Why do Exterior Angles add up to 360 degrees?

Source: http://www.fromquarkstoquasars.com/20-gifs-that-teach-you-science-concepts-better-than-your-teacher-probably-can/

This is the real reason why exterior angles of a polygon add up to 360 degrees! If you shrink the polygon (which doesn’t affect the sum of exterior angles), the exterior angles eventually meet at a point, and the sum of angles at a point is 360 degrees.

This is some cool math to think about!

Featured book:

Math Appeal: Mind-Stretching Math Riddles

NEW YORK TIMES bestselling author Greg Tang challenges kids to solve problems creatively in this follow-up to MATH FOR ALL SEASONS.

In this book you’ll learn to see
How very clever you can be.

解密神算超人

Math Online Tom Circle

解密神算超人

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Universum Survey (Please help to do!)

Thanks for being a loyal reader of Mathtuition88.com!

Do you want to know which employer is the most suitable for you?

Please help to do this survey at:

http://uledge.co/ambWWCY

Alternate URL: https://surveys.universumglobal.com/markets/singapore/distributions/ambWWCY

It is a Career Test by Universum. Participate to reveal your career type and discover the optimal employers for you!

Thanks for your help once again!

Einstein for Kids (Book)

Just to introduce a book that is based on an essay I wrote 10 years ago when I was a teenager. I decided to repackage it as a book published on Lulu.com:

Einstein, Relativity and Light for Kids

(Lulu Paperback link)

Einstein, Relativity and Light for Kids: A book about Einstein, Relativity, and Light for Children. Includes an award-winning 2000 word Essay on “What happens if Light slows down”. Written when the author was 17 years old.

An Ebook version is also available here:

Excerpt:

What happens if light slows down – A Beginner’s Guide to Relativity and Light

In the beginning God created the heavens and the earth. And God said, “Let there be light,” and there was light. Light is one of the most ubiquitous things that we see, and it is also one of the oldest – it existed since the beginning of mankind. However, light is also mysterious in that no one really understands what it is and how it is rectilinearly propagated. Nevertheless, the speed of light plays an important part in physics, and it is one of the more often quoted constant. What will happen then, if the speed of light suddenly changes from 300000000m/s to a fraction of its original self –3000 m/s? (It is theoretically possible to slow down light to such a speed, by shining a beam of light through a medium with a refractive index of 100,000.)

Hope this book will be useful to anyone trying to learn more about Einstein through a novel way! What will happen when light slows down? Read the book to find out! 🙂

Beautiful Map of Mathematics

Source: https://plus.google.com/114134834346472219368/posts/hs79fnxkjis?pid=6037478677752714706&oid=101584889282878921052

Mathematistan

This is a really beautiful Map of Mathematics (Mathematistan, a pun on Mathematics and Afghanistan), where one can see all the various branches of Maths, and how they combine together.

I also learnt a new word: Califate, which means an Islamic state led by a supreme religious and political leader known as a caliph – i.e. “successor” – to Muhammad.

Featured book:

Guide to LaTeX (4th Edition)

Published Nov 25, 2003 by Addison-Wesley Professional. Part of the Tools and Techniques for Computer Typesetting series. The series editor may be contacted at frank.mittelbach@latex-project.org. LaTeX is the text-preparation system of choice for scientists and academics, and is especially useful for typesetting technical materials. This popular book shows you how to begin using LaTeX to create high-quality documents. The book also serves as a handy reference for all LaTeX users. In this completely revised edition, the authors cover the LaTeX2ε standard and offer more details, examples, exercises, tips, and tricks. They go beyond the core installation to describe the key contributed packages that have become essential to LaTeX processing.

Inside, you will find:

Complete coverage of LaTeX fundamentals, including how to input text, symbols, and mathematics; how to produce lists and tables; how to include graphics and color; and how to organize and customize documents
Discussion of more advanced concepts such as bibliographical databases and BIBTeX, math extensions with AMS-LaTeX, drawing, slides, and letters
Helpful appendices on installation, error messages, creating packages, using LaTeX with HTML and XML, and fonts
An extensive alphabetized listing of commands and their uses

New to this edition:

More emphasis on LaTeX as a markup language that separates content and form–consistent with the essence of XML
Detailed discussions of contributed packages alongside relevant standard topics
In-depth information on PDF output, including extensive coverage of how to use the hyperref package to create links, bookmarks, and active buttons

As did the three best-selling editions that preceded it, Guide to LaTeX, Fourth Edition, will prove indispensable to anyone wishing to gain the benefits of LaTeX.

