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.

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 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).

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)

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.

Magic Hexagon

Watch this video about the interesting Magic Hexagon (a relative of the magic square), and why there is only one magic hexagon in the world!

Featured book:

Speedsolving the Cube: Easy-to-Follow, Step-by-Step Instructions for Many Popular 3-D Puzzles

3×3 Matrix Inverse Shortcut (Fastest Method)

Source: http://www.math.columbia.edu/~bayer/LinearAlgebra/pdf/inverse.pdf

Backup copy (In case of broken link): 3×3 Matrix Inverse

This is an excellent method for finding the inverse of a 3×3 matrix, probably the fastest and easiest method for 3×3.

(Row reduction is better for 4×4 matrices and above.)

This method was first introduced to me by my student!

Featured book:

Linear Algebra and Its Applications, 4th Edition

Education News: School Starts Too Early?

Source: http://www.scientificamerican.com/article/school-starts-too-early/?&WT.mc_id=SA_WR_20140827

School Starts Too Early

The later high school classes start in the morning, the more academic performance improves

Aug 19, 2014 |By Mark Fischetti

Parents, students and teachers often argue, with little evidence, about whether U.S. high schools begin too early in the morning. In the past three years, however, scientific studies have piled up, and they all lead to the same conclusion: a later start time improves learning. And the later the start, the better.Biological research shows that circadian rhythms shift during the teen years, pushing boys and girls to stay up later at night and sleep later into the morning. The phase shift, driven by a change in melatonin in the brain, begins around age 13, gets stronger by ages 15 and 16, and peaks at ages 17, 18 or 19.

Does that affect learning? It does, according to Kyla Wahlstrom, director of the Center for Applied Research and Educational Improvement at the University of Minnesota. She published a large study in February that tracked more than 9,000 students in eight public high schools in Minnesota, Colorado and Wyoming. After one semester, when school began at 8:35 a.m. or later, grades earned in math, English, science and social studies typically rose a quarter step—for example, up halfway from B to B+.

Read more at: http://www.scientificamerican.com/article/school-starts-too-early/?&WT.mc_id=SA_WR_20140827

You are welcome to leave your comments below!

Featured book:

How Children Succeed: Grit, Curiosity, and the Hidden Power of Character

Linear Algebra Resources

1) https://www.math.okstate.edu/~binegar/3013-S99/3013-l16.pdf

(How to diagonalize matrices)

2) http://www.sosmath.com/matrix/expo/expo.html

(How to find exponential of matrices)

3) http://en.wikipedia.org/wiki/Matrix_differential_equation

(Matrix Differential Equations)

4) http://www.ucl.ac.uk/CNDA/courses/coursework/JNF.pdf

(Jordan Normal Form)

5) http://www.math.niu.edu/~fletcher/jnf.pdf

(Jordan Normal Form for 2×2 Matrices)

6) http://www.math.colostate.edu/~gerhard/M345/CHP/ch9_6.pdf

(Matrix Exponential with repeated Eigenvalues)

7) http://www.math.vt.edu/people/afkhamis/class_home/notes/F08W12.pdf

Click to access matrixexp.pdf

(Matrix Differential equation with repeated Eigenvalues)

8) http://www.math.utah.edu/~gustafso/2250matrixexponential.pdf

(Matrix Exponential of Block Diagonal)

Featured book:

Linear Algebra and Its Applications, 4th Edition
Linear algebra is relatively easy for students during the early stages of the course, when the material is presented in a familiar, concrete setting. But when abstract concepts are introduced, students often hit a brick wall. Instructors seem to agree that certain concepts (such as linear independence, spanning, subspace, vector space, and linear transformations), are not easily understood, and require time to assimilate. Since they are fundamental to the study of linear algebra, students’ understanding of these concepts is vital to their mastery of the subject. David Lay introduces these concepts early in a familiar, concrete Rⁿ setting, develops them gradually, and returns to them again and again throughout the text so that when discussed in the abstract, these concepts are more accessible.

Rational Parametrization of a Circle & Pythagorean Triples

Today, we discuss an interesting topic rarely taught in school: The rational parametrization of a unit circle. That is, how to find the x coordinates and y coordinates of a circle expressed as a rational function? The usual parametrization of a circle is (cos t, sin t).

unit circle

We consider the straight line (l), passing through the point A(1,0) and the point (0,h).

The gradient of this line is $\displaystyle m=\frac{h-0}{0-1}=-h$ .

The y-intercept of this line is $h$ .

Hence the equation of line l is $\boxed{y=-hx+h}$ .

We know the equation of the unit circle is $\boxed{x^2+y^2=1}$ .

