Proof that Square Root of Two is Irrational [Rare Constructive Proof] (YouTube Video)

This is my second ever YouTube Video, and it is about a rarely seen proof (Constructive) that Square Root of Two is irrational.

The video is based on my earlier post on: Constructive Proof that Square Root of Two is irrational

Thanks for watching! Please speed up the video according to your preference! You can speed up either 2x or 1.5x for best effect.

Featured Book:

The Irrationals: A Story of the Numbers You Can’t Count On

The ancient Greeks discovered them, but it wasn’t until the nineteenth century that irrational numbers were properly understood and rigorously defined, and even today not all their mysteries have been revealed. In The Irrationals, the first popular and comprehensive book on the subject, Julian Havil tells the story of irrational numbers and the mathematicians who have tackled their challenges, from antiquity to the twenty-first century. Along the way, he explains why irrational numbers are surprisingly difficult to define–and why so many questions still surround them. Fascinating and illuminating, this is a book for everyone who loves math and the history behind it.

Intellectual wealth

An interesting news to share:


Billionaire Ronnie Chan rather be mathematician or scientist if he could live life over

Billionaire Ronnie Chan Chi-chung seems to have it all figured out. Were the Hang Lung Properties chairman to live his life over again, it would not be as a businessman — he’d be a mathematician or scientist instead.

Chan, who offered this little gem during a speech at the Hang Lung Mathematics Awards ceremony, said he may have more material wealth than famed mathematician Yau Shing-tung but much less intellectual wealth.

As co-founder of the awards, which were set to encourage secondary school students to pursue maths and sciences, Chan urged youngsters to go the extra mile and become mathematicians or scientists as they can contribute more to society than what a businessman can.

Read more at:

Hopefully this can encourage students currently studying Maths, be it O Level Maths, JC H1 or H2 Maths, or even University Maths!

Featured Book:

The Shape of Inner Space: String Theory and the Geometry of the Universe’s Hidden Dimensions

New Scientist
“It is a testimony to [Yau’s] careful prose (and no doubt to the skills of co-author Steve Nadis) that this book so compellingly captures the essence of what pushes string theorists forward in the face of formidable obstacles. It gives us a rare glimpse into a world as alien as the moons of Jupiter, and just as fascinating…. Yau and Nadis have produced a strangely mesmerizing account of geometry’s role in the universe.”

“Physicists investigate one cosmos, but mathematicians can explore all possible worlds. So marvels Fields medalist Shing-Tung Yau…. Relating how he solved a major theoretical problem in string theory in the 1970s, Yau explains how the geometries of the vibrating multidimensional strings that may characterize the Universe have implications across physics.”

Mathematicians prove the Umbral Moonshine Conjecture

Source: Science Daily

Mathematicians prove the Umbral Moonshine Conjecture

Date: December 15, 2014

Source: Emory University

Summary: Monstrous moonshine, a quirky pattern of the monster group in theoretical math, has a shadow — umbral moonshine. Mathematicians have now proved this insight, known as the Umbral Moonshine Conjecture, offering a formula with potential applications for everything from number theory to geometry to quantum physics.

“We’ve transformed the statement of the conjecture into something you could test, a finite calculation, and the conjecture proved to be true,” says Ken Ono, a mathematician at Emory University. “Umbral moonshine has created a lot of excitement in the world of math and physics.”

Co-authors of the proof include mathematicians John Duncan from Case Western University and Michael Griffin, an Emory graduate student.

“Sometimes a result is so stunningly beautiful that your mind does get blown a little,” Duncan says. Duncan co-wrote the statement for the Umbral Moonshine Conjecture with Miranda Cheng, a mathematician and physicist at the University of Amsterdam, and Jeff Harvey, a physicist at the University of Chicago.

Ono will present their work on January 11, 2015 at the Joint Mathematics Meetings in San Antonio, the largest mathematics meeting in the world. Ono is delivering one of the highlighted invited addresses.

Read more at: Science Daily

Featured Book:

Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics (Cambridge Monographs on Mathematical Physics)


“An excellent introduction to this area for anyone who is looking for an informal survey… written in a lively and readable style.”
R.E. Boucherds, University of California at Berkeley for the Bulletin of the AMS

“It is written in a breezy, informal style which eschews the familiar Lemma-Theorem-Remark style in favor of a more relaxed and continuous narrative which allows a wide range of material to be included. Gannon has written an attractive and fun introduction to what is an attractive and fun area of research.”
Geoffrey Mason, Mathematical Reviews

“Gannon wants to explain to us “what is really going on.” His book is like a conversation at the blackboard, with ideas being explained in informal terms, proofs being sketched, and unknowns being explored. Given the complexity and breadth of this material, this is exactly the right approach. The result is informal, inviting, and fascinating.”
Fernando Q. Gouvea, MAA Reviews

Matchstick Problem

matchstick quiz

Translation: How do we move only 1 matchstick to make the equation valid?