The accompanying CD-ROM is part of the TeX Live set distributed by TeX Users Groups, containing a full LaTeX installation for Windows, MacOSX, and Linux, as well as many extensions, including those discussed in the book.

Holder’s Inequality for Lp Spaces

These are two excellent videos explaining Holder’s Inequality for Lp Spaces:

Featured book:

Divine Proportions: Rational Trigonometry to Universal Geometry

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.

Love Math Graphs!

How to remember graphs?

Many students have a hard time remembering how graphs look like.

Here is a humorous cartoon (suitable for Valentine’s Day) on how some graphs look like!

Remember, Maths is not just about exams, homework, or getting A1/A2. Maths, above all, is about the LOVE of learning and thinking.

$math graphs$

Featured book:

Love and Math: The Heart of Hidden Reality

A New York Times Science Bestseller

What if you had to take an art class in which you were only taught how to paint a fence? What if you were never shown the paintings of van Gogh and Picasso, weren’t even told they existed? Alas, this is how math is taught, and so for most of us it becomes the intellectual equivalent of watching paint dry.

In Love and Math, renowned mathematician Edward Frenkel reveals a side of math we’ve never seen, suffused with all the beauty and elegance of a work of art. In this heartfelt and passionate book, Frenkel shows that mathematics, far from occupying a specialist niche, goes to the heart of all matter, uniting us across cultures, time, and space.

Love and Math tells two intertwined stories: of the wonders of mathematics and of one young man’s journey learning and living it. Having braved a discriminatory educational system to become one of the twenty-first century’s leading mathematicians, Frenkel now works on one of the biggest ideas to come out of math in the last 50 years: the Langlands Program. Considered by many to be a Grand Unified Theory of mathematics, the Langlands Program enables researchers to translate findings from one field to another so that they can solve problems, such as Fermat’s last theorem, that had seemed intractable before.

At its core, Love and Math is a story about accessing a new way of thinking, which can enrich our lives and empower us to better understand the world and our place in it. It is an invitation to discover the magic hidden universe of mathematics.

The important thing is to keep thinking

This is a really inspirational story to me. “The important thing is to keep thinking.”

keep thinking

Source: http://www.reigndesign.com/blog/doing-it-with-twins-the-twin-prime-conjecture/

Now, I want you to imagine for a moment that you live in the United States, to be exact: New Hampshire. You’re a recruiter at the University of New Hampshireand your job is to hire the best people to become professors and lecturers.

Now suppose one day you get an application from this guy, Zhang Yitang, a 50-something mathematician. Since getting his PhD from Purdue, he’s struggled to find an academic job, working as a motel clerk and a Subway sandwich maker. I wouldn’t blame you if you passed over him.

It turns out if you had skipped Zhang Yitang, you’d have been making a big mistake, because a few weeks ago this 57-year old Chinese mathematician made headlines around the world when he proved a result in number theory which has been challenging mathematicians for years.

Featured book:

How Not to Be Wrong: The Power of Mathematical Thinking

The Freakonomics of math—a math-world superstar unveils the hidden beauty and logic of the world and puts its power in our hands

The math we learn in school can seem like a dull set of rules, laid down by the ancients and not to be questioned. In How Not to Be Wrong, Jordan Ellenberg shows us how terribly limiting this view is: Math isn’t confined to abstract incidents that never occur in real life, but rather touches everything we do–the whole world is shot through with it.

Math allows us to see the hidden structures underneath the messy and chaotic surface of our world. It’s a science of not being wrong, hammered out by centuries of hard work and argument. Armed with the tools of mathematics, we can see through to the true meaning of information we take for granted: How early should you get to the airport? What does “public opinion” really represent? Why do tall parents have shorter children? Who really won Florida in 2000? And how likely are you, really, to develop cancer?

How Not to Be Wrong presents the surprising revelations behind all of these questions and many more, using the mathematician’s method of analyzing life and exposing the hard-won insights of the academic community to the layman–minus the jargon. Ellenberg chases mathematical threads through a vast range of time and space, from the everyday to the cosmic, encountering, among other things, baseball, Reaganomics, daring lottery schemes, Voltaire, the replicability crisis in psychology, Italian Renaissance painting, artificial languages, the development of non-Euclidean geometry, the coming obesity apocalypse, Antonin Scalia’s views on crime and punishment, the psychology of slime molds, what Facebook can and can’t figure out about you, and the existence of God.

