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Fix4you Locksmith

Best Locksmith in Israel:

Speedy, fast, effective locksmith specializing in door repair, safety lock, repair of door handles and more.

Locksmith is very important in the event of an emergency, e.g. you lost the keys or your lock is damaged.


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Being Good At Maths Can Help Your Kids Start Their Own Business

Every child is unique; every human with an entirely different set of fingerprints. Whether your child has the potential to be an engineer or a boss of their own, one thing remains inextricably part of life – math.

Take it from me, a 22-year-old business owner who used to flunk math. Your kids will hate maths until they start to see that they can actually do and get better at it. From F9 in A Maths, I studied hard and studied right to achieve A2. Your kids can do it with the right guidance and the right motivation.

And I’m not referring to scoring better than your neighbours or nephews and nieces. Your kids don’t care about that – and you shouldn’t too. Focus on the important things.

What are they? In this article, I’ll be sharing with you, parents, what will motivate your kids to want to be good at maths that are not grades.

Help Your Kids Be Their Own Boss

Many kids have entrepreneurial blood and a knack for sales. Look at how well they persuade you to get what they want. Remember the times where they came up with the most ingenious of ideas – you were surprised; thought it could even become a business.

If your kids have a tendency to solve problems in the most creative ways possible and have a sense of leadership, you should get excited. They might be the next CEO of Amazon!

Math is crucial in starting and running a business. Whether it is running an eCommerce store (which is very profitable in 2019 and beyond) using platforms like or selling their artistic services online, math is a must.

Kids are motivated by things other than grades

You might think that this isn’t exciting for your kids.

Maybe they never thought it would be possible… Maybe all they knew was just the linear way of life…

What if they were interested in starting their own mini business? What if they got excited to be good at maths, motivated by the fact that they could apply to real life?

Now that I’ve sparked something in you, you might be thinking ‘Why does running a business need math?’

Being Good At Maths Can Help Your Kids Start Their Own Business

Reason #1: Doing their taxes

I wish my parents taught me how to do my taxes and to avoid ‘savings plans’ by a certain insurance company. These are the important things that will matter to your kids when they are adults – something that is missing in schools and not taught by most parents.

Being good at math will ensure that they don’t get lost in all the numbers, percentages, appreciation, depreciation, interest, and more. With a good foothold in working with numbers, your kids will not hate doing their taxes and avoid any penalties that come with incorrect tax payment or filing.

Reason #2: Ensure they are always profiting

Starting their own business means they have no one to tell them how much they need to charge to have a good amount of profit. No teacher to tell them that they are spending way too much on unnecessary things and blowing out their expenditure. They have no tutor to guide them saying ‘to meet your financial goals and achieve sustainability in Singapore, you need to get $XXX a year.’

Running a business well demands good math skills to calculate and plan the finances.

Now, your kids have a teacher, they have you and a math tutor. Bring math in real-life situations and you’ll start to see an increase of drive in their eyes.


Perhaps it’s time to unveil the a little bit more about reality to them. It could get them thrilled to head on to ace that math paper!

Author: Jamie

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How to speak to Human Singtel Customer Officer

If you try calling the Singtel hotline 1688, you will spend most of your time to navigate the maze of options trying to speak to a human Singtel Customer Officer. I have not managed to successfully find the option to speak to a human Singtel Chat Agent through the hotline, if you know the method please comment below!

Also, do not bother speaking to “Shirley” on the Contact Us page ( It is quite clear Shirley is not a human, but just a bot.

The Trick to Chat with a Human Singtel Customer Officer

  1. Go to the Contact Us page.
  2. Click on Ask Shirley
  3. Type “can i chat with customer service officer”.
  4. Type in the category of your question, e.g. “billing”, “voucher”, etc.
  5. Whatever Shirley says, just click “Submit Enquiry”
  6. Bingo! You are now connected to a Human Chat Agent!

From trial and error, this is the easiest way to get to speak to a Human Singtel Customer Service Officer/ Chat Agent. There is no direct link to chat with a Singtel Customer Representative.

For the above method, you don’t even need to login to your Singtel account. But you may be asked your name and NRIC for verification purposes.

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Chinese Math Olympiad (Primary) Trick vs Singapore Modeling Math

Math Online Tom Circle

Both Methods (I) & (II) below does not use Algebra, which is not taught to primary school kids until secondary school (above 12 or 13 years old).