(“Not Equals” sign \neq is not allowed)

This is a really tricky question… Hint: Need to think in Chinese, this is a Chinese joke 😛

Featured Post:

Motivational Books for Students (Educational)

The Motivational Books recommended in the above post seems to be very popular with readers on this site. Many readers (presumably parents) have bought the books! Buy a Christmas present for yourself this Xmas. 🙂

College pays students for getting a ‘C’ in math

Interesting idea. Definitely a motivation for most students!


  • Hillsborough Community College will pay up to $1,800 in cash to students who make a C or higher in three semesters of math classes.
  • Students also have the to win free textbooks.
  • HCC funds part of this experiment with additional funding coming from a George Soros organization.

Florida community college is trying to inspire students to finish their degree by doling out up to $1,800 in cash to students who make a C or higher in three semesters of math courses.

The program, called Mathematic Access Scholarship Program (MAPS), has run at Hillsborough Community College (HCC) for the past three years and is spearheaded by Manpower Demonstration Research Corporation (MDRC).

“We hoped the incentives would inspire behaviors that would lead to increased student success.”

Featured book:

Great Jobs for Math Majors, Second ed. (Great Jobs For… Series)

“What can I do with a degree in math?”

You’ve worked hard for that math degree. Now what? Sometimes, the choice of careers can seem endless. The most difficult part of a job search is starting it. This is where Great Jobs for Math Majors comes in. Designed to help you put your major to work, this handy guide covers the basics of a job search and provides detailed profiles of careers in math. From the worlds of finance and science to manufacturing and education, you’ll explore a variety of job options for math majors and determine the best fit for your personal, professional, and practical needs.

Do you want to be an actuary? Work in the banking industry? Program computers? In this updated edition, you’ll find:

  • Job-search basics such as crafting résumés and writing cover letters
  • Self-assessment exercises to help determine your professional fit
  • Investigative tools to help you find the perfect job
  • Networking tips to get your foot in the door before your résumé is even sent
  • True tales from practicing professionals about everyday life on the job
  • Current statistics on earnings, advancement, and the future of the profession
  • Resources for further information, including journals, professional associations, and online resources

Book donation drive to help low-income families (Singapore)

Do you have any old textbooks that you don’t need anymore? Check out a meaningful way to share a textbook by NTUC.

Source: NTUC Website

Photo: FairPrice Share-A-Textbook 2014 is officially launched! Show your support by donating your textbooks at any FairPrice stores (including Finest supermarkets, Xtra hypermarkets and FairPrice Xpress stores at Esso service stations). Collection closes 7 December 2014.

FairPrice Share-A-Textbook 2014 is officially launched! Show your support by donating your textbooks at any FairPrice stores (including Finest supermarkets, Xtra hypermarkets and FairPrice Xpress stores at Esso service stations). Collection closes 7 December 2014.

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

Exams are over! Book disproves evolution using Math!

Finally, after an arduous year, the exams are over (in Singapore). 🙂

It is now time for a period of rest and relaxation, to get ready for 2015, where everything starts again.

Students who are looking for an early headstart (a 1 month headstart is rather valuable) in subjects other than mathematics can check out Startutor: The best tuition agency in Singapore.

Recently, I chanced upon this book:

The Organized Universe: Exclusive Scientific Proof That Darwinism Is a Fraud

What strikes me is that the author is using a Mathematical Law (Benford’s Law) to prove that Darwinism is a fraud. It is pretty unique since other books usually rely on proofs based on physics, biology or chemistry.

The Theory of Darwinian Evolution is a Fraud! Finally, a long awaited secret of the Universe. For the first time, Man has scientific proof showing a pattern to all life. Can this be possible? In this ground breaking first book, veteran researcher and inventor/scientist Karl Dahlstrom and co-author C. Phillip Clegg have written an exceptionally readable, mind expanding and almost equation free account of how an accidental discovery of an obscure mathematical theorem came to overturn an established scientific paradigm. The best theories of the universe are beautiful, simple and profound just as Einstein’s famous formula, E=MC2. This book is elegant in its simplicity and yet complex in its scope of re-defining an ordered universe. This book zeros in on a 150 year old controversy of Darwinian Evolution and tears it apart with laser like precision such that the very core of that ‘theory’ is – to borrow a phrase from Descartes – “rent asunder”. There is nothing new under the sun. Since the Big Bang, an organizing principle of matter, embedded in the natural laws of physics, and eventually life itself, has manifested itself in a precise pattern such that it is found in literally everything – from the composition of meteorites to seawater, soil, and gasses here on earth as well as in tabulated data on everything from stock prices to river lengths. This discovery, called Benford’s Law, simply states that in any tabulated data set of natural numbers, the frequency of occurrence of the digits 1 through 9, as the first significant digit, will conform to a set pattern: 1 will occur 30.x% of the time; “2” will occur 17.x%, etc. with the frequency of each succeeding number being less than the preceding number. What does this scientific law of nature have to do with Darwin’s Evolution Theory?

