Test if you have Mathematical Mind ?
Do not scroll to below to see the answer until you have tried …
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Answer:
Since 1 = 11,
of course 11 = 1
That is the mathematical meaning of ‘=’
Hua Luogeng (华罗庚) urged using the daily 10-20 mins intervals while waiting for buses, queues, idle times, make it at least 1 hour a day to read Math books which you carry along with you.
Hua advised on speedy self-learning Math :
1) Choose the Best book on the Topic written by the Master (say, Abstract Algebra), read completely and do the exercises.
2) Read other reference books. Read only those new topics not covered in 1).
If not much new things, return them to bookshelf. This way speed up reading many books in short time.
3) Then read International renown Math Journals.
Beware 90% are copy-cats or rubbish by University lecturers to meet their yearly publishing quota. Only < 10% are masterpieces.
4) Pick one topic to do your independent research.
5) Discuss with friends with better knowledge in the field.
This way you can be a Master in…
View original post 7 more words
华罗庚 《数论导引 》序言
Preface on “Introduction to Number Theory” by Hua Luogeng (1950).
“Math evolved from concrete to abstract, the former is the source of inspiration of the latter. One cannot just study the abstract definitions and theorems without going back to the source of concrete examples, which prove valuable applications in Physics and other sciences.”
“Mathematics, in essence, is about the study of Shapes and Numbers. From Shapes give rise to the Geometrical Intuition, from Numbers give the Relationship and Concepts ”
2003/7/13 台大访问笔记则要:
http://blog.sina.cn/dpool/blog/s/blog_c24597bf0101ctdp.html
突破瓶頸: “先上对车, 后补上票”
Holistic Approach to Attack Math :
新酒进旧瓶, 可以突破: 勤能补拙
10岁的启蒙书:
现代”科举”考场失意:
文学与数学相通:Intuition
Ref: 白居易写给元稹的《与元九书》
如何教好数学?
Shimura Modular Form:
好书推荐: 华罗庚的《数论导引》 , 华的剑桥老师Hardy…
解析数论 Analytic Number Theory:
选对导师和有兴趣的题目
On Riemann Hypothesis:
World Cup fans need math to figure out scenarios
Read more: http://www.dailymail.co.uk/wires/ap/article-2662058/World-Cup-fans-need-math-figure-scenarios.html#ixzz35Rw7VwhW
Follow us: @MailOnline on Twitter | DailyMail on Facebook
RIO DE JANEIRO (AP) — Every four years, the World Cup forces fans to remember their math lessons.
Working out what each team needs from its final match to finish in the top two of a group and advance to the knockout rounds takes some algebra knowledge and powers of prediction.
After Brazil and Mexico played to a scoreless draw on Tuesday, the calculation became clear: Both teams just need to draw in their next matches to advance with five points in Group A. Croatia, which beat Cameroon Wednesday, would get to six points by beating Mexico. So a draw with Cameroon would still get Brazil through with five points. If Mexico beats Croatia, Brazil would advance even if it loses. But if Mexico and Croatia draw, and Brazil loses — then it gets complicated with tiebreakers.
Read more: http://www.dailymail.co.uk/wires/ap/article-2662058/World-Cup-fans-need-math-figure-scenarios.html#ixzz35RwFh9c3
Follow us: @MailOnline on Twitter | DailyMail on Facebook
Featured book:
The Math of Sports: Integrating Math in the Real World (Integrating Math in the Real World Series)
Ever wondered if there is an alternative way to cutting cake so that it can stay fresh and softer in the refrigerator?
This is how!
Featured book:
The Calculus Lifesaver: All the Tools You Need to Excel at Calculus (Princeton Lifesaver Study Guide)
For many students, calculus can be the most mystifying and frustrating course they will ever take. The Calculus Lifesaver provides students with the essential tools they need not only to learn calculus, but to excel at it.