(I) Ancient Chinese Arithmetic Method:

例2:The sister is 13 years old this year, while the brother is 9 years old.
When the total of their ages is 40, how old will they be?


Trick: Age difference does not change over time, they + or – in tandem. When 2 ages change, the multiple between them also changes.

(II) Singapore Modeling Math:


Compare the Ancient Chinese Arithmetic (I) & Singapore Modeling Math (II):

  • (II) is better for young kids to understand visually the logic behind,
  • (I) is more a memorised trick but poor math education pedagogy. In [例 2] it is very hard for kids to understand why the 2 steps (40+4) & (40 – 4), although they lead to…

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English Fun Math

Math Online Tom Circle

Different Maths:
1. English (UK/USA) Math is playfully tricky (Math Olympiad, Math Puzzles eg. this Numberpile video)

2. French Math is abstractly theoretical (Group theory explains Symmetry, Rubik’s cube game,… )

3. Chinese Math (Abacus, “Chicken-Rabbits Problem” 鸡兔问题, Singapore Modeling Math) is applied & procedural (明朝 13CE invented “Determinant” 行列式 before the 19th CE “Matrix” by UK Sylvester,Chinese Remainder Theorem 韩信点兵).

4. Indian Math (Vedic Math , Factorization, Ramanujian-Style Math) is mysteriously spiritual.

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AI with Advanced Math helps in discovering new drugs

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Why French excel in math ?

Math Online Tom Circle

Mathematics and quantitative finance, France

Since 1990, there have been 22 winners of the Fields Medal, widely regarded as the Nobel Prize of mathematics. Thirteen came from just two countries, Russia and France. Russia has more winners (seven), but more than twice the population, so the honours go to France, with six winners.

Cédric Villani, the 2010 Fields Medallist, cited national character. “Maths is an abstract way of looking at the world, which fits well with the French mentality. We apply algebra to everything.” Elite institutions help too. France’s brightest school leavers progress to the grandes écoles, which traditionally educate top scientists, administrators and presidents. For maths, you want Monsieur Villani’s alma mater, the École Normale Supérieure (ENS). All 10 French Fields Medallists learnt there. At ENS, no teacher can stay longer than 10 years. Instead of ancient dons, students get tutors at the forefront…

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Happy Birthday To Urbain Le Verrier, Who Discovered Neptune With Math Alone

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Percentage Mental Calculation

Math Online Tom Circle

4% of 75 = 75% of 4

Mental Steps:
75 x 4 – > 300 – > / 100 – > = 3 (ANS. )
Is it easier ?

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The Laws of Mathematics Zermelo-Fraenkel Axioms Set Theory

Math Online Tom Circle

The Foundation of Modern Math which is based on the “Set Theory”, hit a crisis due to Bertrand Russell “Barber’s Paradox”, until the Zermelo-Fraenkel Axioms came to its rescue.

Here is a humorous example of an ISP Internet Plan violating the 9 Z-F Axioms:

How ISPs Violate the Laws of Mathematics:


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Quora: Is Abstract Math a fluke (偶然) ?

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Math Online Tom Circle

Challenge: Put ’15’ and ’14’ in the right sequence (Impossible! )


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“Essential mathematical GIFs that will make mathematics finally make sense”

Math Online Tom Circle

“Essential mathematical GIFs that will make mathematics finally make sense” by Ali Kayaspor

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RMM Olympiad 2019

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Best Machine Learning / Deep Learning Books

After some research, I find that the best book on Machine Learning / Deep Learning currently on the market is: Deep Learning with Python.

It is written by François Chollet, who is the creator of the famous Keras library (Google Tensorflow). The most popular language for machine learning is currently Python, which is the subject of this book. (Another language is R, but many people find R to be less user-friendly as compared to Python).

The book is suitable for those who are studying deep learning from scratch, with zero prior knowledge. Currently, machine learning is at the peak of its hype, with many employers seeking to utilize machine learning in one way or another. It is very useful for students to pick up this skill for potential usage in the future.

Deep Learning with Python

A key competitor to Keras/Tensorflow is Pytorch, developed by Facebook. It is supposedly even more user-friendly than Keras/Tensorflow. However, the drawback is that it is too new and few books/tutorials exist.