By solving the two simultaneous equations (boxed), we get a quadratic formula:

$x^2(1+h^2)-2h^2x+h^2-1=0$

Solving the above using the quadratic formula gives us $\displaystyle x=\frac{h^2-1}{1+h^2}$ .

Using $\boxed{y=-hx+h}$ , we get $\displaystyle y=\frac{2h}{1+h^2}$ .

Hence, $\displaystyle \boxed{ (\frac{h^2-1}{1+h^2},\frac{2h}{1+h^2})}$ is a parametrization of the unit circle.

We can use this to generate Pythagorean triples! Simply choose a value of h, say, $h=8$ .

Then $\displaystyle x=\frac{h^2-1}{1+h^2}=\frac{63}{65}$ .

$\displaystyle y=\frac{2h}{1+h^2}=\frac{16}{65}$ .

Substituting into $\boxed{x^2+y^2=1}$ , and multiplying by the denominator, we get the Pythagorean Triple $16^2+63^2=65^2$ . Interesting?

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.

Fix a Wobbly Table (with Math)

Fix your wobbly table with just a small tweak – but why does this work?
Reddit discussion: http://www.reddit.com/r/BradyHaran/co…
Featuring Matthias Kreck from the University of Bonn.

Featured book:

The Princeton Companion to Mathematics

This is a one-of-a-kind reference for anyone with a serious interest in mathematics. Edited by Timothy Gowers, a recipient of the Fields Medal, it presents nearly two hundred entries, written especially for this book by some of the world’s leading mathematicians, that introduce basic mathematical tools and vocabulary; trace the development of modern mathematics; explain essential terms and concepts; examine core ideas in major areas of mathematics; describe the achievements of scores of famous mathematicians; explore the impact of mathematics on other disciplines such as biology, finance, and music–and much, much more.

Unparalleled in its depth of coverage, The Princeton Companion to Mathematics surveys the most active and exciting branches of pure mathematics, providing the context and broad perspective that are vital at a time of increasing specialization in the field. Packed with information and presented in an accessible style, this is an indispensable resource for undergraduate and graduate students in mathematics as well as for researchers and scholars seeking to understand areas outside their specialties.

Maths Tuition Centre: Circle Fraction Question (PSLE)

Solution:

The total area is made up of 4 semicircles and four of the area X. (4S+4X)

The unshaded area is made up of 1 semicircle and one of the area X. (1S+1X)

Hence, the fraction of the figure unshaded is 1/4.

ICM2014 — opening ceremony

First Female Mathematician to get Fields’ Medal: Maryam Mirzakhani!

http://www.youtube.com/watch?v=pOtj7VxbFMk

Featured book:

Modern Mathematics in the Light of the Fields Medals

This small book demonstrates the evolution of certain areas of modern mathematics by examining the work of past winners of the Fields Medal, the “Nobel Prize” of mathematics. Foreword by Freeman Dyson.

Gowers's Weblog

I’d forgotten just how full the first day of an ICM is. First, you need to turn up early for the opening ceremony, so you end up sitting around for an hour and half or so before it even starts. Then there’s the ceremony itself, which lasts a couple of hours. Then in the afternoon you have talks about the four Fields Medallists and the Nevanlinna Prize winner, with virtually no breaks. Then after a massive ten minutes, the Nevanlinna Prize winner talks about his (in this case) own work, about which you have just heard, but in a bit more detail. That took us to 5:45pm. And just to round things off, Jim Simons is giving a public lecture at 8pm, which I suppose I could skip but I think I’m not going to. (The result is that most of this post will be written after it, but right…

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Private Candidate Tuition & Ad hoc O Level Maths Tuition

Private Candidate Tuition

Taking your Math exam as a Private Candidate?

Currently we are available to tutor private candidates on weekday mornings.

Contact our friendly tutor Mr Wu: email at mathtuition88@gmail.com

Ad hoc O Level Maths Tuition

Nowadays students are often so busy that regular weekly tuition may not be an option. (Due to stay back in school, extra lessons etc.)

We offer Ad hoc O Level Maths Tuition at our tuition centre at Bishan. Students can come for tuition when they need to ask questions or clarify concepts.

Fees will be charged per lesson; this will fit the students’ busy schedule perfectly.

Visit this page to learn more: https://mathtuition88.com/maths-tuition-centre/

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.

Featured book:

Divine Proportions: Rational Trigonometry to Universal Geometry

Author: NJ Wildberger

Algebraic Topology Video by Professor N J Wildberger

This is the Introductory lecture to a beginner’s course in Algebraic Topology given by N J Wildberger of the School of Mathematics and Statistics at UNSW in 2010.

This first lecture introduces some of the topics of the course and three problems.

If you are curious about how to make the interesting flap of paper (Problem 1), the solution can be found here. 🙂

Featured book:

Divine Proportions: Rational Trigonometry to Universal Geometry

Author: N J Wildberger

Math Trivia: Sum of all the Roulette Numbers is 666 !

roulette4

The numbers on a roulette are from 1 to 36, including one zero (European style) or two zeroes (American style).

What is the sum of all the numbers on the roulette wheel? The answer is actually 666, the “number of the beast”!

$\boxed{1+2+3+\cdots +35+36=(37)(36)/2=666}$

Interesting? Perhaps this is a warning that gambling is not good?!

Check out this interesting Math video too:

Featured book:

Secrets of Mental Math: The Mathemagician’s Guide to Lightning Calculation and Amazing Math Tricks
These simple math secrets and tricks will forever change how you look at the world of numbers.

Secrets of Mental Math will have you thinking like a math genius in no time. Get ready to amaze your friends—and yourself—with incredible calculations you never thought you could master, as renowned “mathemagician” Arthur Benjamin shares his techniques for lightning-quick calculations and amazing number tricks. This book will teach you to do math in your head faster than you ever thought possible, dramatically improve your memory for numbers, and—maybe for the first time—make mathematics fun.

Yes, even you can learn to do seemingly complex equations in your head; all you need to learn are a few tricks. You’ll be able to quickly multiply and divide triple digits, compute with fractions, and determine squares, cubes, and roots without blinking an eye. No matter what your age or current math ability, Secrets of Mental Math will allow you to perform fantastic feats of the mind effortlessly. This is the math they never taught you in school.

Reviews:

“The clearest, simplest, most entertaining, and best book yet on the art of calculating in your head.” —Martin Gardner, author of Mathematical Magic Show and Mathematical Carnival

Filipino street kid math genius

Check out this video of a Filipino street kid, who can calculate square roots without a calculator, and even knows the concepts of perfect squares and imaginary numbers!

Compare and contrast with this video:

No matter whether you are good or not so good at Math, it is never too late to learn! It is always possible to improve in Math.

Featured book:

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

World’s Biggest Number that is actually used in a proof (Graham’s Number)

Check out this video on Graham’s Number — The World’s Biggest Number actually used in a mathematical proof! (Featuring Ron Graham himself!)

Featured book:

Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks

Magical Mathematics reveals the secrets of amazing, fun-to-perform card tricks–and the profound mathematical ideas behind them–that will astound even the most accomplished magician. Persi Diaconis and Ron Graham provide easy, step-by-step instructions for each trick, explaining how to set up the effect and offering tips on what to say and do while performing it. Each card trick introduces a new mathematical idea, and varying the tricks in turn takes readers to the very threshold of today’s mathematical knowledge. For example, the Gilbreath Principle–a fantastic effect where the cards remain in control despite being shuffled–is found to share an intimate connection with the Mandelbrot set. Other card tricks link to the mathematical secrets of combinatorics, graph theory, number theory, topology, the Riemann hypothesis, and even Fermat’s last theorem.

Diaconis and Graham are mathematicians as well as skilled performers with decades of professional experience between them. In this book they share a wealth of conjuring lore, including some closely guarded secrets of legendary magicians. Magical Mathematics covers the mathematics of juggling and shows how the I Ching connects to the history of probability and magic tricks both old and new. It tells the stories–and reveals the best tricks–of the eccentric and brilliant inventors of mathematical magic.Magical Mathematics exposes old gambling secrets through the mathematics of shuffling cards, explains the classic street-gambling scam of three-card monte, traces the history of mathematical magic back to the thirteenth century and the oldest mathematical trick–and much more.

Come join INSINC! Stand to win $$ for off-peak trips on MRT and LRT. More than $2,500,000 paid out so far!!

This is for Singaporean readers of my blog. Thanks for your reading and support. 🙂

Here is a good recommendation for those who take the train regularly, with chances to win cash prizes.

Take the train (MRT or LRT) regularly? You can win $$ for off-peak trips on MRT and LRT! More than $2,500,000 paid out so far!!

The advancement and perfection of mathematics are intimately connected with the prosperity of the state — Napoleon Bonaparte

Source: http://www.thehindu.com/features/education/careers/stay-ahead-with-math/article6252546.ece

From search engines to big data and cloud services, math plays a key role in IT applications. Read on to know more about the opportunities math has to offer.

In this increasingly digital world, mathematics is everywhere. It is wise to keep track of the myriad opportunities that would be laid open by mathematics education.