All of the material in this user-friendly study guide has been proven to get results. The book arose from Adrian Banner’s popular calculus review course at Princeton University, which he developed especially for students who are motivated to earn A’s but get only average grades on exams. The complete course will be available for free on the Web in a series of videotaped lectures. This study guide works as a supplement to any single-variable calculus course or textbook. Coupled with a selection of exercises, the book can also be used as a textbook in its own right. The style is informal, non-intimidating, and even entertaining, without sacrificing comprehensiveness. The author elaborates standard course material with scores of detailed examples that treat the reader to an “inner monologue”–the train of thought students should be following in order to solve the problem–providing the necessary reasoning as well as the solution. The book’s emphasis is on building problem-solving skills. Examples range from easy to difficult and illustrate the in-depth presentation of theory.
The Calculus Lifesaver combines ease of use and readability with the depth of content and mathematical rigor of the best calculus textbooks. It is an indispensable volume for any student seeking to master calculus.
教授會武術，流氓也擋不住 – 川大教授課堂利用數學公式劈磚:
This science professor uses kungfu to demonstrate 2 simple Physics mathematical formulas :
2 formulas:
Impulse : http://www.physicsclassroom.com/class/momentum/u4l1b.cfm
Lever : http://www.theclevver.com/theory.htm
Tolstoy’s “War and Peace”:
“Only by taking an infinitesimal small unit for observation (the differential of history, that is, the individual tendencies of men) and attaining to the art of integrating them (that is, finding the sum of these infinitesimals) can we hope to arrive at the laws of history.”
Having trouble with Angles questions?
Try out this practice booklet (questions compiled from Prelims) with Primary 6 PSLE Angle questions!
Does your child dread Math and avoid it like the plague?
Looking for a book that is fun yet suitable for young kids?
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We all know it’s wonderful to read bedtime stories to kids, but what about doing math? Many generations of Americans are uncomfortable with math and numbers, and too often we hear the phrase, “I’m just not good at math!” For decades, this attitude has trickled down from parents to their kids, and we now have a culture that finds math dry, intimidating, and just not cool.
Bedtime Math wants to change all that. Inside this book, families will find fun, mischief-making math problems to tackle—math that isn’t just kid-friendly, but actually kid-appealing. With over 100 math riddles on topics from jalapeños and submarines to roller coasters and flamingos, this book bursts with math that looks nothing like school. And with three different levels of challenge (wee ones, little kids, and big kids), there’s something for everyone. We can make numbers fun, and change the world, one Bedtime Math puzzle at a time.
Recently, I saw that many people searched the following terms on Google and landed on my website:
Let me try to answer the above questions:
Usually teachers will set the mid-year exams and the prelims at a (much) higher level than the actual O Levels. This is the current trend, which may result in many people failing the mid-year exam. The idea may be to motivate students to study harder and avoid being complacent with their results. Do not be demoralized by failing the exam! On the contrary, do reevaluate your study strategies, and strive to improve your knowledge and technique in mathematics.
The way to be good at additional mathematics is the same as the way to be good at piano, chess, and virtually any human endeavour. The key to improving is practice! Practice with understanding is the key. Would you imagine to be possible to improve in playing the piano without practicing the song? Improve in badminton without training? Definitely not! Similarly, improving in additional mathematics is not possible without practice. This is why the Ten Year Series is such a popular book: it is indeed the most useful book you can buy for studying Additional Mathematics.
Practicing with understanding helps with Application of Concepts, Increase Speed, Accuracy, which all helps in being good at additional mathematics.
In addition, during the practice sessions, try to practice checking for careless mistakes. It will help tremendously in improving your grades. Practicing with understanding means that we need to understand the method used, to the extent that if the teacher sets a slightly different question we are still able to do it. This is the secret to being good at additional maths. 🙂
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Dear Readers of Mathtuition88.com blog,
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Please try out the above! 🙂
Sincere thanks and have a nice week.
Source: http://blogs.scientificamerican.com/observations/2014/06/11/world-cup-prediction-mathematics-explained/
The World Cup is back, and everyone’s got a pick for the winner. Gamblers have been predicting the outcome of sporting contests since the first foot race across the savannah, but in recent years a unique type of statistical analysis has taken over the prediction business. Everyone from Goldman Sachs to Bloomberg to Nate Silver’s FiveThirtyEight has an online World Cup predictor that uses numbers, not hunches, to generate precise probabilities for match outcomes. Goldman Sachs, for instance, gives host nation Brazil a 48.5 percent chance of winning it all; FiveThirtyEight puts the odds at 45 percent while Bloomberg Sports has concluded there’s just a 19.9 percent chance of a triumph for the Seleção.