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Alexa Ranking Below 1 Million

Thanks to all readers, Mathtuition88 has reached Alexa Rank of below 1 Million!

Current Global Rank: 983,883

SG Rank: 11,701

This is the first time has broken the 1000,000 Alexa Rank barrier.

For ranking, the lower the better. For instance Google is ranked number 1, and YouTube is number 2. Facebook is ranked number 3.

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Powerpac Ceramic Cooker Review

This should be the cheapest ceramic cooker on the market. PowerPac is a reliable Singaporean brand that has been on the market for quite some time.

At the price of $29.90, the Ceramic Cooker even comes with a free stainless steel pot with cover (CNY Promo). Unbeatable price! It is also free delivery.

It is ideal for home steamboat / hotpot cooking. By the way, ceramic cooker can cook all types of pots and pans (unlike induction cookers).


[S$29.90](▼51%)[PowerPac Promo]PowerPac Ceramic Cooker (Any Pot) 2000 Watts (PPIC833) //Suitable for any use with any types of pots


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Free Philips Air Fryer (Enfagrow Promo)

Official URL: Enfagrow Homepage

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Group Theorems: Lagrange, Sylow, Cauchy

Math Online Tom Circle

1. Lagrange Theorem:
Order of subgroup H divides order of Group G

Converse false:
having h | g does not imply there exists a subgroup H of order h.
Example: Z3 = {0,1,2} is not subgroup of Z6
although o(Z3)= 3 which divides o(Z6)= 6

if h = p (prime number),
2. Cauchy Theorem: if p | g
then G contains an element x (so a subgroup) of order p.
$latex x^{p} = e $ ∀x∈ G

3. Sylow Theorem :
for p prime,
if p^n | g
=> G has a subgroup H of order p^n:
$latex h= p^{n}$

Conclusion: h | g
Lagrange (h) => Sylow (h=p^n) => Cauchy (h= p, n=1)

Trick to Remember:

g = kh (god =kind holy)
=> h | g
g : order of group G
h : order of subgroup H…

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陈省身 SS Chern – “The 2nd Gauss”

Math Online Tom Circle

陈省身香港科技大学 演讲经典 (Highly Recommended – spoken in English & Chinese)

他的女婿是香港科大校长。现任校长是 Prof 陈繁星 Tony F. Chan , the ‘Search Algorithm Thesis’ supervising Prof of the 2 Google founders in Stanford University, Applied Math Department)

Key Points:

  1. SS Chern won the “Wolf Prize” in the same year with the Hungarian “Vagabond” 流浪汉 Mathematician Erdos Paul.
  2. He was appointed the Scientific Advisor of President Reagan, during which Chern recommended to build the USA Center of Math in Berkeley University. After retirement, Chern built a similar center in China at his alma mater Nankai University 南开大学 where he graduated in 1930.
  3. SS Chern was an assistant lecturer of Prof Yang WuZhi 杨武之, who liked to invite Chern to his house for dinners, where he saw his 8-year old son Yang ZhenNing 杨振宁 (Nobel Prize Physics 1958).
  4. SS Chern has guided great PhD students eg. James Simons (Billionaire Financial Investor applying…

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70-million Bounded Gap Between Primes

Math Online Tom Circle

Since Ancient Greek :

  1. Euclid had proved there are infinite primes.
  2. Sieve of Eratosthenes to enumerate the primes.
  3. Recent time 3 Mathematicians GPY attempted another Sieve method to find the bounded gap(N) of primes in infinity, but stuck at one critical step.
  4. Dr. YiTang Zhang 张益唐 (1955 -) spent 7 years in solitude after failure in academic career, in 2013 during a 10-min walk at the deer backyard of his friend’s house, he found an Eureka solution for the GPY’s critical step: $latex boxed { epsilon = frac {1} {168}}&fg=aa0000&s=2 $ which gave the first historical bounded Gap (N) from an infinity large number to a limit of 70 million.


  • Chinese love the number “8” ba which sounds like the word prosperity 发 fa (in Cantonese) . He could have instead used 160, so long as $latex epsilon $ is small.
  • The UltimateGoal of the Bounded…

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Betta Fish Tricks

Betta fish are quite intelligent and curious. I trained my Betta fish to do some simple tricks.