I have read the preview on Amazon and it seems pretty interesting. 🙂

Also, just a reminder that the Free Career Analysis Quiz by renowned survey company Universum is still active for a limited time only! Do check it out before it ends.



MI H2 Maths Prelim Solutions 2010 Paper 2 Q7 (P&C)

MI H2 Maths Prelim 2010 Paper 2 Q7 (P&C)

7) Two families are invited to a party. The first family consists of a man and both his parents while the second family consists of a woman and both her parents. The two families sit at a round table with two other men and two other women.
Find the number of possible arrangements if
(i) there is no restriction, [1]
(ii) the men and women are seated alternately, [2]
(iii) members of the same family are seated together and the two other women must be seated separately, [3]
(iv) members of the same family are seated together and the seats are numbered. [2]


(i) (10-1)!=9!=362880

(ii) First fix the men’s sitting arrangement: (5-1)!

Then the remaining five women’s total number of arrangements are: 5!

Total=4! x 5!=2880

(iii) Fix the 2 families (as a group) and the 2 men: (4-1)! x 3! x 3!

(3! to permute each family)

By drawing a diagram, the two women have 4 slots to choose from, where order matters: ^4 P_2

Total = (4-1)! \times 3! \times 3! \times ^4 P_2 = 2592


We first find the required number of ways by treating the seats as unnumbered: (6-1)!\times 3!\times 3! =4320

Since the seats are numbered, there are 10 choices for the point of reference, thus no. of ways = 4320 \times 10 =43200

Maths Tutor Singapore, H2 Maths, A Maths, E Maths

If you or a friend are looking for maths tuition: o level, a level, IB, IP, olympiad, GEP and any other form of mathematics you can think of

Experienced, qualified (Raffles GEP, Deans List, NUS Deans List, Olympiads etc) and most importantly patient even with the most mathematically challenged.

so if you are in need of the solution to your mathematical woes, drop me a message!

Tutor: Mr Wu



H2 Maths Tuition: Complex Numbers Notes

H2 Maths: Complex Numbers 1 Page Notes



Cartesian Form

Draw diagram first, then find the appropriate quadrant and use

(can use GC to double check)

Polar Form

Exponential Form

When question involvespowers, multiplication or division, it may be helpful toconvert to exponential form.

Please write Ƶ and 2 differently.

De Moivre’s Theorem

Equivalent to

Memory tip: Notice that arg behaves similarly to log.

Locusof z is aset of pointssatisfying certain given conditions.

in English means:The distance between (the point representing)and (the point representing)

Means the distance offromis a constant,.

So this is acircular loci.

Centre:, radius =

means that the distance offromis equal to its distance from

In other words, the locus is theperpendicular bisectorof the line segment joiningand.

represents ahalf-linestarting frommaking an anglewith the positive Re-axis.

(Exclude the point (a,b) )

Common Errors

– Some candidates thought thatis the same asand thatis the same as.

– The “formula”for argumentsdoes not workfor points in the 2ndand 3rdquadrant.

– Very many candidates seem unaware that their calculators will work in radians mode and there were many unnecessary “manual” conversions from degrees to radians.

H2 Maths Tuition: Foot of Perpendicular (from point to plane) (Part II)

This is a continuation from H2 Maths Tuition: Foot of Perpendicular (from point to line) (Part I).

Foot of Perpendicular (from point to plane)

From point (B) to Plane ( p)


Equation (I):

Where does F lie?

F lies on the plane  p.

\overrightarrow{\mathit{OF}}\cdot \mathbf{n}=d

Equation (II):





Final Step

Substitute Equation (II) into Equation (I) and solve for k.


[VJC 2010 P1Q8i]

The planes \Pi _{1} and \Pi _{2} have equations \mathbf{r\cdot(i+j-k)}=6 and \mathbf{r\cdot(2i-4j+k)}=-12 respectively. The point A  has position vector  \mathbf{{9i-7j+5k}} .

(i) Find the position vector of the foot of perpendicular from  A to \Pi _{2} .


Let the foot of perpendicular be F.

Equation (I)

\overrightarrow{\mathit{OF}}\cdot  \left(\begin{matrix}2\\-4\\1\end{matrix}\right)=-12

Equation (II)


Subst. (II) into (I)


Solve for k,  k=-3 .


H2 Maths Tuition

If you are looking for Maths Tuition, contact Mr Wu at: