The Mystery of e^Pi-Pi (Very Mysterious Number)

If you have a calculator, check out the value of $e^\pi-\pi$. It is 19.99909998…

Why is it so close to the integer 20? Is it a coincidence (few things in Math are coincidence), or is it a sign of something deeper? e and Pi are two very fundamental numbers in Math, and the very fact that $e^\pi-\pi\approx 20$ may well mean something.

This was observed by a few mathematicians (Conway, Sloane, Plouffe, 1988) many years ago, but till this day there is no answer.

Do give it a thought!

Featured book:

Pi: A Biography of the World’s Most Mysterious Number

We all learned that the ratio of the circumference of a circle to its diameter is called pi and that the value of this algebraic symbol is roughly 3.14. What we weren’t told, though, is that behind this seemingly mundane fact is a world of mystery, which has fascinated mathematicians from ancient times to the present. Simply put, pi is weird. Mathematicians call it a “transcendental number” because its value cannot be calculated by any combination of addition, subtraction, multiplication, division, and square root extraction.

In this delightful layperson’s introduction to one of math’s most interesting phenomena, Drs. Posamentier and Lehmann review pi’s history from prebiblical times to the 21st century, the many amusing and mind-boggling ways of estimating pi over the centuries, quirky examples of obsessing about pi (including an attempt to legislate its exact value), and useful applications of pi in everyday life, including statistics.

This enlightening and stimulating approach to mathematics will entertain lay readers while improving their mathematical literacy.

Chinese maths teachers show way (Math News)

Latest Math News: Chinese maths teachers show way. Seems like Asia (especially China) is really going to be a world power in the next century, in terms of both education and economy. Students really need to improve their Chinese and Math skills in order to take advantage of this future trend!

CHINESE maths teachers have been sharing their tips on teaching the subject to teachers in Hucknall.

The teachers, from Shanghai, were part of a pioneering exchange programme to improve lessons in the subject.

On Monday, Hillside Primary School welcomed 29 teachers, the first group to come to England as part of the exchange between the Department for Education and the Shanghai Municipal Education Commission.

Two of the visitors will spend three weeks at the school.

In September, 71 top maths teachers from England travelled to Shanghai.

One was Tom Isherwood, a maths teacher at Hillside Primary and Nursery School.

He said: “The visit to Shanghai had a profound effect on the way I approach teaching mathematics.

Featured book:

Carson Dellosa Singapore Math Bundle Gr 1-2

Get a complete math curriculum in one with this specially bundled package of Singapore Math learning. Singapore Math is one of the leading math programs in the world! Each grade-appropriate set includes level A and B of the Singapore Math Practice series, 70 Must-Know Word Problems, Mental Math, and Step-by-Step Problem Solving. So, jump start your math learning today!

Are you a Careerist, Entrepreneur, Harmoniser, Idealist, Hunter, Internationalist, or Leader?

Are you a Careerist, Entrepreneur, Harmoniser, Idealist, Hunter, Internationalist, or Leader?

Find out by doing this Free Career Analysis by renowned survey company Universum:
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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!

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What is tested for A Maths (Additional Maths) Exam

Just to share an A Maths question that is likely to come out for 2014 A Maths Exam. It is the brand new topic just added this year: Sum and difference of cubes.

$\boxed{\alpha^3+\beta^3=(\alpha+\beta)(\alpha^2-\alpha\beta+\beta^2)}$

$\boxed{\alpha^3-\beta^3=(\alpha-\beta)(\alpha^2+\alpha\beta+\beta^2)}$

Attached below is a practice question on Alpha Cube+Beta Cube question that may be likely to come out this year! After all, if it is just added in the syllabus it is highly likely that they will test it.

Alpha Beta Cube Question

Good luck for the exam!

Maths Tuition

For Mathematics Tuition, contact Mr Wu at:

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Highly Recommended Math Books for University Self Study

Recently, a viewer of my website asked if I was able to suggest any undergraduate level university textbooks for self study that follows the university curriculum.

Self-study is challenging but not impossible. Choosing a good and appropriate book of the right level is of crucial importance. For instance, for beginners to Calculus, I wouldn’t recommend Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics) by Rudin. It is simply too difficult for beginners or even intermediate students. Any book by Bourbaki is also not suitable for beginners, for instance.

Update: I recently found a book that is a better alternative to Rudin: Mathematical Analysis, Second Edition by Apostol! Many online sources have very positive reviews on Apostol’s Analysis book. I have read it and found it much more readable than Rudin.

I would like to suggest the following books (mainly for Pure Mathematics). Ideally, the motivated student is able to self study and obtain the knowledge equivalent to a 4 Year course at a university.

The recommendations are divided into Year 1, Year 2, Year 3 and Year 4.

If you have any other recommendations, please feel free to comment below!

Year 1

Introduction to Pure Math and Proofs:

How to Prove It: A Structured Approach

Calculus:

Thomas’ Calculus (12th Edition)

Linear Algebra:

Linear Algebra and Its Applications, 4th Edition

Multivariable Calculus:

Thomas’ Calculus, Multivariable (13th Edition)

Year 2

Linear Algebra II (Second Year Course):

Linear Algebra, 4th Edition

Analysis I:

Introduction to Real Analysis

Abstract Algebra I:

A First Course in Abstract Algebra (3rd Edition)

This book will be an introduction to Group Theory.

Probability:

Introduction to Probability, 2nd Edition

Analysis II:

Calculus, 4th edition

(Note: Despite the title “Calculus”, this book is actually a rather rigorous book on Analysis, suitable as a second course textbook)

Complex Analysis I:

Complex Variables and Applications (Brown and Churchill)

Year 3

ODE (Ordinary Differential Equations):

Ordinary Differential Equations (Dover Books on Mathematics)

Algebra II:

Abstract Algebra, 3rd Edition

Algebra II will usually be a course on Rings, Modules.