Where do these numbers come from? All statistical analysis must start with data, and these soccer prediction engines skim results from former matches. A fair bit of judgment is necessary here. Big international soccer tournaments only come around every so often, so the analysts have to choose how to weight team performance in lesser events such as international “friendlies,” where nothing of consequence is at stake. The modelers also have to decide how far back to pull data from—does Brazil’s proud soccer history matter much when its oldest player is 34?—and how to rate the performance of individual players during their time playing for club teams such as Manchester United or Real Madrid.
Wherever the data comes from, the modeler now has to incorporate it into a model. Frequently, the modeler translates the question of “who is going to win?” into the form “how many goals will team X score against team Y?” And for this, she relies [PDF] on a statistical tool called a bivariate Poisson regression.
Read more at: http://blogs.scientificamerican.com/observations/2014/06/11/world-cup-prediction-mathematics-explained/
Statistics and mathematics is useful after all! Only time will tell if the prediction is correct.
Featured book:
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This inexpensive paperback provides a brief, simple overview of statistics to help readers gain a better understanding of how statistics work and how to interpret them correctly. Each chapter describes a different statistical technique, ranging from basic concepts like central tendency and describing distributions to more advanced concepts such as t tests, regression, repeated measures ANOVA, and factor analysis. Each chapter begins with a short description of the statistic and when it should be used. This is followed by a more in-depth explanation of how the statistic works. Finally, each chapter ends with an example of the statistic in use, and a sample of how the results of analyses using the statistic might be written up for publication. A glossary of statistical terms and symbols is also included.
National University of Singapore (21st),
Nanyang Technological University (91-100th)
http://www.timeshighereducation.co.uk/world-university-rankings/2014/reputation-ranking
Asia Ranking:
NUS (2nd) after The University of Tokyo (1st).
http://www.timeshighereducation.co.uk/world-university-rankings/2013-14/regional-ranking/region/asia
Source: http://www.bbc.com/news/magazine-27835311
It’s puzzling but true that in any group of 23 people there is a 50% chance that two share a birthday. At the World Cup in Brazil there are 32 squads, each of 23 people… so do they demonstrate the truth of this mathematical axiom?
Imagine the scene at the Brazilian football team’s hotel. Hulk and Paulinho are relaxing after another stylish win. Talk turns from tactics to post World Cup plans.
“It’ll be one party after another,” says Hulk, confidently assuming Brazilian victory on home soil. “First the World Cup, then my birthday a couple of weeks later.”
“Your birthday’s in July?” replies Paulinho. “Me too – 25 July, when’s yours?
“No way, exactly the same day!” exclaims Hulk incredulously. “What are the chances of that?”
With 365 days in a regular year, most people’s intuitive answer would probably be: “Pretty small.”
But in this case our intuition is wrong – and the proof of that is known as the birthday paradox.
Also read our earlier post on Understanding the Birthday Paradox!
Featured book:
The Math of Sports: Integrating Math in the Real World (Integrating Math in the Real World Series)
DCI Matte Black Powder Coated Metal Mathematics Blackboard Pop Quiz Clock, 11-1/2″ Diameter
If you can read this clock, you are without a doubt a geek. Each hour is marked by a simple math problem. Solve it and solve the riddle of time. Matte black powder coated metal. Requires 1 AA battery (not included). 11-1/2″ Diameter.
Who knew there was such an interesting clock? 😛
Also interesting is this mug below. If you can read it, it shows you are well versed in Mathematics!
I Ate Some Pie and It Was Delicious Mug– Perfect For Any Math Nerd!!– Funny High Quality Coffee Mug!! (11oz, Black I Ate Sum Pi)
Suppose a party has six people. Consider any two of them. They might be meeting for the first time—in which case we will call them mutual strangers; or they might have met before—in which case we will call them mutual acquaintances. The theorem says:
In any party of six people either at least three of them are (pairwise) mutual strangers or at least three of them are (pairwise) mutual acquaintances.