Ellenberg pulls from history as well as from the latest theoretical developments to provide those not trained in math with the knowledge they need. Math, as Ellenberg says, is “an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength.” With the tools of mathematics in hand, you can understand the world in a deeper, more meaningful way. How Not to Be Wrong will show you how.

The Three Square Geometry Problem – Numberphile

Watch this interesting video about the “Three Square Geometry Problem”!

Theoretically, a fifth-grader or P5/PSLE student can solve it! The featured solution is truly brilliant and requires one to “think out of the box”.

Featured book:

Tutor in a Book’s Geometry

Need help with Geometry? Designed to replicate the services of a skilled private tutor, the new and improved Tutor in a Book’s Geometry is at your service! TIB’s Geometry is an extremely thorough, teen tested and effective geometry tutorial.

TIB’s Geometry includes more than 500 of the right, well-illustrated, carefully worked out and explained proofs and problems. Throughout TIB’s Geometry, there is ongoing, specific guidance as to the most effective solution and test taking strategies. Recurring patterns, which provide solutions to proofs, are pointed out, explained and illustrated using the visual aids that students find so helpful. Also included are dozens of graphic organizers, which help students understand, remember and recognize the connections between concepts.

TIB’s author Jo Greig intended this book to level the playing field between the students who have tutors and those that don’t. As a long time, very successful private mathematics tutor and teacher, Jo Greig knew exactly how best to accomplish this! TIB’s Geometry 294 pages are packed with every explanation, drawing, hint and memory tool possible! Not only does it have examples of the right proofs and problems, it also manages to impart every bit of the enthusiasm that great tutors impart to their private tutoring students. Ms. Greig holds a bachelors’ degree in mathematics. Dr. J. Shiletto, the book’s mathematics editor, holds a Ph.D in mathematics.

Challenging O Level Trigonometry Question (A Maths)

A reader of our Mathtuition88.com blog asked the following Maths question:

Given that $\sin x+\sin y=a$ and $\cos x+\cos y=a$ , where $a\neq 0$ , express $\sin x+\cos x$ in terms of $a$ .

This is a rather challenging question, since there are many options to start. Which formula(s) should we use? Factor formula? R-formula? Give it a try first if you want to have a challenge.

Solution:

It turns out we can write:

$\sin y=a-\sin x$

$\cos y=a-\cos x$

Then, use $\sin^2 y+\cos ^2 y=1$

$(a-\sin x)^2+(a-\cos x)^2=1$

Expanding,

$a^2-2a\sin x+\sin^2 x+a^2-2a\cos x+\cos^2 x=1$

Rearranging,

$2a^2-2a(\sin x+\cos x)+1=1$

$2a(a-(\sin x+\cos x))=0$

Since $a\neq 0$ , we have $a-(\sin x+\cos x)=0$ .

Thus, $\boxed{\sin x+\cos x=a}$ .

Featured book:

Schaum’s Outline of Trigonometry, 5th Edition: 618 Solved Problems + 20 Videos (Schaum’s Outline Series)

Tough Test Questions? Missed Lectures? Not Enough Time?

Fortunately, there’s Schaum’s. This all-in-one-package includes more than 600 fully solved problems, examples, and practice exercises to sharpen your problem-solving skills. Plus, you will have access to 20 detailed videos featuring Math instructors who explain how to solve the most commonly tested problems–it’s just like having your own virtual tutor! You’ll find everything you need to build confidence, skills, and knowledge for the highest score possible.

More than 40 million students have trusted Schaum’s to help them succeed in the classroom and on exams. Schaum’s is the key to faster learning and higher grades in every subject. Each Outline presents all the essential course information in an easy-to-follow, topic-by-topic format. You also get hundreds of examples, solved problems, and practice exercises to test your skills.

This Schaum’s Outline gives you

618 fully solved problems to reinforce knowledge
Concise explanations of all trigonometry concepts
Updates that reflect the latest course scope and sequences, with coverage of periodic functions and curve graphing.

Fully compatible with your classroom text, Schaum’s highlights all the important facts you need to know. Use Schaum’s to shorten your study time–and get your best test scores!