“The advancement and perfection of mathematics are intimately connected with the prosperity of the state,” said Napolean Bonaparte. While there may be several opinions regarding Napolean as a leader, this statement holds indisputably true even today.

Read more at: http://www.thehindu.com/features/education/careers/stay-ahead-with-math/article6252546.ece

Featured book:

The Princeton Companion to Mathematics

[News] NUS makes it easier for students in three faculties to qualify for honours

More than 80 or 90 per cent of students on four-year direct honours programmes at publicly-funded universities here graduate with honours or the equivalent. But only 60 per cent of those in the three-year arts and social sciences, business and science degree courses at the National University of Singapore (NUS) qualify for the fourth year of study, which allows them to graduate with honours.

To close the gap, NUS is lowering the grade to qualify for the honours year in these three schools, which are among the larger faculties in the university and take in some 3,600 students a year. This means another 10 to 15 per cent – 400 to 500 students- from these three faculties can move on to the fourth year to study for their honours.

Previously, students in the three faculties require a Cumulative Average Point (CAP) of 3.5 and above to qualify for honours study. With the change, they need only 3.2. NUS, though, will stick to its policy of keeping the the three plus one structure. Students who fail to notch up a score of at least 3.2 will have to exit the course.

NUS Provost Tan Eng Chye said the university decided to lower the requirement as the quality of students has gone up over the years. Students need As and Bs to enter most of the courses now. Last year, for example, students needed a ABB to enter the arts and social sciences course and those entering business needed triple As.

Source: NUS makes it easier for students in three faculties to qualify for honours

Featured book:

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

On time management

Terence Tao (Famous Mathematician and Field’s Medalist) on advice on time management. Suitable for students too!

Subscribe to our free newsletter by our Maths Tuition Centre: https://mathtuition88.com/free-newsletter/

Featured book:

The 7 Habits of Highly Effective Teens

With more than five million copies in print all around the world, The 7 Habits of Highly Effective Teens is the ultimate teenage success guide—now updated for the digital age.

Imagine you had a roadmap—a step-by-step guide to help you get from where you are now, to where you want to be in the future. Your goals, your dreams, your plans…they are all within reach. You just need the tools to help you get there.

That’s what Sean Covey’s landmark book, The 7 Habits of Highly Effective Teens, has been to millions of teens: a handbook to self-esteem and success. Now updated for the digital age, this classic book applies the timeless principles of the 7 Habits to the tough issues and life-changing decisions teens face. In an entertaining style, Covey provides a simple approach to help teens improve self-image, build friendships, resist peer pressure, achieve their goals, and get along with their parents, as well as tackle the new challenges of our time, like cyberbullying and social media. In addition, this book is stuffed with cartoons, clever ideas, great quotes, and incredible stories about real teens from all over the world.

An indispensable book for teens, as well as parents, teachers, counselors, or any adult who works with teens, The 7 Habits of Highly Effective Teenshas become the last word on surviving and thriving as a teen and beyond.

“If The 7 Habits of Highly Effective Teens doesn’t help you, then you must have a perfect life already.”–Jordan McLaughlin, Age 17

What's new

Prodded by several comments, I have finally decided to write up some my thoughts on time management here. I actually have been drafting something about this subject for a while, but I soon realised that my own experience with time management is still very much a work in progress (you should see my backlog of papers that need writing up) and I don’t yet have a coherent or definitive philosophy on this topic (other than my advice on writing papers, for instance my page on rapid prototyping). Also, I can only talk about my own personal experiences, which probably do not generalise to all personality types or work situations, though perhaps readers may wish to contribute their own thoughts, experiences, or suggestions in the comments here. [I should also add that I don’t always follow my own advice on these matters, often to my own regret.]

I can…

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Mini-monomath

Excellent and educational post by famous Mathematician Timothy Gowers on how to solve Math (Olympiad) problems.

(Post is at the bottom of this article)

Many students often give up immediately when facing a difficult maths problem. However, if students persist on for some time, usually they can come up with a solution or at least an idea on how to solve the problem. That is a great achievement already!

Never give up, even when your Maths question looks like this!

Quote: What I wrote gives some kind of illustration of the twists and turns, many of them fruitless, that people typically take when solving a problem. If I were to draw a moral from it, it would be this: when trying to solve a problem, it is a mistake to expect to take a direct route to the solution. Instead, one formulates subquestions and gradually builds up a useful bank of observations until the direct route becomes clear. Given that we’ve just had the football world cup, I’ll draw an analogy that I find not too bad (though not perfect either): a team plays better if it patiently builds up to an attack on goal than if it hoofs the ball up the pitch or takes shots from a distance. Germany gave an extraordinary illustration of this in their 7-1 defeat of Brazil.