Featured book:
This is the first appearance of
$latex boxed {pi = 3 }$
in any human’s text in history.
45 feet / 15 feet = 3
Just to share a video here:
It is a video of a girl who once did a math quiz and totally blanked out for the whole quiz. However, it turned out that her teacher did not actually ask for the quiz back, and gave her as much time as she wanted to complete the quiz. Under the relaxed circumstances, she completed the quiz and got a ‘C’. (big improvement from totally blank).
Then, she went to UCLA (very good school in US), and became a mathematics major, and wrote the book that is listed below the video!
Truly inspiring. For some kids, too much pressure may result in Math anxiety and totally blank out, while for other kids a little bit of pressure is needed to ensure that they do take studies seriously. Need to find the perfect balance for each child.
**The book the girl above wrote is: Math Doesn’t Suck: How to Survive Middle School Math Without Losing Your Mind or Breaking a Nail
Dear Readers,
Thanks for following our Maths Blog.
We are glad to introduce to you a Free Trial of Amazon Prime worth $99!
Random Math Fact:
Did you know:
Euler’s “lucky” numbers are positive integers n such that m2 − m + n is a prime number for m = 0, …, n − 1.
Leonhard Euler published the polynomial x2 − x + 41 which produces prime numbers for all integer values of x from 0 to 40. Obviously, when x is equal to 41, the value cannot be prime anymore since it is divisible by 41. Only 6 numbers have this property, namely 2, 3, 5, 11, 17 and 41.
Source: http://en.wikipedia.org/wiki/Lucky_numbers_of_Euler
The French Mathematician and Physicist Joseph Fourier proved
e is irrational,
Another French mathematician Charles Hermite went further: e belongs to another mathematical world:
e is transcendental.
Hermite’s German student Lindermann followed the same method, proved:
pi is also transcendental.
Anyone who has taken high school math is familiar with the constant 
Today we are going to prove that e is in fact irrational! We will go through Joseph Fourier‘s famous proof by contradiction. The maths background we need is to know the power series expansion: 
(Reference: http://en.wikipedia.org/wiki/Proof_that_e_is_irrational)
Suppose to the contrary that e is a rational number, so 
Using the power series formula mentioned above, we have
Multiply both sides by 
Now, we split the sum into two parts:
Rearranging,
Now, denote 
We now prove that 
This inequality is strict for every 
We have that 
Interesting? 🙂
Featured book:
Euler: The Master of Us All (Dolciani Mathematical Expositions, No 22)
Did you know the constant e is sometimes called Euler’s number?
Learn more about Euler in this wonderful book. Rated 4.9/5 stars, it is one of the highest rated books on the whole of Amazon.
Leonhard Euler was one of the most prolific mathematicians that have ever lived. This book examines the huge scope of mathematical areas explored and developed by Euler, which includes number theory, combinatorics, geometry, complex variables and many more. The information known to Euler over 300 years ago is discussed, and many of his advances are reconstructed. Readers will be left in no doubt about the brilliance and pervasive influence of Euler’s work.
Watch this video for another proof that e is irrational!
Featured book:
Match Wits With Mensa: The Complete Quiz Book
This is the #1 Top-Selling book recommended on my website! It includes Mathematical Logic Puzzles from Mensa. Highly recommended for gifted children. Parents, if your child is gifted and you want to stretch his or her learning potential, you may want to buy this book as it is the most complete quiz book on the market. It doesn’t matter whether you are in the Gifted Education Programme, as long as you have an interest in logic puzzles this book is for you.
Maths and Science is essentially about logical thinking, so logic puzzles will directly benefit studies in maths and science. Above all, logic puzzles are meant to be fun and a good and healthy pastime.
Puzzle fans have bought more than 650,000 copies of the Mensa Genius Quiz series—the only books that let readers “match wits with Mensa,” comparing how well they do against members of the famous high-IQ society. Here, in a giant omnibus edition, are four best-selling titles: The Mensa Genius Quiz Books 1 & 2, The Mensa Genius Quiz-A-Day Book, and The Mensa Genius ABC Book. Here are more than 800 fun mindbenders to exercise every part of your brain—word games, trivia, logic riddles, number challenges, visual puzzles—plus tips on how to improve your thinking skills. All the puzzles have been tested by members of American Mensa, Ltd., and include the percentage of Mensa testers who could solve each one, so that you can score yourself against some of the nation’s fittest mental athletes.