Schaum’s Outlines–Problem Solved.

Fire HD Kids Edition Tablet: Educational Review

Shop Amazon – Introducing Fire HD Kids Edition – Everything Kids Love. Everything Parents Want.

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.

What are Friedman numbers?

What are Friedman numbers? Watch this video to find out!

Most amazing thing is that as numbers get bigger, the likelihood that they are Friedman numbers actually increase! (Friedman numbers have “density one”!)

Featured book:

First Steps for Math Olympians: Using the American Mathematics Competitions (Problem Books) (MAA Problem Book Series)

Any high school student preparing for the American Mathematics Competitions should get their hands on a copy of this book! A major aspect of mathematical training and its benefit to society is the ability to use logic to solve problems. The American Mathematics Competitions (AMC) have been given for more than fifty years to millions of high school students. This book considers the basic ideas behind the solutions to the majority of these problems, and presents examples and exercises from past exams to illustrate the concepts. Anyone taking the AMC exams or helping students prepare for them will find many useful ideas here. But people generally interested in logical problem solving should also find the problems and their solutions interesting. This book will promote interest in mathematics by providing students with the tools to attack problems that occur on mathematical problem-solving exams, and specifically to level the playing field for those who do not have access to the enrichment programs that are common at the top academic high schools. The book can be used either for self-study or to give people who want to help students prepare for mathematics exams easy access to topic-oriented material and samples of problems based on that material. This is useful for teachers who want to hold special sessions for students, but it is equally valuable for parents who have children with mathematical interest and ability. As students’ problem solving abilities improve, they will be able to comprehend more difficult concepts requiring greater mathematical ingenuity. They will be taking their first steps towards becoming math Olympians!

Carnival of Mathematics

Mathtuition88.com will be the host of the next Carnival of Mathematics! (Submission site: http://www.aperiodical.com/carnival-of-mathematics)

I will now be receiving submissions for Carnival 115.

Firstly, let’s have a discussion on what is so special about the number 115. David Brooks has kindly provided a PDF (Input for Carnival of Math) which the following information is sourced from.

The “Mathematical Association of America” (http://maanumberaday.blogspot.com/2009/11/115.html) notes that:

115 = 5 x 23.

115 = 23 x (2 + 3).

115 has a unique representation as a sum of three squares: 3² + 5² + 9² = 115.

115 is the smallest three-digit integer, abc, such that (abc)/(a*b*c) is prime: 115/5 = 23.

STS-115 was a space shuttle mission to the International Space Station flown by the space shuttle Atlantis on Sept. 9, 2006.

Some other interesting Trivia about 115 include:

115 is the emergency telephone number when calling in Iran. 🙂

115 is the number of cardinals who actually participated to vote for the 265^th Pope succeeding the Pope John Paul II in April 2005, even though 117 cardinals were eligible.

Featured posts:

1) How Many Colored Tetrominoes?

Permalink URL:
http://mrburkemath.blogspot.com/2014/09/how-many-colored-tetrominoes.html

Title of post:
How Many Colored Tetrominoes?

Post Author:
Christopher J. Burke

This is a very interesting link about tetrominoes! If you are not sure what are tetrominoes, it is perfectly ok! Just go to the website link above and you will find out!

Question: How many different colored tetrominoes are there if we allow only four colors total?

Second question: What the heck is a tetromino?

Dominoes are a great game with rectangle tiles, composed of two adjacent squares with certain numbers of pips on them. A tetromino is a group of four adjacent squares, each sharing at least one side with at least one other square. In other words, those little falling shapes made popular in the game Tetris, and all of its knock-off variations, as seen below:

tetromino

2) Using expected frequencies when teaching probability

Summary: The use of the term ‘expected frequencies’ is novel and not widely known in mathematics education. The basic idea is very simple: instead of saying “the probability of X is 0.20 (or 20%)”, we would say “out of 100 situations like this, we would expect X to occur 20 times”.

To learn about this more intuitive and novel way of using expected frequencies to teach probability, visit the site at http://understandinguncertainty.org/using-expected-frequencies-when-teaching-probability.

3) Kettle and Cake Logic

Sigmund Freud tells the tale of a man accused of breaking his neighbour’s kettle. He mounts a three-stranded defence :

1. “I never borrowed it in the first place!”
2. “And anyway it was already broken when I did!”
3. “In any case, it was fine when I returned it!”