Featured book (by Timothy Gowers):

The Princeton Companion to Mathematics

This is a one-of-a-kind reference for anyone with a serious interest in mathematics. Edited by Timothy Gowers, a recipient of the Fields Medal, it presents nearly two hundred entries, written especially for this book by some of the world’s leading mathematicians, that introduce basic mathematical tools and vocabulary; trace the development of modern mathematics; explain essential terms and concepts; examine core ideas in major areas of mathematics; describe the achievements of scores of famous mathematicians; explore the impact of mathematics on other disciplines such as biology, finance, and music–and much, much more.

Unparalleled in its depth of coverage, The Princeton Companion to Mathematics surveys the most active and exciting branches of pure mathematics, providing the context and broad perspective that are vital at a time of increasing specialization in the field. Packed with information and presented in an accessible style, this is an indispensable resource for undergraduate and graduate students in mathematics as well as for researchers and scholars seeking to understand areas outside their specialties.

Features nearly 200 entries, organized thematically and written by an international team of distinguished contributors

Presents major ideas and branches of pure mathematics in a clear, accessible style

Defines and explains important mathematical concepts, methods, theorems, and open problems

Introduces the language of mathematics and the goals of mathematical research

Covers number theory, algebra, analysis, geometry, logic, probability, and more

Traces the history and development of modern mathematics

Profiles more than ninety-five mathematicians who influenced those working today

Explores the influence of mathematics on other disciplines

Includes bibliographies, cross-references, and a comprehensive index

Contributors incude:

Graham Allan, Noga Alon, George Andrews, Tom Archibald, Sir Michael Atiyah, David Aubin, Joan Bagaria, Keith Ball, June Barrow-Green, Alan Beardon, David D. Ben-Zvi, Vitaly Bergelson, Nicholas Bingham, Béla Bollobás, Henk Bos, Bodil Branner, Martin R. Bridson, John P. Burgess, Kevin Buzzard, Peter J. Cameron, Jean-Luc Chabert, Eugenia Cheng, Clifford C. Cocks, Alain Connes, Leo Corry, Wolfgang Coy, Tony Crilly, Serafina Cuomo, Mihalis Dafermos, Partha Dasgupta, Ingrid Daubechies, Joseph W. Dauben, John W. Dawson Jr., Francois de Gandt, Persi Diaconis, Jordan S. Ellenberg, Lawrence C. Evans, Florence Fasanelli, Anita Burdman Feferman, Solomon Feferman, Charles Fefferman, Della Fenster, José Ferreirós, David Fisher, Terry Gannon, A. Gardiner, Charles C. Gillispie, Oded Goldreich, Catherine Goldstein, Fernando Q. Gouvêa, Timothy Gowers, Andrew Granville, Ivor Grattan-Guinness, Jeremy Gray, Ben Green, Ian Grojnowski, Niccolò Guicciardini, Michael Harris, Ulf Hashagen, Nigel Higson, Andrew Hodges, F. E. A. Johnson, Mark Joshi, Kiran S. Kedlaya, Frank Kelly, Sergiu Klainerman, Jon Kleinberg, Israel Kleiner, Jacek Klinowski, Eberhard Knobloch, János Kollár, T. W. Körner, Michael Krivelevich, Peter D. Lax, Imre Leader, Jean-François Le Gall, W. B. R. Lickorish, Martin W. Liebeck, Jesper Lützen, Des MacHale, Alan L. Mackay, Shahn Majid, Lech Maligranda, David Marker, Jean Mawhin, Barry Mazur, Dusa McDuff, Colin McLarty, Bojan Mohar, Peter M. Neumann, Catherine Nolan, James Norris, Brian Osserman, Richard S. Palais, Marco Panza, Karen Hunger Parshall, Gabriel P. Paternain, Jeanne Peiffer, Carl Pomerance, Helmut Pulte, Bruce Reed, Michael C. Reed, Adrian Rice, Eleanor Robson, Igor Rodnianski, John Roe, Mark Ronan, Edward Sandifer, Tilman Sauer, Norbert Schappacher, Andrzej Schinzel, Erhard Scholz, Reinhard Siegmund-Schultze, Gordon Slade, David J. Spiegelhalter, Jacqueline Stedall, Arild Stubhaug, Madhu Sudan, Terence Tao, Jamie Tappenden, C. H. Taubes, Rüdiger Thiele, Burt Totaro, Lloyd N. Trefethen, Dirk van Dalen, Richard Weber, Dominic Welsh, Avi Wigderson, Herbert Wilf, David Wilkins, B. Yandell, Eric Zaslow, Doron Zeilberger