Dyscalculia specialist Ronit Bird talks about the difficulties some children have in developing number sense and learning basic arithmetic. She explains some of the common symptoms and indicators for dyscalculia and offers suggstions for how parents can help their children at home. For more information on Dyscalculia please visit http://www.ronitbird.com/
Featured book:
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-Karen Jones, Chartered Educational Psychologist, The Educational Guidance Service
With over 200 activities and 40 games this book is designed to support learners aged 6 to 14 years, who have difficulty with maths and numbers. Ronit Bird provides a clear explanation of dyscalculia, and presents the resources in a straightforward fashion.
This is the clearest and most interesting explanation of the Monty Hall Problem I have ever seen:
What is the Monty Hall Problem? It is basically a game show with 3 doors. Behind one of the doors is a car, while behind the other two doors are two goats. Most people will want to get the car of course.
The player gets a chance to choose one of the doors. Then, the host will open a door which contains a goat. Now, the player is allowed two choices: either stick to his original choice, or switch to the other unopened door. Which choice is better?
Watch the video to find out!
Featured book:
The Monty Hall Problem: The Remarkable Story of Math’s Most Contentious Brain Teaser
Mathematicians call it the Monty Hall Problem, and it is one of the most interesting mathematical brain teasers of recent times. Imagine that you face three doors, behind one of which is a prize. You choose one but do not open it. The host–call him Monty Hall–opens a different door, always choosing one he knows to be empty. Left with two doors, will you do better by sticking with your first choice, or by switching to the other remaining door? In this light-hearted yet ultimately serious book, Jason Rosenhouse explores the history of this fascinating puzzle. Using a minimum of mathematics (and none at all for much of the book), he shows how the problem has fascinated philosophers, psychologists, and many others, and examines the many variations that have appeared over the years. As Rosenhouse demonstrates, the Monty Hall Problem illuminates fundamental mathematical issues and has abiding philosophical implications. Perhaps most important, he writes, the problem opens a window on our cognitive difficulties in reasoning about uncertainty.
| PSLE tweaks will come but as part of broader changes to education system: Heng Straits Times SINGAPORE – Changes being made to the Primary School Leaving … be done in the light of the broader changes to Singapore’s education system, … |
| SMU to broaden learning for freshmen Straits Times Freshmen entering the Singapore Management University (SMU) in August next year will go through a revamped syllabus, in the university’s bid to … |
| MOE to focus on tertiary, secondary education before turning to PSLE Channel News Asia SINGAPORE: With the Character and Citizenship Education syllabus being rolled out in all schools, the Ministry of Education (MOE) will tilt its focus … |
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A very useful inequality in Mathematics is the AM-GM Inequality.
The arithmetic mean of numbers 
The geometric mean of numbers ![\boxed{\sqrt[n]{x_1\cdot x_2 \cdots x_n}}](https://s0.wp.com/latex.php?latex=%5Cboxed%7B%5Csqrt%5Bn%5D%7Bx_1%5Ccdot+x_2+%5Ccdots+x_n%7D%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
The AM-GM Inequality states that:
For any nonnegative numbers
![\displaystyle\boxed{\frac{x_1+x_2+\cdots+ x_n}{n}\geq\sqrt[n]{x_1\cdot x_2 \cdots x_n}}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cboxed%7B%5Cfrac%7Bx_1%2Bx_2%2B%5Ccdots%2B+x_n%7D%7Bn%7D%5Cgeq%5Csqrt%5Bn%5D%7Bx_1%5Ccdot+x_2+%5Ccdots+x_n%7D%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
Let say we have three (nonnegative) numbers a, b, c that add up to 30, i.e. 
