.
When students encounter a difficult Math problem, there are two ways to approach it. Are you not smart enough to solve it … or have you just not solved it yet? The mindset the student adopts can make a huge difference in the learning effectiveness.
The key point is that there is great power of believing that you can improve. Adopting a growth mindset enables students to transcend their initial limitations and improve to a new level.
Featured Book:
Self-theories: Their Role in Motivation, Personality, and Development (Essays in Social Psychology)
Source: Science Daily
Date: December 15, 2014
Source: Emory University
Summary: Monstrous moonshine, a quirky pattern of the monster group in theoretical math, has a shadow — umbral moonshine. Mathematicians have now proved this insight, known as the Umbral Moonshine Conjecture, offering a formula with potential applications for everything from number theory to geometry to quantum physics.
“We’ve transformed the statement of the conjecture into something you could test, a finite calculation, and the conjecture proved to be true,” says Ken Ono, a mathematician at Emory University. “Umbral moonshine has created a lot of excitement in the world of math and physics.”
Co-authors of the proof include mathematicians John Duncan from Case Western University and Michael Griffin, an Emory graduate student.
“Sometimes a result is so stunningly beautiful that your mind does get blown a little,” Duncan says. Duncan co-wrote the statement for the Umbral Moonshine Conjecture with Miranda Cheng, a mathematician and physicist at the University of Amsterdam, and Jeff Harvey, a physicist at the University of Chicago.
Ono will present their work on January 11, 2015 at the Joint Mathematics Meetings in San Antonio, the largest mathematics meeting in the world. Ono is delivering one of the highlighted invited addresses.
Read more at: Science Daily
Featured Book:
Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics (Cambridge Monographs on Mathematical Physics)
“An excellent introduction to this area for anyone who is looking for an informal survey… written in a lively and readable style.”
R.E. Boucherds, University of California at Berkeley for the Bulletin of the AMS
“It is written in a breezy, informal style which eschews the familiar Lemma-Theorem-Remark style in favor of a more relaxed and continuous narrative which allows a wide range of material to be included. Gannon has written an attractive and fun introduction to what is an attractive and fun area of research.”
Geoffrey Mason, Mathematical Reviews
“Gannon wants to explain to us “what is really going on.” His book is like a conversation at the blackboard, with ideas being explained in informal terms, proofs being sketched, and unknowns being explored. Given the complexity and breadth of this material, this is exactly the right approach. The result is informal, inviting, and fascinating.”
Fernando Q. Gouvea, MAA Reviews
Are you curious what do Mathematicians eat for breakfast? 🙂
Featured Book:
The Math of Food (Integrating Math in the Real World Series)
How can math help you improve your diet?
Sharpens math skills from whole-number operations through basic algebra and geometry
Builds problem-solving and critical-thinking skills
Includes teacher notes, concepts and skills covered, relevant Internet sites, and more
In O Level E Maths, we learn how to bisect an angle using compass and straightedge (ruler). However, is it possible to trisect an angle?
It turns out it is impossible! This took 2000 years to prove, and requires the use of a very difficult theory called Galois Theory.
Check out this interesting video on trisecting angles:
It turns out it is possible to trisect angles using Origami though:
Featured Book:
Galois’ Theory Of Algebraic Equations
Galois’ Theory of Algebraic Equations gives a detailed account of the development of the theory of algebraic equations, from its origins in ancient times to its completion by Galois in the nineteenth century. The main emphasis is placed on equations of at least the third degree, i.e. on the developments during the period from the sixteenth to the nineteenth century. The appropriate parts of works by Cardano, Lagrange, Vandermonde, Gauss, Abel and Galois are reviewed and placed in their historical perspective, with the aim of conveying to the reader a sense of the way in which the theory of algebraic equations has evolved and has led to such basic mathematical notions as “group” and “field”. A brief discussion on the fundamental theorems of modern Galois theory is included. Complete proofs of the quoted results are provided, but the material has been organized in such a way that the most technical details can be skipped by readers who are interested primarily in a broad survey of the theory. This book will appeal to both undergraduate and graduate students in mathematics and the history of science, and also to teachers and mathematicians who wish to obtain a historical perspective of the field. The text has been designed to be self-contained, but some familiarity with basic mathematical structures and with some elementary notions of linear algebra is desirable for a good understanding of the technical discussions in the later chapters.