Gowers's Weblog

The title of this post is a nod to Terry Tao’s four mini-polymath discussions, in which IMO questions were solved collaboratively online. As the beginning of what I hope will be a long exercise in gathering data about how humans solve these kinds of problems, I decided to have a go at one of this year’s IMO problems, with the idea of writing down my thoughts as I went along. Because I was doing that (and doing it directly into a LaTeX file rather than using paper and pen), I took quite a long time to solve the problem: it was the first question, and therefore intended to be one of the easier ones, so in a competition one would hope to solve it quickly and move on to the more challenging questions 2 and 3 (particularly 3). You get an average of an hour and a half per…

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Fundamental Theorem of Algebra – Numberphile

Professor David Eisenbud gives an excellent explanation of the Fundamental Theorem of Algebra!

In high school, we learnt that some quadratic equations (e.g. $x^2+1=0$ ) do not have real roots. However, by the Fundamental Theorem of Algebra, every polynomial equation of degree d has d complex roots! (counting multiplicity)

Featured book:

The Prince of Mathematics: Carl Friedrich Gauss

Learn about the boy who – could read and add numbers when he was three years old, – thwarted his teacher by finding a quick and easy way to sum the numbers 1-100, – attracted the attention of a Duke with his genius, and became the man who… – predicted the reappearance of a lost planet, – discovered basic properties of magnetic forces, – invented a surveying tool used by professionals until the invention of lasers. Based on extensive research of original and secondary sources, this historical narrative will inspire young readers and even curious adults with its touching story of personal achievement.

Maths Tuition Centre: University Math Textbooks Review

Calculus Math Textbooks Review

Most students taking science related courses like Engineering or Physics need to study at least one semester of Calculus. Calculus can be a rather difficult subject, and having a good textbook to learn from is half the battle won! 🙂

We review 3 of the Top Calculus Textbooks on Amazon.com:

Calculus

The Larson CALCULUS program has a long history of innovation in the calculus market. It has been widely praised by a generation of students and professors for its solid and effective pedagogy that addresses the needs of a broad range of teaching and learning styles and environments. Each title is just one component in a comprehensive calculus course program that carefully integrates and coordinates print, media, and technology products for successful teaching and learning.

2)

Calculus, 4th edition

This book by Michael Spivak is strongly recommended for Math Majors, or for students interested in learning the theory behind calculus. Includes the theory of epsilon-delta analysis.

3)

Thomas’ Calculus (13th Edition)
Thomas’ Calculus, Thirteenth Edition, introduces readers to the intrinsic beauty of calculus and the power of its applications. For more than half a century, this text has been revered for its clear and precise explanations, thoughtfully chosen examples, superior figures, and time-tested exercise sets. With this new edition, the exercises were refined, updated, and expanded–always with the goal of developing technical competence while furthering readers’ appreciation of the subject. Co-authors Hass and Weir have made it their passion to improve the text in keeping with the shifts in both the preparation and ambitions of today’s learners.

This text is designed for a three-semester or four-quarter calculus course (math, engineering, and science majors).

Featured Promotion:

Shop Amazon – New Textbooks – Save up to 40%

A Maths List of Formula to Remember

Are you looking for a list of Additional Mathematics (A Maths) Formulas to remember?

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

Updated to include: Supplementary Angles, Complementary Angles, and Half Angle Formulas for Trigonometry

Remember, memorizing the formula is not enough. We need to know how to apply and use the formula! (The next level is to know how to derive the formulas, but that will not be tested in the exams. 🙂 )

Featured book:

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

Do you really really hate Math? Is it your most dreaded subject?

Why not learn to love Math as it is pretty much a compulsory subject until high school? Read this book, it may change your mindset about Math. 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

Disclaimer: We are not related to mathstuition88.com

Recently, it has come to attention that there is another website (mathstuition88.com). (Note the letter ‘s’.)

We would like to reiterate that we are not related, affiliated, or associated in any way to that website.

Our official website (https://mathtuition88.com/) is registered with WordPress.com, and is created by founder Mr Wu.

Our website is also officially ranked on Teach 100, a Daily Ranking of Education Blogs.

Mathematics Memes and Cartoons

Source: http://www.siliconrepublic.com/careers/item/37443-career-memes-of-the-week-m/

This week’s career memes are an ode to mathematicians, the numerical wizards who use their knowledge to solve practical problems in disciplines such as business, commerce, technology, engineering and the sciences.

A mathematician’s job involves performing computations and analysing and interpreting data, reporting conclusions from a data analysis and using those findings to support or improve business decisions, and developing mathematical or statistical models to analyse data.

Many mathematicians work for governments or for private scientific and R&D companies.

Career memes of the week: mathematician

Mathematician meme

Number Theory Math Olympiad Question and Answer

Source: http://www.fen.bilkent.edu.tr/~cvmath/Problem/problem.htm

Check out June’s Math Olympiad Number Theory Problem (from Bilkent University):

Find all triples of positive integers (a, b, c) satisfying $(a^3+ b)(b^3 + a) = 2^c$ .

Give it a try, and then click on the solution to check your answer!

Featured book:

104 Number Theory Problems: From the Training of the USA IMO Team

This challenging problem book by renowned US Olympiad coaches, mathematics teachers, and researchers develops a multitude of problem-solving skills needed to excel in mathematical contests and in mathematical research in number theory. Offering inspiration and intellectual delight, the problems throughout the book encourage students to express their ideas in writing to explain how they conceive problems, what conjectures they make, and what conclusions they reach. Applying specific techniques and strategies, readers will acquire a solid understanding of the fundamental concepts and ideas of number theory.

Welcome to the Teach100 community!

Recently, I added the Maths Blog to the Teach100 website. Glad to know that the blog has been approved!

“Thank you for submitting Singapore Maths Tuition to the Teach100! Your blog has been approved and is currently ranked at #427 of 601 blogs. Congratulations! We recently reached our 500th blog, and are excited to add your blog to our growing community!”

http://teach.com/teach100/blogs/718-Singapore-Maths-Tuition

Teach.com

A Mathematician’s Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form

A Mathematician’s Lament is a short book on the pedagogics and philosophy of mathematics by Paul Lockhart, originally a research mathematician but for many years a math teacher at a private school. Characterised as a strongly worded opinion piece arguing for an intuitive and heuristic approach to teaching and the importance of mathematics teaching reforms, the book frames learning mathematics as an artistic and imaginative pursuit which is not reflected at all in the way the subject is taught in the American educational system.

The book was developed from a 25-page essay that was written in 2002, originally circulated in typewritten manuscript copies, and subsequently on the Internet.

Compound Interest (O Levels Maths Tuition)

Compound Interest

“Compound interest is the eighth wonder of the world. He who understands it, earns it … he who doesn’t … pays it.” – Albert Einstein

The formula for Compound interest is:

Total Amount = $\displaystyle\boxed{P(1+\frac{i}{100})^n}$

Where P=Principal amount (starting amount of money)

i = interest rate (in percent)

n = number of times compounded

We will illustrate this using an example:

Compound Interest Example Question

Source: Admiralty Secondary School Preliminary Examination 2011 Paper 2

Q: The cash price of a sports car is $420,000.

Mr Lionel buys it on compound interest loan terms. He pays a down payment of $300,000 and the balance at the end of 5 years with a compound interest rate of 5% per annum. Calculate the amount that Mr Lionel has to pay at the end of 5 years.

Solution:

Firstly, we have to find out what is the balance. The balance would be $420,000-$300,000=$120,000.

That is the Principal amount, i.e. P=120,000. The interest rate, i=5. n=5 since the number of times compounded is 5 (once each year).

Hence, Total Amount = $\displaystyle\boxed{P(1+\frac{i}{100})^n=120000(1+\frac{5}{100})^5=153153.79}$

In conclusion, he has to pay $153153.79 at the end of 5 years.

How is Compound Interest the Eighth Wonder of the World?

Imagine you have an amount of $1000. (P=1000)

And you manage to find a bank that pays 10% compound interest per annum. (i=10)

What happens after 50 years? (n=50)

Using the formula, Total Amount = $\displaystyle\boxed{P(1+\frac{i}{100})^n=1000(1+\frac{10}{100})^{50}=117390.85}$

The amount would become around $117,000! Isn’t it amazing? This is why Maths is useful and fun.

Check out our Cool Math page for more Math fun facts!

Maths Tuition Centre 2014

https://mathtuition88.com/group-tuition/ https://mathtuition88.com/

Maths Tuition @ Bishan starting in 2014.

Secondary 4 O Level E Maths and A Maths.

Patient and Dedicated Maths Tutor (NUS Maths Major 1st Class Honours, Dean’s List, RI Alumni)

Email: mathtuition88@gmail.com

Secondary 4 O Level E Maths and A Maths Group Tuition, Bishan

https://mathtuition88.com/group-tuition/
https://mathtuition88.com/

Maths Tuition @ Bishan starting in 2014.

Secondary 4 O Level E Maths and A Maths.

Patient and Dedicated Maths Tutor (NUS Maths Major 1st Class Honours, Dean’s List, RI Alumni)

Email: mathtuition88@gmail.com

Maths Challenge

Hi, do feel free to try out our Maths Challenge (Secondary 4 / age 16 difficulty):

Source: Anderson E Maths Prelim 2011

If you have solved the problem, please email your solution to mathtuition88@gmail.com .

(Include your name and school if you wish to be listed in the hall of fame below.)

Students who answer correctly (with workings) will be listed in the hall of fame. 🙂

Hall of Fame (Correct Solutions):

1) Ex Moe Sec Sch Maths teacher Mr Paul Siew

2) Queenstown Secondary School, Maths teacher Mr Desmond Tay

3) Tay Yong Qiang (Waiting to enter University)

On the road to make math fun: An army major who quit to become a mathematics teacher

Source: http://www.telegraphindia.com/1120625/jsp/calcutta/story_15629755.jsp#.Uq7JOJVDGDk

On the road to make math fun

MITA MUKHERJEE

Madanlal Baldevraj Ghai during the city leg of his tour. Picture by Sayantan Ghosh

An army major who quit to become a mathematics teacher has embarked on a self-funded tour of the country to promote the subject.

Madanlal Baldevraj Ghai, 70, stayed in a dormitory at Howrah station to keep costs down during the three days he spent in Calcutta recently, meeting officials of the primary and secondary board and the school education department to offer suggestions on how to make the study of mathematics more interesting.

“India has produced brilliant mathematicians not just in the Vedic and medieval ages but also in modern times. Unfortunately, for quite a few years, not many students have been pursuing the subject at the higher level, which has resulted in a decline in the number of top-quality mathematicians,” the former teacher at PMN College in Rajpura, Punjab, told Metro.

“We, the elderly mathematics teachers, need to reach out to students and guardians in every corner of the country to dispel the misconception that mathematics is dry and boring,” added Ghai, who has an MPhil in the subject and is pursuing his PhD at Punjabi University, Patiala.

His 50-day tour was also prompted by the Prime Minister declaring 2012 as the year of mathematics as a tribute to Srinivasa Ramanujan, the autodidact mathematician who died in 1920 at the age of 32.

Mr Heng, however, noted that how well a child does in school depends on how motivated he is.

Source: http://www.channelnewsasia.com/news/singapore/parents-urged-to-consider/898332.html

Minister for Education Heng Swee Keat has said parents should consider other factors apart from a school’s previous year cut-off point (COP) when helping their P6 children decide on which secondary school to choose.

Minister for Education Heng Swee Keat (Photo: MOE)

SINGAPORE: Minister for Education Heng Swee Keat has said parents should consider other factors apart from a school’s previous year cut-off point (COP) when helping their P6 children decide on which secondary school to choose.

Writing on his Facebook page, Mr Heng said it would be good for parents to have an open talk with their children to know what type of secondary school they are interested in.

…

Mr Heng, however, noted that how well a child does in school depends on how motivated he is.

So he encourages parents to carefully consider the kind of environment that will best motivate their children, and enable them to develop themselves fully in the next four to five years.

Some children, he said, are late developers and the right environment helps them thrive.

Mr Heng urged parents to think of how best they can help their children develop confidence and enjoy the space to discover his talents and passions.

Continue reading at http://www.channelnewsasia.com/news/singapore/parents-urged-to-consider/898332.html

Recommended Maths Olympiad Books for Self Learning / Domain Test

Math Olympiad Books are useful for GEP/DSA preparation. It is also useful for the latest type of test called Domain Tests, which is basically a subject test (Math included) for entry into top secondary schools like the Raffles / Hwa Chong family. There are different subject domains (depending on the school), ranging from General domain / Academic domain / CCA domain.

A First Step to Mathematical Olympiad Problems (Mathematical Olympiad Series)

The Art of Problem Solving, Vol. 1: The Basics

The first book is written by Professor Derek Holton. Prof Holton writes a nice column for a Math magazine, which gives out books as prizes to correct solutions.

GEP Math Olympiad Books

If you are searching for GEP Math Olympiad Books to prepare for the GEP Selection Test, you may search for Math Olympiad Books for Elementary School. Note that Math Olympiad Books for IMO (International Mathematics Olympiad) are too difficult even for a gifted 9 year old kid!

A suitable book would be The Original Collection of Math Contest Problems: Elementary and Middle School Math Contest problems. It covers the areas of Algebra, Geometry, Counting and Probability, and Number Sense, over 500 examples and problems with fully explained solutions.