(Note: You can use this book for learning Galois Theory too)

Differential Geometry:

Differential Geometry of Curves and Surfaces

Year 4

Number Theory:

An Introduction to the Theory of Numbers

Galois Theory:

Abstract Algebra, 3rd Edition

(Note: Same textbook as for Algebra II)

PDE (Partial Differential Equations):

A First Course in Partial Differential Equations: with Complex Variables and Transform Methods (Dover Books on Mathematics)

Functional Analysis:

Introductory Functional Analysis with Applications

Topology:

Topology (2nd Edition)

Measure and Integration:

The Elements of Integration and Lebesgue Measure

Congratulations for reaching the bottom of this long list!

All the best for your studies in Mathematics. 🙂

Check out this interesting Math Olympiad Integer Sequence Question! (September 2014 Math Problem of the Month)

While the books in this series are primarily designed for AMC competitors, they contain the most essential and indispensable concepts used throughout middle and high school mathematics. Some featured topics include key concepts such as equations, polynomials, exponential and logarithmic functions in Algebra, various synthetic and analytic methods used in Geometry, and important facts in Number Theory.

The topics are grouped in lessons focusing on fundamental concepts. Each lesson starts with a few solved examples followed by a problem set meant to illustrate the content presented. At the end, the solutions to the problems are discussed with many containing multiple methods of approach.

I recommend these books to not only contest participants, but also to young, aspiring mathletes in middle school who wish to consolidate their mathematical knowledge. I have personally used a few of the books in this collection to prepare some of my students for the AMC contests or to form a foundation for others.

By Dr. Titu Andreescu
US IMO Team Leader (1995 – 2002)
Director, MAA American Mathematics Competitions (1998 – 2003)
Director, Mathematical Olympiad Summer Program (1995 – 2002)
Coach of the US IMO Team (1993 – 2006)
Member of the IMO Advisory Board (2002 – 2006)
Chair of the USAMO Committee (1996 – 2004)

I love this book! I love the style, the selection of topics and the choice of problems to illustrate the ideas discussed. The topics are typical contest problem topics: divisors, absolute value, radical expressions, Veita’s Theorem, squares, divisibility, lots of geometry, and some trigonometry. And the problems are delicious.

Although the book is intended for high school students aiming to do well in national and state math contests like the American Mathematics Competitions, the problems are accessible to very strong middle school students.

The book is well-suited for the teacher-coach interested in sets of problems on a given topic. Each section begins with several substantial solved examples followed by a varied list of problems ranging from easily accessible to very challenging. Solutions are provided for all the problems. In many cases, several solutions are provided.

By Professor Harold Reiter
Chair of MATHCOUNTS Question Writing Committee.
Chair of SAT II Mathematics committee of the Educational Testing Service
Chair of the AMC 12 Committee (and AMC 10) 1993 to 2000.

Gmail iOS update adds iPhone 6 support and a math joke

Just to share, interesting Math Joke by Google Gmail!

You may need to brush up on your math if it seems odd to you that Google just updated the iOS Gmail app to version 3.1415926.

But Google is definitely not the type of company that turns down an opportunity to make a math joke, even when that joke is as simple as naming an app update after pi.

Besides that, the Gmail for iOS update has a single improvement: support for the iPhone 6 and iPhone 6 Plus.

Featured:

You may want to buy this Spigen Tough Armor Case if you are buying the new “flexible” iPhone 6 / 6 plus!

iPhone 6 Plus Case, Spigen® [KICK-STAND] iPhone 6 Plus (5.5) Case Protective [Tough Armor] [Gunmetal] Dual Layer EXTREME Protection Cover Heavy Duty Kick-Stand Feature Case for iPhone 6 Plus (5.5) (2014) – Gunmetal (SGP11053)

• TPU + Polycarbonate = Dual Protection
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• Drop Protection with AIR CUSHION Corners
• Built in Kick-Stand for Hands-Free Viewing
• Compatible with Apple iPhone 6 Plus (5.5″) Only – 2014

Fire HD Kids Edition Tablet: Educational Review

As an Amazon Affiliate, Mathtuition88 is proud to introduce the Fire HD Kids Edition:

 All-new Fire HD tablet—with 1 year of Amazon FreeTime Unlimited, Kid-Proof Case, and a 2-year worry-free guarantee—up to 95 in savings A real tablet, not a toy—A quad-core processor for great performance, a vivid HD display, front and rear-facing cameras, and Dolby Digital Audio Built for even the toughest kids—Enjoy the peace of mind with an unprecedented 2-year worry-free guarantee—if they break it, we’ll replace it for free. No questions asked Don’t worry about the bill—The Kids Edition includes a year of Amazon FreeTime Unlimited so kids get unlimited access to 5,000 books, movies, TV shows, educational apps, and games—at no additional cost. Best-in-class parental controls—Create individual profiles for each of your children. Personalize screen time limits, educational goals, and age-appropriate content Kid-Proof Case—Durable, lightweight case to protect against drops and bumps caused by kids at play. This is a potential good alternative to the Ipad. Ipad is more for games, while the Fire HD Kids Edition Tablet is more educational, with a hand-curated subscription of over 5,000 kid-friendly books, movies, TV shows, educational apps, and games. It will be out soon this October 2014! Pre-order now by clicking this link: Click here to Pre-order. AlgTop1: One-dimensional objects This is a continuation of the series of Algebraic Topology videos. Previous post was AlgTop 0. Professor Wildberger is an interesting speaker. He holds some unorthodox views, for instance he doesn’t believe in “real numbers” or “infinite sets”. Nevertheless, his videos are excellent and educational. Highly recommended to watch! The basic topological objects, the line and the circle are viewed in a new light. This is the full first lecture of this beginner’s course in Algebraic Topology, given by N J Wildberger at UNSW. Here we begin to introduce basic one dimensional objects, namely the line and the circle. However each can appear in rather a remarkable variety of different ways. Author: NJ Wildberger This revolutionary book establishes new foundations for trigonometry and Euclidean geometry. It shows how to replace transcendental trig functions with high school arithmetic and algebra to dramatically simplify the subject, increase accuracy in practical problems, and allow metrical geometry to be systematically developed over a general field. This new theory brings together geometry, algebra and number theory and sets out new directions for algebraic geometry, combinatorics, special functions and computer graphics. The treatment is careful and precise, with over one hundred theorems and 170 diagrams, and is meant for a mathematically mature audience. Gifted high school students will find most of the material accessible, although a few chapters require calculus. Applications include surveying and engineering problems, Platonic solids, spherical and cylindrical coordinate systems, and selected physics problems, such as projectile motion and Snell’s law. Examples over finite fields are also included. A1 marks for A Maths / E Maths How many marks to get A1 for A Maths / E Maths for O Levels? The official answer is not released by Cambridge / MOE, but it is definitely not 75 as the papers are subject to the bell curve (using normal distribution). According to popular forum Hardwarezone: Hello! Was wondering how much marks do I have to get in order to get A1… Many have been saying you need to get 90%. Is it really 90% for both Maths? Cambridge has never revealed its score. Was wondering what you hve heard from your teachers or from other reliable sources. Thank you! Appreciate it very much. Ans by a forummer: 90 marks for emaths. 80+ for amaths Now, getting 90 marks for E Maths is no mean feat. But it is possible with practice and the right coaching! Getting 80+ for A Maths is no joke either. If you have taken A Maths before you know how difficult it is, and usually for any test in school more than half the class will fail. We must approach the O Levels with the right positive mindset: 1) It is always possible to improve. No matter how weak the student is in Maths, it is always possible to improve. The key thing is to: 2) Start revision and practice early. The earlier you start revision and practicing Maths, the more chance of improvement you have! 3) Learn to love math and appreciate its beauty, or at least try your best not to hate math. Since Math is pretty much compulsory till JC, why not try to like it? Adopt a positive mindset and you will be able to study for longer hours for Maths, which will translate to a better score in the end. If you are looking to brush up on your A Maths / E Maths skills and learn some tips on scoring during exams, join our weekly group tuition at Bishan! https://mathtuition88.com/group-tuition/ How to prove square root of 2 is irrational (Constructive Approach) In our previous post, we discussed how to prove that the square root of 2 is irrational, using a proof by contradiction. There is a less well-known proof that is a direct constructive approach to proving that the square root of 2 is irrational! We consider an arbitrary rational number $\displaystyle\frac{a}{b}$, and show that the difference between $\sqrt{2}$ and $\displaystyle\frac{a}{b}$ cannot be zero. Hence, the square root of 2 cannot be rational. Firstly, we have: $\displaystyle|\sqrt{2}-\frac{a}{b}|=|\frac{\sqrt{2}b-a}{b}|$ $\displaystyle =|\frac{\sqrt{2}b-a}{b}|\times \frac{\sqrt{2}b+a}{\sqrt{2}b+a}$ (Rationalizing the numerator) $\displaystyle =\frac{|2b^2-a^2|}{\sqrt{2}b^2+ab}$ $\displaystyle =\boxed{\frac{|2b^2-a^2|}{b(\sqrt{2}b+a)}}$ Now, we analyse the numerator. We can write $a=2^\alpha\cdot x$, $b=2^\beta\cdot y$, where $x,y$ are odd. Then $2b^2=2^{2\beta +1}\cdot y^2$, $a^2=2^{2\alpha}\cdot x^2$. Since the largest power of two dividing $2b^2$ is an odd power, whilst for $a^2$ the largest power of two dividing it is an even power, $2b^2$ and $a^2$ cannot be the same number. Hence we have $|2b^2-a^2|\geq 1$. Now, we analyse the denominator. Firstly, we can consider just the rationals $\displaystyle \frac{a}{b}\leq 3-\sqrt{2}\approx 1.59$. Because if $\frac{a}{b}>1.59$, it is clear that $\frac{a}{b}$ is not going to be $\sqrt{2}\approx 1.41$. Rearranging, we have: $\displaystyle \sqrt{2}+\frac{a}{b}\leq 3$. Multiplying throughout by $b$, $\sqrt{2}b+a\leq 3b$. Going back to the original equation (boxed), we can conclude that: $\displaystyle\boxed{\frac{|2b^2-a^2|}{b(\sqrt{2}b+a)}}\geq\frac{1}{b(3b)}=\frac{1}{3b^2}>0$. We have shown constructively that $\sqrt{2}$ is not a rational number! Reference: http://en.wikipedia.org/wiki/Square_root_of_2#Constructive_proof Every math student needs a tool belt of problem solving strategies to call upon when solving word problems. In addition to many traditional strategies, this book includes new techniques such as Think 1, the 2-10 method, and others developed by math educator Ed Zaccaro. Each unit contains problems at five levels of difficulty to meet the needs of not only the average math student, but also the highly gifted. Answer key and detailed solutions are included. Grades 4-12 How to choose a toilet using Mathematics We have seen how to cut a cake using the mathematical way, but did you know Mathematics can also be used when choosing toilets? Here’s how: Build a foundation and focus on what matters most for math readiness with Common Core Math 4 Today: Daily Skill Practice for fourth grade. This 96-page comprehensive supplement contains standards-aligned reproducible activities designed to focus on critical math skills and concepts that meet the Common Core State Standards. Each page includes 16 problems to be completed during a four-day period. The exercises are arranged in a continuous spiral so that concepts are repeated weekly. An assessment for the fifth day is provided for evaluating students’ understanding of the math concepts practiced throughout the week. Also included are a Common Core State Standards alignment matrix and an answer key. World Cup fans need math to figure out scenarios World Cup fans need math to figure out scenarios 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. The Scientific (Mathematical) Way to Cut a Cake 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! 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. • Serves as a companion to any single-variable calculus textbook • Informal, entertaining, and not intimidating • Informative videos that follow the book–a full forty-eight hours of Banner’s Princeton calculus-review course–is available at Adrian Banner lectures • More than 475 examples (ranging from easy to hard) provide step-by-step reasoning • Theorems and methods justified and connections made to actual practice • Difficult topics such as improper integrals and infinite series covered in detail • Tried and tested by students taking freshman calculus Primary 6 PSLE Angles Questions and Solutions Having trouble with Angles questions? Try out this practice booklet (questions compiled from Prelims) with Primary 6 PSLE Angle questions! P6 Angles Questions Solutions Does your child dread Math and avoid it like the plague? Looking for a book that is fun yet suitable for young kids? Featured book: Our mission: to make math a fun part of kids’ everyday lives. 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. How to be good in Additional Mathematics Recently, I saw that many people searched the following terms on Google and landed on my website: 1. Why is the mid-year exams difficult and many people fail it? 2. How to be good in additional mathematics. Let me try to answer the above questions: Why is the mid-year exams difficult and many people fail it? 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. How to be good in additional 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. 🙂 From a well-known actress, math genius and popular contestant on “Dancing With The Stars”—a groundbreaking guide to mathematics for middle school girls, their parents, and educators World Cup Math Prediction: Brazil 48.5% Chance of Winning!!! 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: 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. World Cup Math World Cup Math: Birthday Paradox 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! Can you read this Math Clock? 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) Theorem on friends and strangers 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: Very Inspirational Math Video Just to share a video here: Very Inspirational Math Video 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 Free Trial: Amazon Prime Dear Readers, Thanks for following our Maths Blog. We are glad to introduce to you a Free Trial of Amazon Prime worth99!

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.

Why is e irrational?

Anyone who has taken high school math is familiar with the constant $\boxed{e=2.718281828\cdots}$.

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: $\displaystyle \boxed{e=\sum_{n=0}^{\infty}\frac{1}{n!}}$. The proof is slightly tricky so stay focussed!

Suppose to the contrary that e is a rational number, so $\displaystyle e=\frac{a}{b}$.

Using the power series formula mentioned above, we have $\displaystyle\sum_{n=0}^\infty \frac{1}{n!}=\frac{a}{b}$

Multiply both sides by $b!$, $\displaystyle \sum_{n=0}^{\infty}\frac{b!}{n!}=\frac{ab!}{b}=a(b-1)!$

Now, we split the sum into two parts:

$\displaystyle \sum_{n=0}^b \frac{b!}{n!}+\sum_{n=b+1}^\infty \frac{b!}{n!}=a(b-1)!$

Rearranging,

$\displaystyle \sum_{n=b+1}^\infty \frac{b!}{n!}=a(b-1)!-\sum_{n=0}^b \frac{b!}{n!}$

Now, denote $\displaystyle x=\sum_{n=b+1}^\infty \frac{b!}{n!}>0$. $x$ is an integer since both $\displaystyle a(b-1)!$ and $\displaystyle\sum_{n=0}^b \frac{b!}{n!}$ are integers and their difference (which is x) will be an integer.

We now prove that $x<1$. For all terms with $n\geq b+1$ we have the upper estimate

$\displaystyle\begin{array}{rcl} \frac{b!}{n!}&=&\frac{1\times 2\times \cdots \times b}{1\times 2\times \cdots \times b \times (b+1) \times \cdots \times n}\\ &=&\frac{1}{(b+1)(b+2)\cdots (b+(n-b))}\\ &\leq& \frac{1}{(b+1)^{n-b}} \end{array}$

This inequality is strict for every $n\geq b+2$. Changing the index of summation to $k=n-b$ and using the formula for the infinite geometric progression $S_\infty = \frac{a}{1-r}$, we obtain:

$\displaystyle x=\sum_{n=b+1}^\infty \frac{b!}{n!} < \sum_{n=b+1}^\infty \frac{1}{(b+1)^{n-b}}=\sum_{k=1}^\infty \frac{1}{(b+1)^k}=\frac{\frac{1}{b+1}}{1-\frac{1}{b+1}}=\frac{1}{b}\leq 1$

We have that $x$ is an integer but $0. This is a contradiction (since there is no integer strictly between 0 and 1), and so $e$ must be irrational. (QED)

Interesting? 🙂

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!

Dyscalculia — A Parent’s Guide

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/

‘The new dyscalculia toolkit has a great introduction that is broken down into manageable chunks, brilliant explanations and interesting reading. The new tables explain what each game entails at the start of the book, making planning and using the toolkit much easier and effective especially if short on time! Very enjoyable to read, and highly recommended’
-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.

The Monty Hall Problem

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!

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.

Singapore Education News

 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 …

Featured Product:

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• Written by experienced math teachers and a United States Chess Champion for K-8 supplemental math learning and K-8 math practice
• Made in USA: Includes tournament classic chess set, interactive coloring math comic book and colored pencils
• Suitable for complete beginners to chess and children at all levels of math ability, from underachievers to gifted students
• With contribution from the Harry Potter chess consultant, American International Master Jeremy Silman, creator of the Harry Potter chess scene in Harry Potter and the Sorcerer’s Stone (Warner Bros. Pictures, 2001)

AM-GM inequality

A very useful inequality in Mathematics is the AM-GM Inequality.

The arithmetic mean of numbers $x_1, x_2, \cdots, x_n$ is $\displaystyle \boxed{\frac{x_1+ x_2+\cdots+x_n}{n}}$.

The geometric mean of numbers $x_1, x_2, \cdots, x_n$ is $\boxed{\sqrt[n]{x_1\cdot x_2 \cdots x_n}}$.

The AM-GM Inequality states that:

For any nonnegative numbers $x_1, x_2, \cdots, x_n$,

$\displaystyle\boxed{\frac{x_1+x_2+\cdots+ x_n}{n}\geq\sqrt[n]{x_1\cdot x_2 \cdots x_n}}$, and equality holds if and only if $x_1=x_2=\cdots=x_n$.

How to Apply?

Let say we have three (nonnegative) numbers a, b, c that add up to 30, i.e. $a+b+c=30$. Can we know what is the largest possible product $abc$?

Yes! Using the AM-GM inequality we have just learnt above, we know $\displaystyle \frac{a+b+c}{3}\geq \sqrt[3]{abc}$.

$\displaystyle 10\geq \sqrt[3]{abc}$

Cubing both sides, we have, $\displaystyle abc\leq 10^3=1000$.

Also, the AM-GM inequality tells us that there is equality only when $a=b=c$, i.e. $a=b=c=10$. Hence, the largest possible product $abc$ is 1000.

Featured book from Amazon:

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

Math Handheld Computer Game

Featured Item:

Educational Insights Math Whiz

Is your child disinterested in Math? Looking for some fun and educational Math games?

Math Whiz plays like a video game and teaches like electronic flash cards. This portable ELA quizzes kids on addition, subtraction, multiplication and division, AND works as a full-function calculator at the press of a button. Problems are displayed on the LCD screen. Features eight skill levels, as well as lights and sounds for instant feedback. Two AAA batteries required (not included).

The Boy With The Incredible Brain – Autism Math Documentary

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.

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.

Coordinate Geometry Notes

Check out these Formulas for:

• Distance between 2 points
• Midpoint between 2 points
• and more

Featured book: The Ultimate book of Formulas (2000+ Formulas!)

Schaum’s Outline of Mathematical Handbook of Formulas and Tables, 4th Edition: 2,400 Formulas + Tables (Schaum’s Outline Series)

This Schaum’s Outline gives you

• More than 2,400 formulas and tables
• Covers elementary to advanced math topics
• Arranged by topics for easy reference

Chinese Lucky Numbers – Numberphile

8 and 6 are lucky but 4 is unlucky… if you’re Chinese!

Featuring Xiaohui Yuan from the University of Nottingham.

Website: http://www.numberphile.com/

Brady John Haran is an Australian independent film-maker and video journalist who is known for his educational videos and documentary films produced for BBC News and for his YouTube channels. (http://en.wikipedia.org/wiki/Brady_Haran)

Highly recommended to subscribe to Numberphile on Youtube for fun and interesting Math videos!

Featured book:

Number: The Language of Science

Number is an eloquent, accessible tour de force that reveals how the concept of number evolved from prehistoric times through the twentieth century. Tobias Dantzig shows that the development of math—from the invention of counting to the discovery of infinity—is a profoundly human story that progressed by “trying and erring, by groping and stumbling.” He shows how commerce, war, and religion led to advances in math, and he recounts the stories of individuals whose breakthroughs expanded the concept of number and created the mathematics that we know today.

– Rated 4.5/5 on Amazon

News: Singapore Education Ranked Third in World

 Singapore takes third spot in global education rankings Straits Times Teacher Anthony Tan conducting an English lesson with a class of Primary 6 pupils at Woodlands Primary School. Singapore’s education system has … Singapore offers Saudi Arabia help in education Arab News PROPOSAL: Singapore Senior Minister of State Lee Yi Shyan with Mazen Batterjee, vice chairman of the JCCI, on Wednesday. (AN photo by Irfan … In Singapore, Training Teachers for the ‘Classroom of the Future’ Education Week News Welcome to the Classroom of the Future—a mock-up housed by Singapore’s National Institute of Education (NIE) to demonstrate what learning might … Singapore Polytechnic Assists CDIO Implementation At Malaysia’s Polytechnic Bernama PUTRAJAYA, May 6 (Bernama) — Singapore Polytechnic is assisting Malaysia on the implemention of innovative engineering education framework … Lift education standards: Linfox boss The Australian “Most of our graduates are now coming out of Thailand, Vietnam, Singapore and China because they are just so well educated,” he said. “I can get … In search of education The News International Unless we start investing massively in education, science, technology and innovation, as was done by Singapore, Korea, Malaysia, China and others, … Sultanate, Singapore and the Indian Ocean Oman Daily Observer These are thoughtful words from your education minister (Heng Swee Keat), … a pragmatism which incidentally I believe we share with Singapore. Direct School Admission not meant to lower academic standards TODAYonline In Singapore, there is no compromising a good education. Having a talent does not give a student the licence not to pursue academic excellence. NAFA inspires The Hindu The safe and comfortable cosmopolitan environment Nanyang Academy of Fine Arts, Singapore makes it the perfect destination for education abroad. Japan’s Education Minister visits SMU Perspectives@SMU Singapore Management University (SMU) received a special guest on its campus on 3 May 2014 – Japan’s Minister of Education, Culture, Sports, …

Monster Group

Check out this Youtube video on the Monster Group (related to Group Theory, a branch in Mathematics)

In the mathematical field of group theory, the monster group M or F1 (also known as the FischerGriess monster, or the Friendly Giant) is a group of finite order. (See Wikipedia: http://en.wikipedia.org/wiki/Monster_group)

Featured book:

The Symmetries of Things

This book is written by John Conway, one of the mathematicians who worked on the Monster Group. Rated highly on Amazon.

Start with a single shape. Repeat it in some way—translation, reflection over a line, rotation around a point—and you have created symmetry.

Symmetry is a fundamental phenomenon in art, science, and nature that has been captured, described, and analyzed using mathematical concepts for a long time. Inspired by the geometric intuition of Bill Thurston and empowered by his own analytical skills, John Conway, with his coauthors, has developed a comprehensive mathematical theory of symmetry that allows the description and classification of symmetries in numerous geometric environments.

This richly and compellingly illustrated book addresses the phenomenological, analytical, and mathematical aspects of symmetry on three levels that build on one another and will speak to interested lay people, artists, working mathematicians, and researchers.