Yes! Using the AM-GM inequality we have just learnt above, we know ![\displaystyle \frac{a+b+c}{3}\geq \sqrt[3]{abc}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7Ba%2Bb%2Bc%7D%7B3%7D%5Cgeq+%5Csqrt%5B3%5D%7Babc%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
Cubing both sides, we have, 
Also, the AM-GM inequality tells us that there is equality only when 
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Competition Math for Middle School
Written for the gifted math student, the new math coach, the teacher in search of problems and materials to challenge exceptional students, or anyone else interested in advanced mathematical problems. Competition Math contains over 700 examples and problems in the areas of Algebra, Counting, Probability, Number Theory, and Geometry. Examples and full solutions present clear concepts and provide helpful tips and tricks. “I wish I had a book like this when I started my competition career.” Four-Time National Champion MATHCOUNTS coach Jeff Boyd “This book is full of juicy questions and ideas that will enable the reader to excel in MATHCOUNTS and AMC competitions. I recommend it to any students who aspire to be great problem solvers.” Former AHSME Committee Chairman Harold Reiter
Introduction to Category Theory 1:
Course Overview:
Category Theory = Abstract Algebra of Functions
Lambda Calculus = Calculus of Functions
Lambda Calculus = Category
History:
$latex cap bigotimes$
Introduction to Category Theory (2) Monoids 么群
Introduction to Category Theory (3)Real lecture begins from here: Categories, Functors, Natural Transformation:
1. Category Definition:
1a) Examples of Categories:
Excellent example on “Natural Transformation“:
Ref: Classic Textbook
范畴论 Category Theory :
Watch the above video to prove that there has to be 12 Pentagons and 20 Hexagons on a Soccer Ball!
The video also teaches us about the beautiful Euler Formula, 
Featured Book:
Euler: The Master of Us All (Dolciani Mathematical Expositions, No 22)
An ideal book for enlivening undergraduate mathematics…he (Dunham) has Euler dazzling us with cleverness, page after page. — Choice
Mathematician William Dunham has written a superb book about the life and amazing achievements of one of the greatest mathematicians of all time. Unlike earlier writings about Euler, Professor Dunham gives crystal clear accounts of how Euler ingeniously proved his most significant results, and how later experts have stood on Euler’s broad shoulders. Such a book has long been overdue. It will not need to be done again for a long long time. — Martin Gardner
William Dunham has done it again! In “Euler: the Master of Us All”, he has produced a masterful portrait of one of the most fertile mathematicians of all time. With Dunham’s beautiful clarity and wit, we can follow with amazement Euler’s strokes of genius which laid the groundwork for most of the mathematics we have today. — Ron Graham, Chief Scientist, AT&T
William Dunham has written a superb book about the life and amazing achievements of one of the greatest mathematicians of all time. Unlike earlier writings about Euler, Dunham gives crystal clear accounts of how Euler ingeniously proved his most significant results, and how later experts have stood on Euler’s broad shoulders. Such a book has long been overdue. It will not need to be done again for a long, long time.Martin Gardner
Dunham has done it again! In “Euler: The Master of Us All,” he has produced a masterful portrait of one of the most fertile mathematicians of all time. With Dunham’s beautiful clarity and wit, we can follow with amazement Euler’s strokes of genius which laid the groundwork for most of the mathematics we have today. — Ronald Graham, Chief Scientist, AT&T
1. Matrix (M): stretch & twist space
2. Vector (v): a distance along some direction
3. M.v = v’ stretched & twisted by M
Some directions are special:-
a) v stretched but not twisted = Eigenvector;
b) The amount of stretch = constant = Eigenvalue (λ)
Let M the matrix, λ its eigenvalue,
v eigenvector.
By definition: M.v = λ.v
v = I.v (I identity matrix)
M.v = λI.v
(M – λI).v=0
As v is non-zero,
1. Determinant (M- λI) =0 => find λ
2. M.v = λ.v => find v
Note1: Why call Eigenvalue ?
From German: “Die dem Problem eigentuemlichen Werte”
= “The values belonging to this problem”
=> eigenWerte = EigenValue
Eigenvalue also called ‘characteristic values’ or ‘autovalues’.
Eigen in English = Characteristic (but already used for Field).