I bought my 5 Gallon tank on Qoo10: Betta Fish Tank. The modern viewpoint is that Betta should not be kept in small bowls or vases (though they might still survive). A 2.5 gallon tank is considered the absolute minimum, with 5 gallon and above being a better choice.

Jumping Trick and Going Through Hoop Trick

Following Pen Trick

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小算盘 大乾坤 Abacus

Math Online Tom Circle

浙江临海市 算盘博物管主人 雷国华 收藏算盘20年,散尽储蓄,感慨涕下不已。

中国的算盘上面2珠 (2×5),下面5珠(5), 共15,加1 = 16 进位。古代是16位 (Hexadecimal) 制, 半斤 八两 (= 1/2 斤 x 16 两 = 8 两)。

日本改良成10位制: 上面1珠 (1×5),下面4珠(4), 共9,加1 = 10 进位。

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3D Rotation Matrices and Examples

The following rotation matrices rotate vectors by an angle \theta in an anticlockwise direction about the x-, y-, or z-axis respectively (the rotated axis points towards the observer).

\begin{aligned}  R_x(\theta)&=\begin{pmatrix}1 &0 &0\\  0 &\cos\theta &-\sin\theta\\  0 &\sin\theta &\cos\theta  \end{pmatrix}\\  R_y(\theta)&=\begin{pmatrix}\cos\theta &0 &\sin\theta\\  0 &1 &0\\  -\sin\theta &0 &\cos\theta  \end{pmatrix}\\  R_z(\theta)&=\begin{pmatrix}  \cos\theta &-\sin\theta &0\\  \sin\theta &\cos\theta &0\\  0 &0 &1  \end{pmatrix}  \end{aligned}

Example 1

Rotating \begin{pmatrix}1\\ 0\\ 0\end{pmatrix} 45^\circ anticlockwise about z-axis:

\begin{aligned}  R_z(45^\circ)\begin{pmatrix}1\\ 0\\ 0\end{pmatrix}&=\begin{pmatrix}  \cos 45^\circ &-\sin 45^\circ &0\\  \sin 45^\circ &\cos 45^\circ &0\\  0 &0 &1  \end{pmatrix}  \begin{pmatrix}1\\ 0\\ 0\end{pmatrix}\\  &=\begin{pmatrix}\sqrt{2}/2\\\sqrt{2}/2\\0\end{pmatrix}.  \end{aligned}

Example 2

Rotating \begin{pmatrix}0\\ 1\\ 0\end{pmatrix} 45^\circ anticlockwise about z-axis:

\begin{aligned}  R_z(45^\circ)\begin{pmatrix}0\\ 1\\ 0\end{pmatrix}&=\begin{pmatrix}  \cos 45^\circ &-\sin 45^\circ &0\\  \sin 45^\circ &\cos 45^\circ &0\\  0 &0 &1  \end{pmatrix}  \begin{pmatrix}0\\ 1\\ 0\end{pmatrix}\\  &=\begin{pmatrix}-\sqrt{2}/2\\\sqrt{2}/2\\0\end{pmatrix}.  \end{aligned}

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十二平均律 12-tone Equal Temperament

Math Online Tom Circle

《十二平均律》 是明朝 "布衣王子" 朱载堉 (1536年-1610年) 发明的,由当时在中国传教的意大利人 利马窦 (Matteo Ricci, 1552 – 1610), 传给欧洲的法国数学家 Marin Mersenne (Mersenne Prime ) 。现代音乐之父 巴哈 Bach 第一个采用,制作世界第一架钢琴有12黑白键,并作曲 《Bach 12-tone Equal Temperament》。

朱载堉把 “1” 到 “i” 的八度 (Octave) 分为等比 (ratio) 距离的12个半音 (half note), 每个音是前音的 $latex sqrt[12] {2 }$,第8高音频率 (frequency) “i” 巧好是第一音 “1”的2倍。他用81行的大算盘算出:

Side Note:

1977年法国大学数学教授在课堂好奇地问我,你们祖先如何解代数?是用算盘吗?当时计算机还不流行,复杂的算法只能用Log Table 或 计算尺 (Slide Rule) 。


Bach 12-tone Equal Temperament


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Richard Dedekind

Math Online Tom Circle

Julius Wilhelmina Richard Dedekind (6 Oct 1831 – 12 Feb 1916)

– Last student of Gauss at Göttingen
– Student and closed friend of Dirichlet who influenced his Mathematical education
– Introduced the word Field (Körper)
– Gave the first university course on Galois Theory
– Developed Real Number ‘Dedekind Cut‘ in 1872
– Accomplished musician
– Never married, lived with his unmarried sister until death
“Whatever provable should be proved.”
– By 1858: still yet established ?
$latex sqrt{2}.sqrt{3} = sqrt{2.3} $
– Gave strong support to Cantor on Infinite Set.