Freud used this as an example of the inconsistent logic of dreamland, although you won’t have to look too far afield in the waking world to find examples of similar reasoning[1].

Sounds interesting? View it at: https://plus.google.com/app/basic/stream/z13swvoqnzeyxtbep22fwvqoaxjlefohb04

4) Math Circle – Billiards

From Math Circle: The reason I picked billiards to feature at this particular moment is because twoof this year’s Fields Medalists study billiards: Maryam Mirzakhani and Artur Avila. To find out more about these amazing mathematicians, see our recent Math Munch post.

Visit http://ichoosemath.com/2014/09/14/math-circle-billiards/ to learn more!

5) The curious reluctance to define prime probability logically

The curious reluctance to define prime probability logically. The title says it all, except stress the point that we need to encourage more reasoning from first principles based on what we individually accept as self-evident, and not on what others believe to be self-evident.

6) Hailstone numbers shape a poem

By JoAnne Growney: One of my favorite mathy poets is Halifax mathematician Robert Dawson — his work is complex and inventive, and fun to puzzle over. Dawson’s webpage at St Mary’s University lists his mathematical activity; his poetry and fiction are available in several issues of the Journal of Humanistic Mathematics and in several postings for this blog (15 April 2012, 30 November 2013, 2 March 2014) and in various other locations findable by Google.
Can a poem be written by following a formula? Despite the tendency of most of us to say NO to this question we also may admit to the fact that a formula applied to words can lead to arrangements and thoughts not possible for us who write from our own learning and experiences. How else to be REALLY NEW but to try a new method? Set a chimpanzee at a typewriter or apply a mathematical formula.
Below we offer Dawson’s “Hailstone” and follow it with his explanation of how mathematics shaped the poem from its origin as a “found passage” from the beginning of Dickens’ Great Expectations.

7) Approximating e using the digits 1–9

Read this article to learn how to approximate e using just the digitis 1-9! ((1 + 9^{–4^{7×6}})^{3^{2^{85}}}. ) Learn how it works and how remarkably accurate it is! The post is written by Richard Green.

Another closely related post is http://www.flyingcoloursmaths.co.uk/estimating-e/ by Flying Colours Maths Blog!

8）Sand Hill-bert Curve

IMG_20140917_190333

What is this about? It is a sand model of the Hilbert Curve, or Hilbert space-filling curve!

Check out http://blog.andreahawksley.com/sand-hill-bert-curve/ to learn more.

9) Decending powers of x

I was in one of my colleagues lessons this week.and he was teaching the class to expand quadratic brackets. As the lesson went on he noticed that a number of pupils had been writing the X squared term, then the constant term then the X term so he pulled the class together to tell them that conventionally we write quadratic equations in decending powers of x. This is excellent practice and something we all should be encouraging, but it made me think “Why decending powers of x?”

Interesting question to ponder!

Read more at: http://cavmaths.wordpress.com/2014/09/26/decending-powers-of-x/

10) Extrapolation Gone Wrong: the Case of the Fermat Primes

Read more at: http://blogs.scientificamerican.com/roots-of-unity/2014/09/26/extrapolation-gone-wrong-the-case-of-the-fermat-primes/

11) Erica Klarreich Profiles an Award-Winning Mathematician

Erica Klarreich interviews a famous recent Fields Medallist Stanford University professor Maryam Mirzakhani at: http://www.theopennotebook.com/2014/09/30/erica-klarreich-profiles-an-award-winning-mathematician/

12) Will Rogers phenomenon

Check out the interesting Will Rogers phenomenon, with application to managing a football team! (http://mathsball.blogspot.com.es/2014/09/impossible-transfer-will-rogers-phenomenon.html)

Featured book:

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

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

Learning Pyramid (How to Learn Maths)

learning_pyramid

The best way to learn maths is actually to teach others. The second best way to learn maths is to practice doing it!

Featured book:

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

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

As the math education crisis in this country continues to make headlines, research continues to prove that it is in middle school when math scores begin to drop—especially for girls—in large part due to the relentless social conditioning that tells girls they “can’t do” math, and that math is “uncool.” Young girls today need strong female role models to embrace the idea that it’s okay to be smart—in fact, it’s sexy to be smart!