Source: Baidu Encyclopedia (Liu Hong)
刘洪(约公元130～210)，字元卓，东汉泰山郡蒙阴县（今山东省临沂市蒙阴县）人，东汉鲁王刘兴后裔，我国古代杰出的天文学家和数学家，珠算发明者，被后世尊为“算圣”。
Liu Hong (Year 130~210) was an ancient Chinese Mathematician who studied astronomy and mathematics. He was the inventor of the art of abacus arithmetic, and hence was given the title of “Sage of Arithmetic”.
Recently, I was fortunate to visit a statue of Liu Hong at his birthplace (modern day Linyi).
(Me beside statue of Liu Hong at Linyi People’s Square, Shandong)
Featured Book:
How To Use A Chinese Abacus: A step-by-step guide to addition, subtraction, multiplication, division, roots and more.
This book will teach you step-by-step how to perform addition, subtraction, division, multiplication, square roots and cube roots on a Chinese abacus. It also explains the ancient ‘extra bead’ method and the ‘suspended bead’ method. Great for both children and adults. Clearly explained with text and pictures throughout every stage of your calculation.
The discriminant of a quadratic polynomial (
Question: 
Solution: http://www.fen.bilkent.edu.tr/~cvmath/Problem/1411a.pdf
One of the above people who answered correctly, Toshihiro Shimizu, is an IMO Gold Medalist from Japan.
Featured Book:
Mathematical Olympiad Challenges
Highly Rated on Amazon!
Hundreds of beautiful, challenging, and instructive problems from algebra, geometry, trigonometry, combinatorics, and number theory
Historical insights and asides are presented to stimulate further inquiry
Emphasis is on creative solutions to open-ended problems
A friend from China gave us a bag of walnuts plucked from their home-grown walnut tree. I decide to count them by applying math:
A stack of walnuts piled in a pyramid, with base layer 6×6 walnuts, above layers 5×5, 4×4, 3×3, 2x 2, and finally top 1 (1×1).
How many walnuts are there in total ? (Answer: 91)
This is simple math but only taught in A-level (with proof by induction).
Hint: Watch free Khan Academy Math lecture to learn more ….
$latex displaystyle boxed {
sum_{1}^{n} k^2 =frac { n (n+1)(2n+1)} {6}
}$
This is a 400-year-old walnut tree: walnut is called “Wise fruit 聪明果”, it looks like human brain, also has proven nutritious benefits to brain.
Recently, I completed an excellent Coursera Course: An Introduction to Functional
Analysis, offered by École Centrale Paris.
Although challenging, it was a fun and interesting course, thanks to the effort by Professors John Cagnol and Anna Rozanova-Pierrat. Functional Analysis is a pretty difficult topic, and it was great to have two good professors explain it.
This course is actually more suitable for students who have some mathematical background, especially in mathematical analysis. New students may find it too hard to follow it from scratch. It is excellent as a review course for students who have taken functional analysis a few years ago but have forgotten some of it.
Featured Book:
Introductory Functional Analysis with Applications
Conrad Wolfram provoked the new idea of Computer-Based Math education:
Teach the ‘Why’ of Maths, leave the ‘How’ to the computer.
How: solve quadratic equation, simultaneous equations, differentiation, integration….
He mentioned Singapore is interested in this new approach of teaching Math ? The O & A Level students can now use scientific calculator in Exams.
Stephen Wolfram: Founder & CEO of Mathematica (UK)
Wolfram Alpha: Knowledge-base Computing using public data on the net and private information.