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HDB Low Water Pressure: Phone Hotline

For those living near the top floor of HDB blocks, you may experience low water pressure since the water tanks are typically built at the rooftop of HDB flats.

The pressure exerted by a column of liquid of height h and density ρ is given by the hydrostatic pressure equation \displaystyle\boxed{p=\rho gh}, where g is the gravitational acceleration.


By this equation, you can understand why those at the top floors experience low water pressure, due to the low height h (from the water tank). There is probably a need for a pump to increase the pressure. For low floors, it is typically ok, the natural pressure should be enough to make the water flow fast.

Water Pressure Low Hotline: 1800-CALL-PUB

The number you should call when experiencing low water pressure for HDB flats is:


My Experience with Low Water Supply

I experienced unbelievably quick action by PUB on Chinese New Year, around 9pm. I called the PUB hotline due to very low water pressure (insufficient for even bathing). Mentally, I was prepared that probably they could only fix it the next working day (which is 2 days later).

To my amazement, a repairman arrived 15-20 minutes later, and proceeded to fix the water tank. The water pressure was back to normal in just half an hour from the time I dialed the hotline. I was quite impressed by the high efficiency.

I am not sure what caused the problem in the first place though. This problem of low water pressure has been reported by Channel News Asia.

HDB Low Water Pressure What To Do

Firstly, you should check with neighbors on the same floor whether they experience the problem of low water pressure. This is to confirm whether the problem is with the water tank pump/pipes, or an isolated problem only pertaining to your unit.

If your neighbors also experience low water pressure, then most likely it is a problem pertaining to the water tank pump/pipes. You should dial the PUB hotline (1800-2255-782) and inform them of the low water pressure. They will liaise with the Town Council and send a technician to investigate. (The service is free of charge.)

If the situation still does not improve, you may wish to purchase this “Wind Storm” High Pressure Shower Head. Basically, it turns a low pressure stream of water into a high pressure spray of water, to the extent that you can feel a “wind” blowing. It also saves water so it will pay for itself in the long run (by reducing your water bills).


[S$11.90](▼77%)[Restock] Wind Storm Shower Head ★Powerful High Pressure Shower heads ★ For Bath


Also see: Free Water Saving Kit (Singapore)

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Dedekind Cut : “Prove 0.9999… = 1”

Math Online Tom Circle

Key Notes:

  1. Rational Numbers Q eg. 1/2, … Complete but non Continuous.
  2. Irrational numbers eg. $latex sqrt {2}$
  3. Real number is Complete and Continuous
  4. Dedekind Cut

The Proof of “0.999… = 1” by using Set Theory :

$latex A subset B$
$latex B subset C$
$latex implies A = B $

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Bit Coin and Block Chain

Math Online Tom Circle

Bit Coin is only one application of Block Chain.

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Happy Chinese New Year

Happy Chinese New Year to all readers of!

Do check out and follow my Facebook page, where I post interesting news on Math/Education.

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Machine Learning is Just Mathematics! Free Machine Learning Resources

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Learn NUMPY in 5 minutes – BEST Python Library!

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Formula for 3×3 Matrix Inverse

Let \displaystyle A=\begin{pmatrix}a &b &c\\  d& e& f\\  g& h& i  \end{pmatrix}.

\displaystyle A^{-1}=\frac{1}{\det(A)}\begin{pmatrix}  ei-fh &ch-bi &bf-ce\\  fg-di &ai-cg &cd-af\\  dh-eg &bg-ah &ae-bd  \end{pmatrix}.

This method is about the same speed as the Gaussian-Elimination method. However, if you have already calculated det(A), using the formula may be slightly faster.

The formula provides an insight on why A is singular (not invertible) if det(A)=0. (Because, if det(A)=0, in the formula we would be dividing by 0 which is not allowed.)

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Men’s wallet with coin compartment Singapore

Despite many cashless initiatives like PayLah, Singapore is not yet a truly cashless country. Coins are still needed to buy coffee or tea at the “Kopitiam”, or to put a $1 coin for pushing a trolley at supermarkets. Even if you carry paper notes, when you receive coins as change you will need a place to put them.

Many wallets do not have coin compartments. On Qoo10, this brand “Arete First” is a Singaporean brand that sells Leather Wallets at a reasonable price, with coin compartment. The quality is good, and it is genuine leather. It can also put a decent number of cards (nine card slots).

Do check it out if you are interested in a wallet with coin compartment.

URL: Arete First Genuine Leather Wallet

[S$19.90](▼88%)[Arete First] Genuine Leather Wallet for Men | RFID blocking | Bifold | Coin compartment


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Horned Nerite Snails

Some video footage of my new Horned Nerite Snails in the Betta Tank. Really amazing and bright colors.

Taken using Canon PowerShot: Canon Camera and Camera Case Singapore

The fish tank is 5 Gallon: Betta Fish Tank

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Imaginary Numbers

Math Online Tom Circle

FINAL EPISODE – Riemann Complex Plane : 4 dimension but viewed in 3 dimensions

Episode 1 – 9 : History

Episode 10: Complex function

Function : 1 input mapped to 1 output

Multi-function : 1 input mapped to (n > 1 ) output

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Math for AI : Gradient Descent

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My most exciting Bishop’s Opening ever

Game URL:

I did make a few serious blunders, hence it is not my best game. But it is certainly quite exciting. My heart pounded furiously as I played 22. f6! sacrificing my queen… If my opponent takes the queen, it will be checkmate in 3. (He didn’t.)

White has just played 22. f6!

My opponent, being 2000 rated, played quite well. He missed punishing my blunder at around move 14.

Currently, my white opening repertoire involves the Bishop’s opening (1. e4 e5 2. Bc4), with a view towards playing f4 whenever possible to transpose into Vienna/King’s Gambit.

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Forest Man of India

Previously, I blogged about the Mountain Man of India:

Mountain Man Motivational Story

Today, I learnt that there is also a Forest Man of India.

By planting one tree a day (for many years), Jadav Payeng (a.k.a Forest Man of India) has planted a forest bigger than Central Park in New York!

This principle of doing small things everyday also applies to students and learning. By doing one Math question per day, at the end of one year, a student would have completed 365 Math questions! This would be a considerable amount that would certainly help to boost the student’s Math grade and also Math understanding.

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Goby and Shrimp Video (GEP Science)

Amazing “teamwork” between a goby fish and a shrimp! Basically, the shrimp is in charge of digging the burrow, but it is almost blind. The goby fish acts as the sentry to alert the shrimp in case of danger.

This video may be useful for GEP science, it may be tested as an example of “symbiotic relationships” in animals. GEP Science syllabus includes a brief introduction to different types of symbiosis (Mutualism, Commensalism and parasitism).

It is also very entertaining and can promote interest in science for children.

Also check out:

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Python Online Courses for Teenagers/Adults

If your child is interested in a Computer Science/Data Science career in the future, do consider learning Python beforehand. Computer Science is getting very popular in Singapore again. To see how popular it is, just check out the latest cut-off point for NUS computer science, it is close to perfect score (AAA/B) for A-levels.

According to many sources, the Singapore job market (including government sector) is very interested in skills like Machine Learning/ Deep Learning/Data Science. It seems that Machine Learning can be used to do almost anything and everything, from playing chess to data analytics. Majors such as accountancy and even law are in danger of being replaced by Machine Learning. Python is the key language for such applications.

I just completed a short course on Python: Python A-Z™: Python For Data Science With Real Exercises! The course fee is payable via Skillsfuture for Singaporeans, i.e. you don’t have to pay a single cent. (You have to purchase it first, then get a reimbursement from Skillsfuture.) At the end, you will get a Udemy certificate which you can put in your LinkedIn profile.

The course includes many things from the basic syntax to advanced visualization of data. It teaches at quite a basic level, I am sure most JC students (or even talented secondary students) with some very basic programming background can understand it.

The best programming language for data science is currently Python. Try not to learn “old” languages like C++ as it can become obsolete soon. Anyway the focus is on the programming structure, it is more or less universal across different languages.icon

Udemy URL: Python A-Z™: Python For Data Science With Real Exercises!