It’s Danica McKellar’s mission to be this role model, and demonstrate on a large scale that math doesn’t suck. In this fun and accessible guide, McKellar—dubbed a “math superstar” by The New York Times—gives girls and their parents the tools they need to master the math concepts that confuse middle-schoolers most, including fractions, percentages, pre-algebra, and more. The book features hip, real-world examples, step-by-step instruction, and engaging stories of Danica’s own childhood struggles in math (and stardom). In addition, borrowing from the style of today’s teen magazines, it even includes a Math Horoscope section, Math Personality Quizzes, and Real-Life Testimonials—ultimately revealing why math is easier and cooler than readers think.

Inequality Olympiad Question and Solution

Let a, b, c be nonnegative real numbers satisfying $a^2+b^2+c^2=1$ . Prove that

$\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\geq\sqrt{7(a+b+c)-3}$

Source: http://www.fen.bilkent.edu.tr/~cvmath/Problem/1408a.pdf

Featured book:

Inequalities: A Mathematical Olympiad Approach

This book is intended for the Mathematical Olympiad students who wish to prepare for the study of inequalities, a topic now of frequent use at various levels of mathematical competitions. In this volume we present both classic inequalities and the more useful inequalities for confronting and solving optimization problems. An important part of this book deals with geometric inequalities and this fact makes a big difference with respect to most of the books that deal with this topic in the mathematical olympiad. The book has been organized in four chapters which have each of them a different character. Chapter 1 is dedicated to present basic inequalities. Most of them are numerical inequalities generally lacking any geometric meaning. However, where it is possible to provide a geometric interpretation, we include it as we go along. We emphasize the importance of some of these inequalities, such as the inequality between the arithmetic mean and the geometric mean, the Cauchy-Schwarz inequality, the rearrangementinequality, the Jensen inequality, the Muirhead theorem, among others. For all these, besides giving the proof, we present several examples that show how to use them in mathematical olympiad problems. We also emphasize how the substitution strategy is used to deduce several inequalities.

A Journey of Mathematics 数学之旅

Math Online Tom Circle

Excellent free course for non mathematicians.

This Philosophical Math course has started half way but past videos are still hosted on the site.

The course is taught by prof Wang of Shanghai Jiaotong Technology University上海交通大学 (SJT), the Alma Mata of former China President Jiang (江泽民), Prime minister Chu (朱镕基), and Prof Qian XueSheng (钱学森) “The Father of Chinese Space and Missile” (China exchanged his country home return with USA FBI for 4 American generals from Korean War prisoners of War) who sent Chinese Taikongnauts (太空人) to space.

SJT was formed initially as the ‘Classe Préparatoire’ (Bachelor degree, post-High school Prep-college) for graduate engineering to MIT, while Qing Hua 清华 University was a prep-college for graduate Science/ Math to Harvard, Chicago, Cornell, etc.

Go to Lesson 3: He explains from a game of Go what is “Space” in maths: Geometrical n-dimensional Space, Linear space, vector space.. why study functional space (in which…

View original post 281 more words

A Simple Brain Theory Endorsed By Bill Gates Claims To Help You Learn Anything

Source: http://www.businessinsider.sg/carol-dwecks-growth-mindset-theory-tweeted-by-bill-gates-2014-8/#.VAxnTPmSx8E

A small psychological change to how we approach challenges can drastically change how successful we are at these tasks.

That’s according to Carol Dweck, a psychology professor at Stanford University, who coined the term “growth mindset” in her 2007 book “Mindset: The New Psychology of Success.”

Microsoft magnate Bill Gates tweeted a video of Dweck explaining the growth mindset earlier this week:

Featured book:

Mindset: The New Psychology of Success

World-renowned Stanford University psychologist Carol Dweck, in decades of research on achievement and success, has discovered a truly groundbreaking idea–the power of our mindset.

Dweck explains why it’s not just our abilities and talent that bring us success–but whether we approach them with a fixed or growth mindset. She makes clear why praising intelligence and ability doesn’t foster self-esteem and lead to accomplishment, but may actually jeopardize success. With the right mindset, we can motivate our kids and help them to raise their grades, as well as reach our own goals–personal and professional. Dweck reveals what all great parents, teachers, CEOs, and athletes already know: how a simple idea about the brain can create a love of learning and a resilience that is the basis of great accomplishment in every area.

Stable Marriage Problem – Numberphile

This is an interesting problem in the topic of combinatorics and graph theory. It can be phrased in the context of arranging stable marriages.