Mathematica: Math tool using Symbolic Functional Language (LISP)
New Kind of Science: Cell Automata
Physics: From Computing World to find new Physical World
Jack Ma of the Alabama.com gives this Arithmetic question to the audience, only 1% get the answer right !
Jack has cash $50, which he uses to buy:
Clothing : $20 (Balance $50-20 = $30)
Shoes: $15 (Balance = $30-15 = $15)
Candy: $9 (Balance = $15- 9 = $6)
Food: $6 (Balance = $6 – 6=$0)
Question:
Add up the Balances = $ 30+15+6 = $51
Where does the extra $1 come from ?
French Concours was influenced by Chinese Imperial Exams (科举ko-gu in ancient Chinese, today in Hokkien dialect) from 7AD till 1910. The French Jesuits working in China during the 16th -18th centuries were the culprits to bring them to France, and Napoleon copied it for the newly established Grande École “École Polytechnique” (a.k.a. X).
The “Bachelier” (or Baccalauréat from Latin-Arabic origin) is the Xiucai (秀才), only with this qualification can a person teach school kids.
With Licencié (ju-ren 举人) a qualification to teach higher education.
Concours was admired in France as meritocratic and fair social system for poor peasants’ children to climb up the upper social strata -” Just study hard to be the top Concours students”! As the old Chinese saying: “十年寒窗无人问, 一举成名天下知” (Unknown poor student in 10 years, overnight fame in whole China once top in Concours).Today, even in France, the top Concours student in École Polytechnique…
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Here are some tips to organize your studying time!
How to Better Organize After School Studying?
The key to achieving success in every academic discipline is practice, practice and more practice. However, this is where many students struggle the most: finding the time and the right methods to study after school. If you want to help your child study better, here is a simple to-do list:
Find the best method for your kid
Children have their own individual studying patterns, some of them prefer to study by themselves, some kids learn more when they participate in study groups, while other children, especially students with learning difficulties, might require a help of a professional tutor. You have to first recognize your child’s needs, try different methods and evaluate the results. Remember that the key to successful studying is regularity: even the best tutor will not be able to help your child, if they meet sporadically. Schedule a time for after school studying every week and check if your child adheres to it.
Take regular breaks
It is good to have a strict studying schedule, but breaks are also important. Regular breaks help boost child’s creativity and approach the task at hand with more enthusiasm.
Intensify as the exam approaches
If the after school studying is meant as a preparation for the upcoming exam, remember to start studying early and intensify as the day of the exam approaches. For example, start from doing simple Singapore Math exercises a few weeks before the day of the exam and progress to more complex issues, while increasing the workload. Never let your child study overnight before the exam, it will only make him feel more tired and stressed out on the day of the exam, plus this kind of behavior supports bad studying habits and false convictions that everything can be mastered within a few hours and there is no need to learn on a day to day basis.
Resource: http://eastwestmath.com
Here is a helpsheet that I created on the topic of Domain and Range of Inverse Functions, a topic no longer in O Levels but still in IP Math.
Featured Book:
韓非子是战国法家, 荀子的高徒, 秦始皇宰相李斯的同学。他说”白马非马”, 即白马不是马, 可以用集合論(Set Theory) 证明:
Let 马 = H = {w, b, r, y …}
w : 白马
b : 黑马
r ：红马
y：黄马
Let 白马 = W = {w}
To prove:
H = W
We must prove:
H ⊂ W and H ⊃ W
From definition we know:
$latex w in H supset W $
$latex H nsubseteq W $
$latex implies H neq W $
白马≠马
白马非马
[QED]
其他例子:
木魚非鱼
Definition: $latex text{Sequence } (a_n) $
has limit a
$latex boxed{forall varepsilon >0, exists N, forall n geq N text { such that } |(a_n) -a| < varepsilon}$
$latex Updownarrow $
$latex displaystyle boxed{ lim_{ntoinfty} (a_n) = a }$
What if we reverse the order of the definition like this:
∃ N such that ∀ε > 0, ∀n ≥ N,
$latex |(a_n) -a| < varepsilon$
This means:
$latex boxed {forall n geq N, (a_n) = a }$
Example:
$latex displaystyle (a_n) = frac{3n^{2} + 2n +1}{n^{2}-n-3}$
$latex displaystyletext{Prove: } (a_n) text { convergent? If so, what is the limit ?}$
Proof:
$latex displaystyle (a_n) = 3 + frac{5n +10}{n^{2}-n-3}$
$latex n to infty, (a_n) to 3$
Let’s prove it.
$latex text {Let } varepsilon >0$
$latex text{Choose N such that } forall n geq N, $
$latex displaystyle |(a_n) -3| = Bigr|frac{5n +10}{n^{2}-n-3}Bigr| < varepsilon$
$latex text{Simplify: }…
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Cédric Villani (Médaille Fields 2010) “Théorème Vivant”:
“La fameuse ligne directe, quand vous recevez un coup de fil du dieu de la mathématique, et qu’une voix résonne dans votre tête. C’est très rare, il faut l’avouer!”
“The famous direct line, when you receive a ‘telephone call’ from the God of the Mathematic, and that a voice resonates in your head. It is very rare, one has to admit.”
屈原 QuYuan (343–278 BCE) Symmetry:
http://en.wikipedia.org/wiki/Qu_Yuan
离騷《天问》
1. “九天之际, 安放安属,
隅隈多有, 谁知其数 ?”
=> 天 (Sky) 和 地 (Earth) must be 2 symmetric spheres.
If 地 (Earth) were flat, then there would be (隅隈) edges and angles at the 天 (Sky) & 地 (Earth) boundary (九天之际).
2. “东西南北, 其修孰多,
南北顺橢, 其衍几何。”
=> 南北顺橢 = The Earth is ellipse (橢), with north-south (南北) slightly flatten.
几何 = Geometry
3. How did QuYuan know this advanced astronomy & geometry in ~ 300 BCE?
屈原
Mozi
墨子 Mozi (468 BCE~ 376 BCE), 2000 years earlier than Newton
“墨子” : “力, 行之所以奋也。”
行: Moving
奋: Acceleration
力: Force is due to acceleration by the moving object.
F ∝ a
F = m.a
Difference Between Good & Bad Mathematicians
S.S. Chern (陈省身): “The former gives many concrete examples, the latter has only abstract theories.”
Triangle
1. Not always true!
Sum of 3 internal angles = Π =180 degrees
depends on which geometry:
Euclidean / non-Euclidean / Riemann
2. True always!
Sum of 3 exterior angles = 2 Π = 360 degrees
Kurt Gödel‘s Mathematical Proof of God’s Existence
Axiom 1: (Dichotomy) A property is positive if and only if its negation is negative.
Axiom 2: (Closure) A property is positive if it necessarily contains a positive property.
Theorem 1. A positive is logically consistent (i.e., possibly it has some instance).
Definition. Something is God-like if and only if it possesses all positive properties.
Axiom 3. Being God-like is a positive property.
Axiom 4. Being a positive property is (logical, hence) necessary.
Definition. A Property P is the essence of x if and and only if x has P and is necessarily minimal.
Theorem 2. If x is God-like, then being God-like is the essence of x.
Definition. NE(x): x necessarily exists if it has an essential property.
Axiom 5. Being NE is God-like.
Theorem 3. Necessarily there is…
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Pascal Wager:
1. We can choose to believe God exists, or we can choose not to so believe.
2. If we reject God and act accordingly, we risk everlasting agony and torment if He does exist (Type I error in Statistics lingo) but enjoy fleeting earthly delights if He doesn’t exist.
3. If we accept God and act accordingly, we risk little if He doesn’t exist (Type II error) but enjoy endless heavenly bliss if He does exist.
4. It’s in our self-interest to accept God’s existence.
5. Therefore God exists!
Mathematical Proof:
Pascal assumed
Probability of God exists = p
Probability God doesn’t exist = 1-p
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Mathtuition88 