Related posts on Python:

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Superposition Principle of Linear Algebra

Superposition Principle

If \mathbf{u} is a solution to \bf{A}\mathbf{x}=\mathbf{b}, and \mathbf{v} is a solution to \bf{A}\mathbf{x}=\mathbf{c}, then \mathbf{u}+\mathbf{v} is a solution to \bf{A}\mathbf{x}=\mathbf{b}+\mathbf{c}.


Properties of Matrix Transpose

1) (A^T)^T=A
2) (A+B)^T=A^T+B^T
3) (kA)^T=kA^T
4) (AB)^T=B^T A^T

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Hangout @ Chinatown

Includes a review on “The Coconut Club”, which serves one of the best Nasi Lemak in Singapore.

Cooking For Fun

Well, today I have one day off and my husband also has time. We planned to go to Chinatown and have lunch there. He wanted to try a nasi lemak restaurant there. It’s the first time for us to go to Chinatown and hunt for non-Chinese food.

Chinatown of Singapore is really close to to the CBD. You can actually walk from Chinatown to CBD and also see blocks of tall buildings from Chinatown.

There are many places to have nice food in Chinatown, such as Amoy Street and Chinatown Food Street.

Chinatown has many traditional buildings of Nanyang style.

The restaurant we went is at Ann Siang Rd.

There are many bars, restaurants and cafes there.

The blue restaurant is the one we were going.

The Coconut Club.

It is not a very big restaurant. Their signature dish is nasi lemak, which can be told from the menu.


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To those starting JC life in today’s Singapore

Excellent post by top GP tutor Mr Steven Ooi. This post sums up the current competitiveness of the Singaporean education system. Similarly for Math, there is a knowledge “arms-race” to the extent that a typical Primary 3 student should know how to calculate 1+2+3+…+100 in case this question pops up in the exam. The textbook syllabus remains just as easy as in the past, but what is tested has become 10 times more difficult.

GP & English Tutor with First Class Honours, NUS

jc singapore

[Writer’s note: While this article is addressed specifically to Singapore-based students, it may hold some truth for students in many other places as well.]

Dear young friend,

I’m writing this to share some thoughts that I used to convey to every one of my students when I was still a full-time tutor in Singapore (I retired in 2016). It is just my personal opinion, formed through years of experience as a student in the 1980s and 1990s, then as a tutor in the 2000s and 2010s.

Observing student life in the 90s and then in the 2010s, I can only conclude that the school system in Singapore has become 10 times more competitive than during my time. At least 10 times. During the 90s, most of my junior college classmates didn’t even attend private tuition. We could hang out at the bowling alley, the arcade or the hawker centre most…

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Malayan Aqua Fern

This “Malayan Aqua Fern” is a very mysterious plant, there is very little information on Google about it.

There are two views on it, based on online writings.

First viewpoint on Malayan Aqua Fern:

The “Malayan Aqua Fern” is none other than Peacock Fern, or Selaginella willdenowii. It is NOT an aquatic plant, and will die if fully submerged under water. This is quite reasonable, based on the pictures of Peacock Fern online, it does look like the pictures of the “Malayan Aqua Fern”. Nonetheless, it can be planted in a crab/terrapin tank where it is not fully submerged under water.

The “Malayan Aqua Fern” may give the illusion of surviving under water when it may be slowly dying (takes around a month to die).

Websites that support this view:

Second view:

The “Malayan Aqua Fern” is an amphibious plant, able to live both submerged or in moist conditions. If this is true, then “Malayan Aqua Fern” would be an incredible aquarium plant as it looks very good in a fish tank.

Websites that support this view:

Is Malayan Aqua Fern Aquatic or Not?

The main question is whether the Malayan Aqua Fern is aquatic or not? Can it grow fully submerged under water? If you have experience with the Malayan Aqua Fern, please leave your comments below!

Malayan Aqua Fern and Betta

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Chromatic Fingering for Moonlight Sonata 3rd Movement

In Beethoven’s Moonlight Sonata 3rd movement, there is a very fast stretch of chromatic notes in bar 187.