How to prove square root of 2 is irrational?

A rational number is a number that can be expressed in a fraction with integers as numerators and denominators.

Some examples of rational numbers are 1/3, 0, -1/2, etc. Now, we know that $\sqrt{2}\approx 1.41421\cdots$.

Is the square root of 2 rational? Or is it irrational (the opposite of rational)? How do we prove it? It turns out we can prove that the square root of two is irrational using a technique called proof by contradiction. (One of the earlier posts on this blog also used proof by contradiction to show that there are infinitely many prime numbers.)

First, we suppose that $\displaystyle\sqrt{2}=\frac{p}{q}$, where $\displaystyle\frac{p}{q}$ is a fraction in its lowest terms.

Next, we square both sides to get $\displaystyle 2=\frac{p^2}{q^2}$.

Hence, $2q^2=p^2$. We can conclude that $p^2$ is even since it is a multiple of 2. Thus, $p$ itself is also even. (the square of an odd number is odd).

Thus, we can write $p=2k$ for some integer k. Substituting this back into $2q^2=p^2$, we get $2q^2=4k^2$, which can be simplified to $q^2=2k^2$.

Hence, $q^2$ is also even, and hence $q$ is also even!

But if both $p$ and $q$ are even, then $\displaystyle\frac{p}{q}$ is not in the lowest terms! (we could divide them by two). This contradicts our initial hypothesis!

Thus, the only possible conclusion is that the square root of two is not a rational number to begin with!

Featured book:

Math Jokes 4 Mathy Folks

Who says math can’t be funny? In Math Jokes 4 Mathy Folks, Patrick Vennebush dispels the myth of the humorless mathematician. His quick wit comes through in this incredible compilation of jokes and stories. Intended for all math types, Math Jokes 4 Mathy Folks provides a comprehensive collection of math humor, containing over 400 jokes.

– Highly rated on Amazon.com

What to do if fail Mid Year Exams?

When the Mid Year Exams are over, students will receive their results nervously. What to do if one fails the Mid Year Exams?

In many schools, it is common to have a significant portion of the school actually failing the Mid Term exam. “40 per cent of his school cohort failed Social Studies and 30 per cent English.” in the school mentioned in the above article.

“Such significant failure rates have become common in schools here when mid-year or preliminary exams roll around, especially for those with a big national exam – PSLE, O or A levels – at the year-end.”

Here are 5 tips on what are the best actions to take for one who fails the Mid Year Exams, especially for Mathematics:

1. Do not be discouraged! Try to maintain a positive attitude on Math. There is still time before the final exams. With proper time management, you will be able to set aside time for revision, which will definitely help.
2. Analyse what went wrong. Are you studying Math the correct way? (i.e. practising with understanding) Are you studying Math just by reading the textbook? (not effective for studying Math as Math needs practice.) Is time management an issue? Or is the main issue careless mistakes?
3. Work out a new study strategy and stick to it religiously. For better results, you need to change your study habits for the better. This may include better time management, or seeking help from Math tutors.
4. O Level Exams are not about intelligence, it is more about good study techniques. The content for O Levels can definitely be mastered by any student given the right amount of time and effort. The key is to put in time and effort to the studies. Even an average student is capable of scoring an A1 in O Levels if he or she works hard. Whereas, a very intelligent but lazy student may not do well for the exams.
5. It is possible to improve tremendously for Maths if you study enough and using the right method. This is a truth that many people can attest to. I have seen students going from fail to A1. Improving one or two grades is also very common.

There are usually two types of students, the ones who are more playful and laidback, and the very perfectionistic student but is prone to stress. For the more playful students, the tough Mid Year exams are actually meant as a wake up call to start studying before it is too late. “‘Papers must be a bit challenging so that they can shake one out of complacency and make one study harder,’ said Mr Lak Pati Singh, 56, principal of St Patrick’s School.”

For students who are too stressed up and already trying their best, the way to improve may be to study more efficiently using the right methods (especially for Maths, the right way to study is practice with understanding). A healthy lifestyle balance may also be very helpful. Again, seeking help from Math tutors may be a choice to be considered, which can alleviate stress from not understanding the subject material.

Studies and Studying: How do top students study?

Check out this post by MIT almost perfect-scorer, on how to study. His secret is to study the material in advance, before the lessons even start! This is really a useful strategy, if implemented correctly. Imagine being in Primary 3 and already knowing the Primary 4 syllabus! Primary 3 Math will be a breeze then. This is one of the reasons why China students are so good at Math – they have already studied it back in China, where the Math syllabus is more advanced!

Do try out this strategy if you are really motivated to improve in your studies. The prime time to do this is during the June and December holidays – take some time to read ahead what is going to be learnt during the next semester.

This is an excerpt of the thread:

I graduated from MIT with a GPA of 4.8 (out of 5.0) in mathematics. I had two non-As, both of which were non-math classes.

That doesn’t imply that I have good study methods, but anyway, here’s how I studied at MIT. My main study method as an undergraduate, for math classes, was knowing a sizable chunk of the material in advance.

This isn’t a method that will work for everybody. I did a lot of mathematics outside of the classroom both in high school and at MIT, and I often saw a substantial portion of the material in a given class before I took it. I can’t emphasize enough how much easier this makes a class, and not just for the reasons you might expect: one of the most valuable things you get out of knowing a lot of the material already is just not being intimidated by it. (And you can get this benefit even if you’ve only seen some of the material before and possibly forgotten some of it too.) You’re much more relaxed, and that makes it easier to process the part of the material that you don’t know.

What that translates to in terms of practical advice is this:

• cultivate a sense of curiosity,
• don’t restrict your learning to the classroom,
• only take classes that actually seem really interesting to you, and
• try to learn something related to those classes the semester before.

None of this is advice for studying for a class you’re taking now, but it’s advice for reducing the extent to which you will need to study for classes you’ll take in the future.

– Qiaochu Yuan

Education News Update

 The Straits Times holds its first Education Forum on Sunday Straits Times The Straits Times’ first Education Forum on May 4, 2014, held at the Singapore Management University’s Mochtar Riady Auditorium. — ST PHOTO: … All 300 places at The Straits Times’ first education forum this Sunday taken up Straits Times Mr David Hoe, an undergraduate at the National University of Singapore (NUS), is one of the speakers at the inaugural The Straits Times Education … Many turn up at E Plus International Education fair The Hindu The aspirants evinced keen interest in countries like Holland, Singapore, New … Official boards of all the countries presented seminars on education … Tuition and divorce The Independent Singapore News In September 2013, The Independent Singapore reported on Senior Minister of State for Education Ms Indranee Rajah’s observation on the perceived … NS committee may propose changes to IPPT management TODAYonline SINGAPORE — Suggestions to improve the management of the Individual … Veterans’ League, which was founded to promote National Education. Should India Embrace Socialism, Singapore Style? Businessinsider India This is because the Singapore government only borrows to develop a … What offers a ray of hope to Indian educators is that Singapore’s education … How does one of the top-performing countries in the world think about technology? The Hechinger Report SINGAPORE—Forty students in bright yellow shirts hunched over their … Investments in education technology have been a key part of Singapore’s … Why Indonesian education is in crisis Jakarta Post Does anyone seriously believe “education” in Indonesia is on par with the west, or even Asian countries like Japan, Korea or Singapore? Ask the … Are you getting a little crazy in your classroom? T.H.E. Journal We have asked Dr. Zachary Walker, an assistant professor at the National Institute of Education, Singapore, an American who is traveling the world … GEMS Education eyes expansion in the region Business Times (subscription) GEMS Education, the world’s largest operator of private schools, aims to … from kindergarten to pre-university, will open in Singapore later this year.

Latest News: Riemann Hypothesis Proved?

Recently, I saw on Arxiv (an online Math journal) that a professor from South-China Normal University, Mingchun Xu, has proved the notoriously difficult Riemann Hypothesis.

Quote: “By using a theorem of Hurwitz for the analytic functions and a theorem due to T.J.Stieltjes and I. Schur, the Riemann Hypothesis has been proved considering the alternating Riemann zeta function. “

More verification is needed to check if it is indeed a proof.

In 1859, Bernhard Riemann, a little-known thirty-two year old mathematician, made a hypothesis while presenting a paper to the Berlin Academy titled  “On the Number of Prime Numbers Less Than a Given Quantity.”  Today, after 150 years of careful research and exhaustive study, the Riemann Hyphothesis remains unsolved, with a one-million-dollar prize earmarked for the first person to conquer it.

Rated: 4.5 stars on Amazon

5 Ways to get Pi on Calculator without pressing the Pi button:

1) 22/7

22/7 is not an exact value for Pi, but it is a pretty good approximation. 22/7=3.142857143… has just a percentage error of 0.04% compared to the actual value of Pi!

Percentage error is calculated by: $\displaystyle\frac{22/7-\pi}{\pi}\times 100\%=0.04\%$

2) 355/113

355/113 is an even better approximation for Pi. 355/113=3.14159292… has merely a percentage error of 0.000008%! This is incredibly accurate for a “relatively” simple fraction like 355/113. 355/113 has a cool Chinese name called “Milü密率, given by the ancient Chinese Mathematician astronomer Zǔ Chōngzhī (祖沖之) who discovered it.

3) 3.14

Using the simple and straightforward 3.14 (0.05% error) may be sufficient for everyday purposes. 🙂

4) $2\sin^{-1}(1)$ or 2 arcsin(1) (Radian Mode)

This relies on the fact that $\sin^{-1}(1)=\pi /2$.

5) $\lim_{n\to\infty}{n\sin(180^\circ/n)}$

We can let n=180 for convenience, and get ${180\sin(1^\circ)\approx 3.14143}$. This is a pretty decent approximation for $\pi$, with just 0.005% error. The approximation gets better as n gets larger.

Featured Book:

Pi: A Biography of the World’s Most Mysterious Number

We all learned that the ratio of the circumference of a circle to its diameter is called pi and that the value of this algebraic symbol is roughly 3.14. What we weren’t told, though, is that behind this seemingly mundane fact is a world of mystery, which has fascinated mathematicians from ancient times to the present. Simply put, pi is weird.

How to become better at Math?

How can I become excellent at math? It really interests me but when I fail I become demotivated and begin to give up.

EDIT: Could anyone suggest books for someone with a math education that just barely touches on high-school Algebra (got into parabolas, rationalizing, some graphing and functions). This is what I am currently doing: attending high school as a Junior.

Some gems of wisdom:

Researchers have shown it takes about ten years to develop expertise in any of a wide variety of areas, including chess playing, music composition, telegraph operation, painting, piano playing, swimming, tennis, and research in neuropsychology and topology.

The key is deliberative practice: not just doing it again and again, but challenging yourself with a task that is just beyond your current ability, trying it, analyzing your performance while and after doing it, and correcting any mistakes. Then repeat. And repeat again.

All the best in your math studies!

Featured book:

Men of Mathematics (Touchstone Book)

Here is the classic, much-read introduction to the craft and history of mathematics by E.T. Bell, a leading figure in mathematics in America for half a century. Men of Mathematics accessibly explains the major mathematics, from the geometry of the Greeks through Newton’s calculus and on to the laws of probability, symbolic logic, and the fourth dimension. In addition, the book goes beyond pure mathematics to present a series of engrossing biographies of the great mathematicians — an extraordinary number of whom lived bizarre or unusual lives. Finally, Men of Mathematics is also a history of ideas, tracing the majestic development of mathematical thought from ancient times to the twentieth century. This enduring work’s clear, often humorous way of dealing with complex ideas makes it an ideal book for the non-mathematician.

