Math Blog

Singapore Education News

PSLE tweaks will come but as part of broader changes to education system: Heng
Straits Times
SINGAPORE – Changes being made to the Primary School Leaving … be done in the light of the broader changes to Singapore’s education system, …
SMU to broaden learning for freshmen
Straits Times
Freshmen entering the Singapore Management University (SMU) in August next year will go through a revamped syllabus, in the university’s bid to …
MOE to focus on tertiary, secondary education before turning to PSLE
Channel News Asia
SINGAPORE: With the Character and Citizenship Education syllabus being rolled out in all schools, the Ministry of Education (MOE) will tilt its focus …

Featured Product:


Yamie Chess School Assistant: K-8 Supplemental Math Learning Toy

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  • Written by experienced math teachers and a United States Chess Champion for K-8 supplemental math learning and K-8 math practice
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  • With contribution from the Harry Potter chess consultant, American International Master Jeremy Silman, creator of the Harry Potter chess scene in Harry Potter and the Sorcerer’s Stone (Warner Bros. Pictures, 2001)

AM-GM inequality

AM-GM inequality

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

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

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

The AM-GM Inequality states that:

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

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

am-gm-inequality

How to Apply?

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

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

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

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

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

Featured book from Amazon:

Competition Math for Middle School

Written for the gifted math student, the new math coach, the teacher in search of problems and materials to challenge exceptional students, or anyone else interested in advanced mathematical problems. Competition Math contains over 700 examples and problems in the areas of Algebra, Counting, Probability, Number Theory, and Geometry. Examples and full solutions present clear concepts and provide helpful tips and tricks. “I wish I had a book like this when I started my competition career.” Four-Time National Champion MATHCOUNTS coach Jeff Boyd “This book is full of juicy questions and ideas that will enable the reader to excel in MATHCOUNTS and AMC competitions. I recommend it to any students who aspire to be great problem solvers.” Former AHSME Committee Chairman Harold Reiter

Introduction to Category Theory 范畴论

tomcircle's avatarMath Online Tom Circle

[Source: ] All lectures & exercises here:
http://ureddit.com/class/36451

image

Introduction to Category Theory 1:

Course Overview:

Category Theory = Abstract Algebra of Functions

Lambda Calculus = Calculus of Functions

Lambda Calculus = Category

History:

image

image

$latex cap bigotimes$

Introduction to Category Theory (2) Monoids 么群


Introduction to Category Theory (3)Real lecture begins from here: Categories, Functors, Natural Transformation:

1. Category Definition:

image

1a) Examples of Categories:
image

Excellent example on “Natural Transformation“:

image

Ref: Classic Textbook

image

范畴论 Category Theory :
image

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How many Pentagons and Hexagons are there on a Soccer Ball?

Watch the above video to prove that there has to be 12 Pentagons and 20 Hexagons on a Soccer Ball!

The video also teaches us about the beautiful Euler Formula, \boxed{V-E+F=2}.

Featured Book:

Euler: The Master of Us All (Dolciani Mathematical Expositions, No 22)

An ideal book for enlivening undergraduate mathematics…he (Dunham) has Euler dazzling us with cleverness, page after page. — Choice

Mathematician William Dunham has written a superb book about the life and amazing achievements of one of the greatest mathematicians of all time. Unlike earlier writings about Euler, Professor Dunham gives crystal clear accounts of how Euler ingeniously proved his most significant results, and how later experts have stood on Euler’s broad shoulders. Such a book has long been overdue. It will not need to be done again for a long long time. — Martin Gardner

William Dunham has done it again! In “Euler: the Master of Us All”, he has produced a masterful portrait of one of the most fertile mathematicians of all time. With Dunham’s beautiful clarity and wit, we can follow with amazement Euler’s strokes of genius which laid the groundwork for most of the mathematics we have today. — Ron Graham, Chief Scientist, AT&T

William Dunham has written a superb book about the life and amazing achievements of one of the greatest mathematicians of all time. Unlike earlier writings about Euler, Dunham gives crystal clear accounts of how Euler ingeniously proved his most significant results, and how later experts have stood on Euler’s broad shoulders. Such a book has long been overdue. It will not need to be done again for a long, long time.Martin Gardner

Dunham has done it again! In “Euler: The Master of Us All,” he has produced a masterful portrait of one of the most fertile mathematicians of all time. With Dunham’s beautiful clarity and wit, we can follow with amazement Euler’s strokes of genius which laid the groundwork for most of the mathematics we have today. — Ronald Graham, Chief Scientist, AT&T

Eigenvector & Eigenvalue

tomcircle's avatarMath Online Tom Circle

1. Matrix (M): stretch & twist space
2. Vector (v): a distance along some direction
3. M.v = v’ stretched & twisted by M

Some directions are special:-
a) v stretched but not twisted = Eigenvector;
b) The amount of stretch = constant = Eigenvalue (λ)

Let M the matrix, λ its eigenvalue,
v eigenvector.
By definition: M.v = λ.v
v = I.v (I identity matrix)
M.v = λI.v
(M – λI).v=0
As v is non-zero,
1. Determinant (M- λI) =0 => find λ
2. M.v = λ.v => find v

Note1: Why call Eigenvalue ?
From German: “Die dem Problem eigentuemlichen Werte”
= “The values belonging to this problem”
=> eigenWerte = EigenValue
Eigenvalue also called ‘characteristic values’ or ‘autovalues’.
Eigen in English = Characteristic (but already used for Field).

Note2: Schrödinger Quantum equation’s Eigenvalue = Maximum probability of electron presence at the orbit…

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Relationship-Mapping-Inverse (RMI)

tomcircle's avatarMath Online Tom Circle

Relationship-Mapping-Inverse (RMI)
(invented by Prof Xu Lizhi 徐利治 中国数学家 http://baike.baidu.com/view/6383.htm)

Find Z = a*b

By RMI Technique:
Let f Homomorphism: f(a*b) = f(a)+f(b)

Let f = log
log: R+ –> R
=> log (a*b) = log a + log b

1. Calculate log a (=X), log b (=Y)
2. X+Y = log (a*b)
3. Find Inverse log (a*b)
4. ANSWER: Z = a*b

Prove:

$latex sqrt{2}^{sqrt{2}^{sqrt{2}}}= 2$

1. Take f = log for Mapping:
$latex logsqrt{2}^{sqrt{2}^{sqrt{2}}} $
$latex = sqrt{2}logsqrt{2}^{sqrt{2}}$
$latex = sqrt{2}sqrt{2}logsqrt{2} $
$latex = 2logsqrt{2} $
$latex = log (sqrt{2})^2 $
$latex = log 2$

2. Inverse of log (bijective):
$latex log sqrt{2}^{sqrt{2}^{sqrt{2}}}= log 2$
$latex sqrt{2}^{sqrt{2}^{sqrt{2}}}= 2$

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Math Handheld Computer Game

Featured Item:

Educational Insights Math Whiz

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

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

 

New Geometry 新几何

tomcircle's avatarMath Online Tom Circle

New Geometry (新几何) invented by Zhang JingZhong (張景中) derived from 2 basic theorems:

1) Triangles internal angles =180º

2) Triangle Area = ½ base * height
=> derive all geometry
=> trigonometry
=> algebra
(These 3 maths are linked, unlike current syllabus taught separately)

The powerful Area (Δ) Proof Techniques:

1) Common Height:
Line AMB, P outside line
Δ PAM / Δ PBM = AM/BM

2) Common 1 Side (PQ):
Lines AB and PQ meet at M
Δ APQ /Δ BPQ = AM/BM

3) Common 1 Angle:
∠ABC=∠XYZ (or ∠ABC+∠XYZ = ∏ )
Δ ABC /Δ XYZ= AB.BC /XY.YZ

These 3 theorems can prove Butterfly and tough IMO problems.

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Butterfly Theorem

tomcircle's avatarMath Online Tom Circle

Butterfly Theorem

In a circle draw a chord PQ with mid-point M. Through M draw 2 chords AB, CD. Join AD, BC cut PQ at X, Y resp. (Butterfly M)

1. Prove: M = mid-point of XY

http://gogeometry.com/GeometryButterfly.html

2. If circle changed to ellipse, still true?

Yes. Affine transformation from circle elongated to ellipse, like distorted image through funny mirror => still MX = MY

Butterfly theorem Butterfly theorem (Pho

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Minimum Memorize in Math – Go by 1st Principle

tomcircle's avatarMath Online Tom Circle

Do not remember these:
$latex boxed {
cos 3A = 4cos^{3} A – 3cos A
}$

$latex boxed {displaystyle
int frac {dx}{sec x}
=
int
frac {1}{sec x}
frac {sec x + tan x}{sec x + tan x}dx
}&fg=aa0000
$

However, it helps, though, to remember:
“Nine Zulu Queens Rule China”
$latex boxed {
mathbb{N}subset mathbb{ Z }subset mathbb{ Q }subset mathbb{ R} subset mathbb{ C }
}&fg=00bb00&s=3
$

How Much Mathematics Should a Student Memorize?

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NUS Math Ranked among Top in Asia

In the latest Quacquarelli Symonds (QS) World University Rankings by Subject (2014), NUS Math is ranked among the best mathematics departments in Asia.

nus ranking

Featured book:

Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States (Studies in Mathematical Thinking and Learning Series)

Chinese students typically outperform U.S. students on international comparisons of mathematics competency. Paradoxically, Chinese teachers receive far less education than U.S. teachers–11 to 12 years of schooling versus 16 to 18 years of schooling.

Studies of U.S. teacher knowledge often document insufficient subject matter knowledge in mathematics. But, they give few examples of the knowledge teachers need to support teaching, particularly the kind of teaching demanded by recent reforms in mathematics education.

This book describes the nature and development of the “profound understanding of fundamental mathematics” that elementary teachers need to become accomplished mathematics teachers, and suggests why such teaching knowledge is much more common in China than the United States, despite the fact that Chinese teachers have less formal education than their U.S. counterparts.

How Much Mathematics Should a Student Memorize?

As Chinese we are good at memorizing poems since young (think of reciting 唐诗300首 – no sweat! ).
By chanting in Hokkien I remember complicated trigo formula:)
Cos 3A = 4Cos^3 A – 3.Cos A
($ 1.3 = $4.3 -$3 with $ =Cos in Hokkien sound).
Once I performed this ‘memory power’ in a lecture hall with me called up by the professor to solve a Physics problem. Halfway in the computation we need to open up the “Cos 3A”, the prof asked the whole class to help me, but I wrote the above down quickly on the white board to the awe of the class. Guess what, when the prof saw it, instead of praising me I got a scolding ! He said “Do not keep unnecessary things in your head”. The prof is a French, he did not know I have Hokkien language advantage… haha.
Anyway, joke aside, try to memorize minimum, go by first principle so that you will never ever forget iin whole life (yes, I remember them now after 40 years).

How Much Mathematics Should a Student Memorize? Part 2, Integral Calculus

This math teacher is excellent in teaching the students to memorize minimum. His example is integrate secant. Most textbooks use a trick ie multiply (sec + tan) above and below, then by substitution. He goes by first principle, change sec = 1/cos, then try to use 2 common trigo sine and cosine, he multiplies cos above & below to make: sec = cos / 1-sin^2,… then integrate by part…

The Boy With The Incredible Brain – Autism Math Documentary

This is the breathtaking story of Daniel Tammet. A twenty-something with extraordinary mental abilities, Daniel is one of the world’s few savants. He can do calculations to 100 decimal places in his head, and learn a language in a week.

He also meets the world’s most famous savant, the man who inspired Dustin Hoffman’s character in the Oscar winning film ‘Rain Man’.

This documentary follows Daniel as he travels to America to meet the scientists who are convinced he may hold the key to unlocking similar abilities in everyone.

Featured books by Daniel Tammet:

Born On A Blue Day: Inside the Extraordinary Mind of an Autistic Savant

Bestselling author Daniel Tammet (Thinking in Numbers) is virtually unique among people who have severe autistic disorders in that he is capable of living a fully independent life and able to explain what is happening inside his head.

He sees numbers as shapes, colors, and textures, and he can perform extraordinary calculations in his head. He can learn to speak new languages fluently, from scratch, in a week. In 2004, he memorized and recited more than 22,000 digits of pi, setting a record. He has savant syndrome, an extremely rare condition that gives him the most unimaginable mental powers, much like those portrayed by Dustin Hoffman in the film Rain Man.


Thinking In Numbers: On Life, Love, Meaning, and Math

The irresistibly engaging book that “enlarges one’s wonder at Tammet’s mind and his all-embracing vision of the world as grounded in numbers.” –Oliver Sacks, MD
THINKING IN NUMBERS is the book that Daniel Tammet, mathematical savant and bestselling author, was born to write. In Tammet’s world, numbers are beautiful and mathematics illuminates our lives and minds. Using anecdotes, everyday examples, and ruminations on history, literature, and more, Tammet allows us to share his unique insights and delight in the way numbers, fractions, and equations underpin all our lives.

 

 

The Mathematical Dialect Quiz

Very interesting Math jokes!

Ben Orlin's avatarMath with Bad Drawings

1
  1. What do you call a rigorous demonstration that a statement is true?
    1. If “proof,” then you’re a mathematician
    2. If “experiment,” then you’re a physicist
    3. If you have no word for this concept, then you’re an economist

2

  1. What do you call a slow, painful, computationally intense method of solving a problem?
    1. If “engineering,” then you’re a mathematician
    2. If “mathematics,” then you’re an engineer

3

View original post 381 more words

Look for 4th Solution ? “The Monkey and Coconuts” Problem

Let me know (in the comment below) if there is a 4th solution – I believe there is a simpler and creative solution.

tomcircle's avatarMath Online Tom Circle

Think of the 4th solution, if any, for this
“Monkey & Coconuts” Problem.
image

It was created by Nobel Physicist Prof Paul Dirac,  which he told another Chinese Nobel Physicist Prof Li ZhengDao (李政道)。
Pro Li wanted to test the Chinese young students in the first China Gifted Children University of 13 year-old kids, none of them could solve this problem (proved they are not so gifted after all for unknown problems :)

The first 2 solutions were solved by Prof Paul Richard Halmos,  the 3rd solved by myself using the Singapore Modelling Math (a modified version of Arithmetics from traditional Math taught in 1970s Chinese Secondary 1 “中学数学” in Singapore).

1st Solution: Higher Math: Sequence

https://tomcircle.wordpress.com/2013/03/30/monkeys-coconuts-problem/

2nd Solution: Linear Algebra: Eigenvalue and Eigenvector
https://tomcircle.wordpress.com/2013/03/30/solution-2-monkeys-coconuts/

3rd Solution: Singapore Modelling Math for PSLE (Primary 6)

https://tomcircle.wordpress.com/2013/03/30/solution-3-best-monkeys-coconuts/

4th Solution:
Any ?

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Modern Algebra (Abstract Algebra) Made Easy

tomcircle's avatarMath Online Tom Circle

UReddit Courses:

Modern Algebra (Abstract Algebra) Made Easy –

Part 0: Binary Operations

Part 1: Group

Part 2: Subgroup

Part 3: Cyclic Group & its Generator

Part 4: Permutations

Part 5: Orbits & Cycles

Part 6 : Cosets & Lagrange’s Theorem

Part 7 : Direct Products / Finite generated Abelian groups

Part 8: Group Homomorphism

Part 9: Quotient Groups

Part 10: Rings & Fields

Part 11: Integral Domain

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Notes on Coordinate Geometry by Hwa Chong Institution

Coordinate Geometry Notes

Source: http://mathace2012.wiki.hci.edu.sg/

Check out these Formulas for:

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

at http://mathace2012.wiki.hci.edu.sg/

formula.jpg

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

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

This Schaum’s Outline gives you

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

 

Monster Group – 196,883 dimensions – “The Voice of God”

tomcircle's avatarMath Online Tom Circle

Monster Group (code name “Moonshine”) is the largest group, discovered by two Cambridge Mathematicians John Conway and Simon Norton.

Monster Group – (1)

Monster Group (2):

John Conway: Life, Death and the Monster (3)

Ref:
1. Simon Norton (1952 -) – an eccentric mathematician who collects all British Railway Train Time Tables.
http://catalogue.nlb.gov.sg/cgi-bin/spydus.exe/ENQ/EXPNOS/BIBENQ?ENTRY=The%20genius%20in%20my%20basement&ENTRY_NAME=BS&ENTRY_TYPE=K&SORTS=DTE.DATE1.DESC%5DHBT.SOVR

image

2.
Finding Moonshine: A Mathematician’s Journey Through Symmetry by Marcus Du Sautoy

image

http://catalogue.nlb.gov.sg/cgi-bin/spydus.exe/FULL/EXPNOS/BIBENQ/6345422/5640834,2

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Chinese Lucky Numbers – Numberphile

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

Featuring Xiaohui Yuan from the University of Nottingham.

Website: http://www.numberphile.com/
Numberphile on Facebook: http://www.facebook.com/numberphile
Numberphile tweets: https://twitter.com/numberphile

Videos by Brady Haran

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

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

Featured book:

Number: The Language of Science

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

– Rated 4.5/5 on Amazon

News: Math, reading performance is stagnant among US 12th-graders, assessment finds

Math, reading performance is stagnant among US 12th-graders, assessment finds
Washington Post
The nation’s high school seniors have shown no improvement in math and reading performance since 2009, and large racial achievement gaps …
Math, reading performance stagnates among US 12th-graders, assessment finds
Business Mirror
WASHINGTON—The United States’s high-school seniors have shown no improvement in math and reading performance since 2009, and large racial …
Social Math: Why Learning Math Involves More Than Writing Numbers
Huffington Post
My lifelong passion for creating better ways to learn math got its start in a school not unlike the Israeli one I visited — and at the very same age.
Math games aim to keep kids sharp over summer
Scranton Times-Tribune
JAKE DANNA STEVENS / STAFF PHOTOGRAPHER Kaylynn Howe, 8, left, and her sister, Destine Howey, play a math game at Bancroft Elementary …
Common Core math gains are worth the pain
New York Daily News
As more than a million New York students in grades 3-8 took the state math exams last week, a small but vocal cadre of parents railed at the new …
Lindblom Math and Science principal among Golden Apple winners
Chicago Tribune
The organization quickly realized that Mather, principal of Lindblom Math and Science Academy In Chicago’s West Englewood neighborhood was the …
Math Learning – A Universal Language?
Huffington Post
Fifth-grade students at Woodward Elementary School had an interesting math assignment this fall: watching college football games. Though …
US students’ reading, math show no progress
TheChronicleHerald.ca
WASHINGTON — In an abysmal showing, only about one-quarter of U.S. high school seniors performed solidly in math in a major assessment known …
Using math in the fight against cancer
WNYT
But a local college professor says many people don’t like math because they don’t see a connection to it. In this Friday’s STEM 13 report, learn how the …
Math Day at Molly Stark honors memory of Gail Harwood
Bennington Banner
BENNINGTON — The Molly Stark School honored a longtime teacher on Monday with “Gail Harwood Math Day.” Harwood passed away in January …

News: Singapore Education Ranked Third in World

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

Monster Group

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

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

Featured book:

The Symmetries of Things

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

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

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

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

 

How to prove square root of 2 is irrational?

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

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

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

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

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

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

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

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

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

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

irrational

Featured book:

Math Jokes 4 Mathy Folks

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

– Highly rated on Amazon.com

 

What to do if fail Mid Year Exams?

Source: http://news.asiaone.com/News/Education/Story/A1Story20080601-68205.html

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

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

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

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

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

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

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

Read more at: http://news.asiaone.com/News/Education/Story/A1Story20080601-68205.html

 

 

More on Linguistic “Half Life”

tomcircle's avatarMath Online Tom Circle

Proto Indo-European and Chinese in the Late Neolithic Age
后新石器时代的原欧-印语与汉语

Tsung-tung Chang[張聰東] 1988:
“Indo-European vocabulary in Old Chinese: A new thesis on the emergence of Chinese language and civilization in the Late Neolithic Age”, Sino-Platonic Papers 7, Philadelphia.

This Chinese scholar wrote the 1988 paper on the Chinese language origin with the proto-Indo-European (proto IE).

Interestingly very similar ‘coincidence’ occurs in 1500 words between Chinese and proto IE:

Take -> 得 tek (ancient Chinese sound as in Fujian dialect today)
Mort -> 殁 mo
See -> 视 see
Cow -> 牛 gu
…

Click to access spp007_old_chinese.pdf

After the Tower of Babel, God confused the human into different languages, but by the linguistic ‘archaeology’ ‘Half Life’ Theory, we can deduce ~ 4,900 years ago the Chinese and the Germanic (English, Denmark, German …) shared the same common linguistic root.

The ancient Chinese scholar Xu Shen许慎(东汉 : 58 CE…

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Grothendieck’s Sheaf (束)

tomcircle's avatarMath Online Tom Circle

Natural Numbers (N) = {1,2,3, 4…}
1-dimension: a Line
2-dimension: a plane
n-dimensional flat space: a Vector Space

Now imagine in a world where we replace every natural number by vector space:
1 by a Line
2 by a Plane
n by a flat space Vector Space

Sum of numbers = Direct sum of vector space.
E.g. Add a 1-D Line to a 2-D Plane = 3-D Space

Product of numbers = Tensor Product (of two vector spaces of respective dimension m & n) with dimension m.n

This new world would be much interesting and richer than the Natural Number world: vector spaces have symmetries, whereas numbers are just numbers with no symmetries.
(Interesting): we can add 1 to 2 in only ONE way (1+2), but there are many ways to embed (add) a Line in a Plane (perpendicular, slanting in any angle, etc).
(Richer): the Lie Group SO(3)…

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Studies and Studying: How do top students study?

Source: http://www.quora.com/Studies-and-Studying/How-do-top-students-study/answer/Qiaochu-Yuan-1

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

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

This is an excerpt of the thread:

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

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

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

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

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

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

– Qiaochu Yuan

Math News: Math student from Nanyang Technological University detects OAuth, OpenID security vulnerability

Is it safe to log in through well known sites such as Facebook and Google? Think again, for Wang Jing, a PhD student in mathematics at the Nanyang Technological University in Singapore, has detected critical security vulnerabilities in the OAuth, OpenID security protocols. (Source: http://phys.org/news/2014-05-math-student-oauth-openid-vulnerability.html) [Second article in the list below]

Forward this information to your friends via the Tweet button below to warn them of the potential danger!

Unique Math Learning Center Opens in Lake Forest: Local After-School Program to Provide …
Chicago Tribune
Mathnasium – The Math Learning Center opened its doors in Lake Forest in March to students looking for math help and math enrichment. The new …
Math student detects OAuth, OpenID security vulnerability
Phys.Org
(Phys.org) —To get right to the point, a doctoral candidate in math has discovered two holes in OAuth and OpenID that could leak data and redirect …
A math lesson for city: Teachers’ contract likely to cost billions
SILive.com
Mayor Bill de Blasio, who visited Staten Island Wednesday evening to speak at an SIEDC cocktail reception at the Hilton Garden Inn, has cleared his …
Math error halts Bank of America’s stock buy back and dividend increase
UPI.com
WILMINGTON, Del., April 28 (UPI) — Bank of America has had to halt its proposed stock buy back and dividend increase because of a math error in its …
Philadelphia girl makes math a game – and excels
Philly.com
Josephine Nyugen, a sixth-grader at St. Cecelia, right, with her father, Joseph, left, plays the math game ‘Into the Vortex’ on the First in Math website.
Math for public works borrowing bill proves tough
St. Cloud Times
Lawmakers from the Minnesota House Capital Investment Committee look around the prison yard near the loading dock inside the Minnesota …
Houghton Mifflin Harcourt takes Go Math! Academy into the home market
Boston Globe
Houghton Mifflin Harcourt, a Boston-based education company, developed the program for students in kindergarten to sixth grade to learn basic math …
Steve Ballmer’s math on Apple innovation doesn’t add up
ZDNet
His math of Apple innovation appears lacking to this longtime Mac user. Let me add a few more of his “tricks” to the list: The Apple II platform. Ballmer …
Math class helps special needs student try to win a wheelchair van
KOB.com
There’s a lot more going on than just addition and subtraction in Mr. Green’s math class. “We just want to help out this family,” Said Cody Green.
Budget math takes a U-turn: Christie blames federal tax law that brought windfall last year
NorthJersey.com
The same federal tax policy that Governor Christie is now blaming for New Jersey’s $807 million budget shortfall helped save his budget last year …

Applied Math in Medicine

tomcircle's avatarMath Online Tom Circle

The young Russian doctor Sergei Arutyunyan was working with patients whose immune systems were rejecting transplanted kidneys.

The doctor has to decide whether to keep or remove it. If they kept the kidney, the patient could die, but if they remove it, the patient would need another long wait (or never) for another kidney.

The mathematician Edward Frankel helped him to analyze the collected data with ‘expert rules’ in a decision tree. (Note: this is like the Artificial Intelligence Rule-based Expert System, except no fuzzy math).

image

Love and Math by Edward Frenkel http://www.amazon.co.uk/dp/0465050743/ref=cm_sw_r_udp_awd_53swtb16779PY

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Education News Update

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

“Turn-off” School Math

tomcircle's avatarMath Online Tom Circle

“…There’s a long history of high caliber mathematicians finding their experiences with school mathematics alienating or irrelevant. “
Read here:
http://lesswrong.com/r/discussion/lw/2uz/fields_medalists_on_school_mathematics/

In Récoltes et Semailles Fields Medalist Alexander Grothendieck describes an experience of the type that Alain Connes mentions:

I can still recall the first “mathematics essay” (math test, or Composition Mathématique) , and that the teacher gave it a bad mark. It was to be a proof of “three cases in which triangles were congruent.” My proof wasn’t the official one in the textbook he followed religiously. All the same, I already knew that my proof was neither more nor less convincing than the one in the book, and that it was in accord with the traditional spirit of “gliding this figure over that one.” It was self-evident that this man was unable or unwilling to think for himself in judging the worth of a train of…

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The Gap of Today’s Math Education: Rigor

tomcircle's avatarMath Online Tom Circle

This professor criticized the lack of rigor in today’s math education, in particular, there exists universally a prevalent ‘ambiguous’ gap between high school and undergraduate math education.
image

I admire his great insight which is obvious to those postwar baby boomer generation.

I remember I was the last Singapore batch or so (early 70s) taking the full Euclidean Geometry course at 15 years old, and strangely in that year of Secondary 3 Math (equivalent to 3ème in Baccalaureate) my (Chinese) school had 2 separate math teacher for Geometry and Elementary/Additional (E./A.) Math.

Guess what ? the Geometry teacher was an Art teacher. It turned out it was a blessing in disguise, as my class of average Math students who hated E./A. Maths all scored 90% distinctions in Geometry. We did not treat Geometry like the other boring maths. The lady Art teacher started on the first day from Euclid’s 5 axioms…

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Riemann Hypothesis Proof

Latest News: Riemann Hypothesis Proved?

Source: http://arxiv.org/abs/1402.5952

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

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

His paper can be downloaded here: http://arxiv.org/pdf/1402.5952v2

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

What is the Riemann Hypothesis about? Watch this Youtube Video:

Featured book:

Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics

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

Rated: 4.5 stars on Amazon