Note2: Schrödinger Quantum equation’s Eigenvalue = Maximum probability of electron presence at the orbit…
View original post 12 more words
Relationship-Mapping-Inverse (RMI)
(invented by Prof Xu Lizhi 徐利治 中国数学家 http://baike.baidu.com/view/6383.htm)
Find Z = a*b
By RMI Technique:
Let f Homomorphism: f(a*b) = f(a)+f(b)
Let f = log
log: R+ –> R
=> log (a*b) = log a + log b
1. Calculate log a (=X), log b (=Y)
2. X+Y = log (a*b)
3. Find Inverse log (a*b)
4. ANSWER: Z = a*b
Prove:
$latex sqrt{2}^{sqrt{2}^{sqrt{2}}}= 2$
1. Take f = log for Mapping:
$latex logsqrt{2}^{sqrt{2}^{sqrt{2}}} $
$latex = sqrt{2}logsqrt{2}^{sqrt{2}}$
$latex = sqrt{2}sqrt{2}logsqrt{2} $
$latex = 2logsqrt{2} $
$latex = log (sqrt{2})^2 $
$latex = log 2$
2. Inverse of log (bijective):
$latex log sqrt{2}^{sqrt{2}^{sqrt{2}}}= log 2$
$latex sqrt{2}^{sqrt{2}^{sqrt{2}}}= 2$
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New Geometry (新几何) invented by Zhang JingZhong (張景中) derived from 2 basic theorems:
1) Triangles internal angles =180º
2) Triangle Area = ½ base * height
=> derive all geometry
=> trigonometry
=> algebra
(These 3 maths are linked, unlike current syllabus taught separately)
The powerful Area (Δ) Proof Techniques:
1) Common Height:
Line AMB, P outside line
Δ PAM / Δ PBM = AM/BM
2) Common 1 Side (PQ):
Lines AB and PQ meet at M
Δ APQ /Δ BPQ = AM/BM
3) Common 1 Angle:
∠ABC=∠XYZ (or ∠ABC+∠XYZ = ∏ )
Δ ABC /Δ XYZ= AB.BC /XY.YZ
These 3 theorems can prove Butterfly and tough IMO problems.
Butterfly Theorem
In a circle draw a chord PQ with mid-point M. Through M draw 2 chords AB, CD. Join AD, BC cut PQ at X, Y resp. (Butterfly M)
1. Prove: M = mid-point of XY
http://gogeometry.com/GeometryButterfly.html
2. If circle changed to ellipse, still true?
Yes. Affine transformation from circle elongated to ellipse, like distorted image through funny mirror => still MX = MY
Butterfly theorem (Pho
Do not remember these:
$latex boxed {
cos 3A = 4cos^{3} A – 3cos A
}$
$latex boxed {displaystyle
int frac {dx}{sec x}
=
int
frac {1}{sec x}
frac {sec x + tan x}{sec x + tan x}dx
}&fg=aa0000
$
However, it helps, though, to remember:
“Nine Zulu Queens Rule China”
$latex boxed {
mathbb{N}subset mathbb{ Z }subset mathbb{ Q }subset mathbb{ R} subset mathbb{ C }
}&fg=00bb00&s=3
$
In the latest Quacquarelli Symonds (QS) World University Rankings by Subject (2014), NUS Math is ranked among the best mathematics departments in Asia.
Chinese students typically outperform U.S. students on international comparisons of mathematics competency. Paradoxically, Chinese teachers receive far less education than U.S. teachers–11 to 12 years of schooling versus 16 to 18 years of schooling.
Studies of U.S. teacher knowledge often document insufficient subject matter knowledge in mathematics. But, they give few examples of the knowledge teachers need to support teaching, particularly the kind of teaching demanded by recent reforms in mathematics education.
This book describes the nature and development of the “profound understanding of fundamental mathematics” that elementary teachers need to become accomplished mathematics teachers, and suggests why such teaching knowledge is much more common in China than the United States, despite the fact that Chinese teachers have less formal education than their U.S. counterparts.
Why radian is preferred in Math? Very good explanation here…
As Chinese we are good at memorizing poems since young (think of reciting 唐诗300首 – no sweat! ).
By chanting in Hokkien I remember complicated trigo formula:)
Cos 3A = 4Cos^3 A – 3.Cos A
($ 1.3 = $4.3 -$3 with $ =Cos in Hokkien sound).