Nice animation and clear explanation! Watch part 2 too for the mathematical explanation.

Featured book:

A Mind For Numbers: How to Excel at Math and Science (Even If You Flunked Algebra)

Whether you are a student struggling to fulfill a math or science requirement, or you are embarking on a career change that requires a higher level of math competency, A Mind for Numbers offers the tools you need to get a better grasp of that intimidating but inescapable field. Engineering professor Barbara Oakley knows firsthand how it feels to struggle with math. She flunked her way through high school math and science courses, before enlisting in the army immediately after graduation. When she saw how her lack of mathematical and technical savvy severely limited her options—both to rise in the military and to explore other careers—she returned to school with a newfound determination to re-tool her brain to master the very subjects that had given her so much trouble throughout her entire life.

In A Mind for Numbers, Dr. Oakley lets us in on the secrets to effectively learning math and science—secrets that even dedicated and successful students wish they’d known earlier. Contrary to popular belief, math requires creative, as well as analytical, thinking. Most people think that there’s only one way to do a problem, when in actuality, there are often a number of different solutions—you just need the creativity to see them. For example, there are more than three hundred different known proofs of the Pythagorean Theorem. In short, studying a problem in a laser-focused way until you reach a solution is not an effective way to learn math. Rather, it involves taking the time to step away from a problem and allow the more relaxed and creative part of the brain to take over. A Mind for Numbers shows us that we all have what it takes to excel in math, and learning it is not as painful as some might think!

How to subtract 12 from 32? (The Common Core Way)

This is an incredibly complicated way to evaluate 32-12.

One wonders if this is a step backward in education.

As Einstein said, “Everything should be made as simple as possible, but not simpler”.

einstein quote

Featured book:

A Mind For Numbers: How to Excel at Math and Science (Even If You Flunked Algebra)

A Basic Multiplication Question (Common Mistake)

Site: http://www.kiasuparents.com/kiasu/forum/viewtopic.php?f=29&t=71476&p=1377930#p1377930

HermioneGranger wrote:

Hi, a desperate question.In solutions of equation, when you multiply 2(2) by two, why is the answer 2(4) and not 4(4)?

TIA

Answer:
Good question!

The theoretical reason is that if we multiply a product a(b) by two, we are actually having two copies of a(b).

Hence, 2 x a(b)= 2 times of a(b) = 2a(b) = a(2b)

For a concrete example, think about what would happen if we multiply 1(1)(1)(1)(1) by 2.

1(1)(1)(1)(1) is just 1, hence the answer should be 1(1)(1)(1)(2)=2, and not 2(2)(2)(2)(2) which is 32.

Featured book:

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

Free Coursera Course: An Introduction to Functional Analysis

Site: https://www.coursera.org/course/functionalanalysis

There is an interesting upcoming course at Coursera, suitable for undergraduates! (Starting 12 September 2014) Join the class if interested, it is free! Functional Analysis is actually a third year course for Math Majors at university. There are some powerful and deep theorems in functional analysis, like the Riesz representation theorem.

About the Course

Functional analysis is the branch of mathematics dealing with spaces of functions. It is a valuable tool in theoretical mathematics as well as engineering. It is at the very core of numerical simulation.

In this class, I will explain the concepts of convergence and talk about topology. You will understand the difference between strong convergence and weak convergence. You will also see how these two concepts can be used.

You will learn about different types of spaces including metric spaces, Banach Spaces, Hilbert Spaces and what property can be expected. You will see beautiful lemmas and theorems such as Riesz and Lax-Milgram and I will also describe Lp spaces, Sobolev spaces and provide a few details about PDEs, or Partial Differential Equations.

Course Syllabus

Week 1: Topology; continuity and convergence of a sequence in a topological space.
Week 2: Metric and normed spaces; completeness
Week 3: Banach spaces; linear continuous functions; weak topology
Week 4: Hilbert spaces; The Riesz representation theorem
Week 5: The Lax-Milgram Lemma
Week 6: Properties of the Lp spaces
Week 7: Distributions and Sobolev Spaces
Week 8: Application: simulating a membrane

Recommended Background

The course is mostly self-contained; however, you need to be familiar with functions, derivatives and integrals. You need to know what A ∩ B means and to know what a proof is. You should be fine if you have taken Calculus II and Algebra II. Students in Europe who have taken 120 ECTS in science should be fine as well.

Because this is an online class, having advanced and non-advanced students in a class will not be a problem; on the contrary we expect a wide range of interesting interactions. However, non-advanced students may have to work a bit more.

Course Format

The class will consist of a series of lecture videos, usually between five and twelve minutes in length. There will be approximately one hour worth of video content per week. Some of the videos contain integrated quiz questions. There will also be standalone quizzes that are not part of the video lectures; you will be asked to solve some problems and evaluate the solutions proposed by your fellow classmates. There will be a final exam.

There will be some additional contents in the form of PDF files.

FAQ

Will I get a Statement of Accomplishment after completing this class?
Yes. Students who successfully complete the class will receive a Statement of Accomplishment signed by the instructor.
What resources will I need for this class?
For this course, all you need is an Internet connection and the time to view the videos, understand the material, discuss the material with fellow classmates, take the quizzes and solve the problems.
What pedagogy will be used?
This MOOC is in English but the math will be taught with a “French Touch”.
What does “teaching math with a French touch” mean?
France has a long-standing tradition where math is addressed from a theoretical standpoint and studied for its implicit value throughout high school and preparatory school for the high-level entrance exams. This leads to a mindset based on proofs and abstraction. This mindset has consequences on problem solving that is sometimes referred to as the “French Engineer”. In contrast, other countries have a tradition where math is addressed as a computation tool.
Does it mean it will abstract and complicated?
The approach will be rather abstract but I will be sure to emphasize the concepts over the technicalities. Above all, my aim is to help you understand the material and the beauty behind it.

Featured book:

Introductory Functional Analysis with Applications

This is the recommended textbook that covers the material in the Coursera Course (and more).

Top 10 U.S. Public Universities

Math Online Tom Circle

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Trade In Your iPhone for Amazon Gift Card!

Are you getting the new iPhone6 soon? If so, you can consider trading in your old iPhone for Amazon Gift Card!

Just to share this amazing deal:

Shop Amazon – Trade In Your iPhone

Happy Teacher’s Day!

Glad to receive some Teacher’s Day cards and presents from my students!

teacher's day card Wishing all teachers a happy Teacher’s Day this Friday, and also wishing students all the best for their upcoming exams.

Featured book:

A Mind For Numbers: How to Excel at Math and Science (Even If You Flunked Algebra)

A Mind For Numbers: How to Excel at Math and Science (Even If You Flunked Algebra)

Featured book:

A Mind For Numbers: How to Excel at Math and Science (Even If You Flunked Algebra)

Mathematician’s Job: High Pay but Lowest Stress

One good reason to study Mathematics – high pay Mathematician job and lowest stress, comparing with other high-pay-high-stress professionals.

Math Online Tom Circle

High pay high stress ?

Not really true … among the top 17 high-paying jobs (yearly earning above US $100,000) in the USA, Mathematician’s job has the lowest stress below 60 (in the scale from 0 no stress to highest stress at 100).

Source:
http://www.businessinsider.sg/high-paying-low-stress-jobs-2014-7/9/#.VAYQrYEZ7qA

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Five things successful teachers do on the first day of class

$math joke$

Featured book:

Comic-Strip Math: Problem Solving: 80 Reproducible Cartoons With Dozens and Dozens of Story Problems That Motivate Students and Build Essential Math Skills

Math + Comics = Learning That’s Fun! Help students build essential math skills and meet math standards with 80 laugh-out-loud comic strips and companion mini-story problems. Each reproducible comic and problem set reinforces a key math skill: multiplication, division, fractions, decimals, measurement, geometry, and more. Great to use for small-group or independent class work and for homework! For use with Grades 3-6.

Funny Video about Math (and other) Majors

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. It’s a book that all teachers from elementary school through college should have in their library. But the humor isn’t just for the classroom-it also appeals to engineers, statisticians, and other math professionals searching for some good, clean, numerical fun. From basic facts (Why is 6 afraid of 7?) to trigonometry (Mathematical puns are the first sine of dementia) and algebra (Graphing rational functions is a pain in the asymptote), no topic is safe. As Professor Jim Rubillo notes, Math Jokes 4 Math Folks is an absolute gem for anyone dedicated to seeing mathematical ideas through puns, double meanings, and blatant bad jokes. Such perspectives help to see concepts and ideas in different and creative ways.