Maclaurin Series Informal Proof

Most students will encounter the Maclaurin Series (also known as the Taylor’s Series centered at zero) when they are studying JC H2 or College Maths. The formula looks pretty intimidating at the start:

$\displaystyle \boxed{f(x)=f(0)+xf'(0)+\frac{x^2}{2!}f''(0)+\cdots+\frac{x^n}{n!}f^{(n)}(0)+\cdots}$

How on earth does one come up with that formula?

However, it turns out it is not that hard to prove the Maclaurin Series informally, or at least to derive the above formula. (The hard part is related to rigorous proof of convergence, etc.)

The idea is to approximate a function by a power series (a kind of infinite polynomial) and then find out what are the coefficients.

So, we assume we can write the function as such:

$\boxed{f(x)=f_0+xf_1+x^2f_2+x^3f_3+\cdots+x^nf_n+\cdots\;(\dagger)}$, where $f_i$ are the coefficients of the polynomial (to be determined).

We also assume that the above equation holds for all $x$.

Then, letting $x=0$, we get $f(0)=f_0$. We have just found the first coefficient!

Next, we differentiate the equation $(\dagger)$ to get:

$f'(x)=f_1+2xf_2+3x^2f_3+\cdots+nx^{n-1}f_n+\cdots$

Letting $x=0$ again, we get: $f'(0)=f_1$.

Now, differentiating the above equation one more time gives us:

$f''(x)=2f_2+6xf_3+\cdots+n(n-1)x^{n-2}f_n+\cdots$

Letting $x=0$ gives $\displaystyle f_2=\frac{f''(0)}{2}$.

Keep on differentiating, and we will see that $\displaystyle f_n=\frac{f^{(n)}(0)}{n!}$.

This is how we get the Maclaurin Series: 🙂

$\displaystyle \boxed{f(x)=f(0)+xf'(0)+\frac{x^2}{2!}f''(0)+\cdots+\frac{x^n}{n!}f^{(n)}(0)+\cdots}$

A Math Formulas Not in the Formula Sheet

Are you looking for a list of A Maths (O Level Additional Maths) Formulae that is not found in the formula list?

Check it out at https://mathtuition88.com/math-notes-worksheets-sale/

The PDF file above contains A Maths Formulae on Algebra, Geometry & Trigonometry, and Calculus (Differentiation and Integration).

We also conduct A Maths Tuition class at Bishan.

SG Education News: Even Saudis are learning Singapore way of Teaching

 Saudis learning the Singapore way of teaching Straits Times Since last October, the National Institute of Education (NIE) has taken leadership trainers from the kingdom under its wing, training them in curriculum … All 300 places at The Straits Times’ first education forum this Sunday taken up Straits Times Mr David Hoe, an undergraduate at the National University of Singapore (NUS), is one of the speakers at the inaugural The Straits Times Education … Singapore Plows Ahead of US With Tech in Schools NBCNews.com In the late 1990s, the Singapore Ministry of Education unveiled its master plan for technology. The first phase was spent building up infrastructure and … Govt mulls more recognition for NSmen in housing, health, education Channel News Asia SINGAPORE: More recognition could be given to National Servicemen (NSmen) in areas such as housing, healthcare and education. Defence … Singapore to beef up nuclear technology expertise Channel News Asia Singapore is beefing up its nuclear technology expertise with a newly-announced programme. The 10-year Nuclear Safety Research and Education … Many turn up at E Plus International Education fair The Hindu The aspirants evinced keen interest in countries like Holland, Singapore, New … Official boards of all the countries presented seminars on education … AWARE’s pushback on more benefits for NSmen ignites debate TODAYonline SINGAPORE — The Government’s plan to enhance housing, healthcare and education benefits for operationally ready national servicemen has …

Introduction to Ricci Flow & Poincare Conjecture

This is an interesting introduction to some extremely advanced Math: Ricci Flow & Poincare Conjecture!

Ricci Flow was used to finally crack the Poincaré Conjecture. It was devised by Richard Hamilton but famously employed by Grigori Perelman in his acclaimed proof. It is named after mathematician Gregorio Ricci-Curbastro.

In this video it is discussed by James Isenberg from the University of Oregon (filmed here at MSRI).

The famed Poincaré Conjecture – the only Millennium Problem cracked thus far.

Math News April 28

 Math wiz monkeys providing researchers with insights into human brain activity Fox News Monkeys trained to solve math problems are providing researchers with new insights into understanding a human learning disability in which children … Math and Science Pay, But High Schoolers Care Less Wall Street Journal (blog) Math and science are the peas and carrots of the jobs market: great for a career future, but resolutely unpopular with the young. Even amid a relatively … Math wrath in Pincher Creek? Pincher Creek Echo Protesters gather during a rally to support a petition calling for math curriculum reform at the Alberta Legislature Building in Edmonton, Alta., … Monkey Math and Other Number-Crunching Critters Discovery News Rhesus monkeys are able to perform math at an advanced level, reports a study this week from Harvard Medical Medical school. The monkeys were … Math department wins national award for exemplary program The Williams record According to Susan Loepp, professor of mathematics, the College’s department is unique in several regards. “Everyone likes math even if they don’t … From math failure to savant: How a mugging made a numbers whiz CTV News Padgett describes the brutal bar attack and his subsequent transformation into a math savant in his new book, Struck by Genius: How a Brain Injury …

E Maths List of Formulae to Memorize (Not in Formula List)

Here is a compilation of the Formulas needed in GCSE O Levels E Maths Exam.

Includes Formulae on Algebra and Numbers, Geometry and Measurement, Statistics and more!

Check it out at https://mathtuition88.com/math-notes-worksheets-sale/

Students in my Maths Tuition class will be taught how to memorize the formulae with understanding, and how to apply them correctly!

Keep calm, and all the best for your mid-year exams.