Once I performed this ‘memory power’ in a lecture hall with me called up by the professor to solve a Physics problem. Halfway in the computation we need to open up the “Cos 3A”, the prof asked the whole class to help me, but I wrote the above down quickly on the white board to the awe of the class. Guess what, when the prof saw it, instead of praising me I got a scolding ! He said “Do not keep unnecessary things in your head”. The prof is a French, he did not know I have Hokkien language advantage… haha.
Anyway, joke aside, try to memorize minimum, go by first principle so that you will never ever forget iin whole life (yes, I remember them now after 40 years).
Differentiate (without memorizing formula)
y=b^x
and
y = log_b (x)
This math teacher is excellent in teaching the students to memorize minimum. His example is integrate secant. Most textbooks use a trick ie multiply (sec + tan) above and below, then by substitution. He goes by first principle, change sec = 1/cos, then try to use 2 common trigo sine and cosine, he multiplies cos above & below to make: sec = cos / 1-sin^2,… then integrate by part…
This is the breathtaking story of Daniel Tammet. A twenty-something with extraordinary mental abilities, Daniel is one of the world’s few savants. He can do calculations to 100 decimal places in his head, and learn a language in a week.
He also meets the world’s most famous savant, the man who inspired Dustin Hoffman’s character in the Oscar winning film ‘Rain Man’.
This documentary follows Daniel as he travels to America to meet the scientists who are convinced he may hold the key to unlocking similar abilities in everyone.
Featured books by Daniel Tammet:
Born On A Blue Day: Inside the Extraordinary Mind of an Autistic Savant
Bestselling author Daniel Tammet (Thinking in Numbers) is virtually unique among people who have severe autistic disorders in that he is capable of living a fully independent life and able to explain what is happening inside his head.
He sees numbers as shapes, colors, and textures, and he can perform extraordinary calculations in his head. He can learn to speak new languages fluently, from scratch, in a week. In 2004, he memorized and recited more than 22,000 digits of pi, setting a record. He has savant syndrome, an extremely rare condition that gives him the most unimaginable mental powers, much like those portrayed by Dustin Hoffman in the film Rain Man.
Thinking In Numbers: On Life, Love, Meaning, and Math
The irresistibly engaging book that “enlarges one’s wonder at Tammet’s mind and his all-embracing vision of the world as grounded in numbers.” –Oliver Sacks, MD
THINKING IN NUMBERS is the book that Daniel Tammet, mathematical savant and bestselling author, was born to write. In Tammet’s world, numbers are beautiful and mathematics illuminates our lives and minds. Using anecdotes, everyday examples, and ruminations on history, literature, and more, Tammet allows us to share his unique insights and delight in the way numbers, fractions, and equations underpin all our lives.
Very interesting Math jokes!
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Let me know (in the comment below) if there is a 4th solution – I believe there is a simpler and creative solution.
Think of the 4th solution, if any, for this
“Monkey & Coconuts” Problem.
It was created by Nobel Physicist Prof Paul Dirac, which he told another Chinese Nobel Physicist Prof Li ZhengDao (李政道)。
Pro Li wanted to test the Chinese young students in the first China Gifted Children University of 13 year-old kids, none of them could solve this problem (proved they are not so gifted after all for unknown problems :)
The first 2 solutions were solved by Prof Paul Richard Halmos, the 3rd solved by myself using the Singapore Modelling Math (a modified version of Arithmetics from traditional Math taught in 1970s Chinese Secondary 1 “中学数学” in Singapore).
1st Solution: Higher Math: Sequence
https://tomcircle.wordpress.com/2013/03/30/monkeys-coconuts-problem/
2nd Solution: Linear Algebra: Eigenvalue and Eigenvector
https://tomcircle.wordpress.com/2013/03/30/solution-2-monkeys-coconuts/
3rd Solution: Singapore Modelling Math for PSLE (Primary 6)
https://tomcircle.wordpress.com/2013/03/30/solution-3-best-monkeys-coconuts/
4th Solution:
Any ?
![\displaystyle 10\geq \sqrt[3]{abc}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+10%5Cgeq+%5Csqrt%5B3%5D%7Babc%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