I was wondering if the most basic fingering is good, or a more advanced fingering involving the 4th or even 5th finger would be better. (Read more on advanced chromatic fingering here:

Some people on Reddit suggest some very fanciful fingerings, including “using LH (Left Hand!!) (faster playing up)4321,4312,4321,4321. So, groups of 4321 with the exception starting on the C. That part goes 4312. Just play all the way up with the LH and smack that top A (was it?) with yer Right Hand”! This fingering is apparently attributed to Alfred Brendel.

It turns out that at least 2 professional pianists seem to use the basic chromatic fingering: 2 13 123 13 123 … etc. They do not use the fourth finger at all. The starting fingering of the first note (G#) may vary.

The first pianist is Valentina Lisitsa. Watch at 0.25 speed to see the fingering at around 6:07 time. Lisitsa seems to use a “special trick” by using her left hand to play the lowest G# in the chromatic range.

The second pianist is Rousseau. You can see his fingering clearly, it is the basic chromatic fingering at around 6:34 time.

Hence, it seems that for Moonlight sonata it is not needed to use the advanced chromatic fingering. Using the pedal during the chromatic (both pianists use it to some extent) gives the illusion of a fast chromatic.

For other pieces like Fantasie Impromptu, the advanced chromatic fingering works well for greater speed.

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BTS Fan Products

BTS is one of the most popular Korean bands. Previously, we wrote on how Scientists Use Math To Discover Why BTS Jin Is The Most Handsome. Their dance steps are also amazing, hence it is no question why they are so popular worldwide. Basically, nowadays top artistes need to be good at dance + singing in order to be popular.

BTS seems very popular locally in Singapore, their concert tickets are often sold out within an hour. By the way, if a youth says that they are in “the Army”, it may not mean that they are soldiers/National Cadet Corps. BTS fans are also known as “the Army”. I once conversed with a taxi driver who told me that his daughter is in RGS, and most of her classmates are fans of K-pop including BTS. Korea is also now a top tourist destination for Singaporeans.

Both Qoo10 and Amazon have BTS fan paraphernalia, with Amazon having a more extensive collection. Most of the items ship to Singapore, with the delivery charge being quite affordable considering the huge distance from America to Singapore.

An example of a BTS cosmetic set from Qoo10.

[S$64.90](▼36%)[VT Cosmetic]【VT X BTS】★THE SWEET SPECIAL EDITION SET(Cushion(Choice 21 or 23 Tint 2 pcs included) /TTBEAUTY


From Amazon, there are collections of BTS jackets, sweaters, and more. We only showcase those that are at least 4 stars out of 5 for the Amazon rating. Basically, any product with substantial ratings of above 4/5 stars on Amazon are quality products. Especially, if they have the “Amazon’s choice” label, they are personally curated and approved by Amazon’s team.

babyHealthy Kpop BTS Love Yourself Hoodie Suga Jimin V Rap-Monster Sweater Jacket Pullover (4.1/5 stars)

BTS Kpop Socks 7 Pairs (Jungkuk, J-Hope, Suga, V, Jin, Jimin, Rap Mon) (4.8/5 stars, Amazon’s choice)

babyHealthy BTS Baseball Jacket Uniform Bangtan Boys Suga Jin Jimin Jung Kook Sweater Coat (4.3/5 stars)

KPOP BTS New Logo Hoodie Bangtan Boys Unisex Sweatershirt JIN SUGA JIMIN V (4.1/5 stars)

BTS – 2018 SEASON’S GREETINGS Set (4.6/5 stars, Amazon’s choice)

BTS 1st Album [Dark and Wild] CD + PhotoCard + PhotoBook K-POP BANGTAN (4.5/5 stars, Amazon’s choice)

Most of BTS videos have 100 million views or more:

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Free American Tourister 25” Luggage (Dumex)

Very good deal here, free 25 inch luggage (American Tourister brand) with purchase of Dumex milk powder!

This is the official Dumex Qoo-10 store, by the way.

[S$325.90](▼23%)[Dumex]Free 25 luggage(worth $220)【Bundle of 6】Dumex Mamil Gold Stage 2/ 3 / 4 / 5 Milk Formula 1.6kg


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Mathematician Couple’s Wedding Math Questions

Math Online Tom Circle

It is fun for the guests who are also mathematicians like the wedding couple…

Why don’t the IT couples do the same with computer questions (eg. private cloud, serverless, containers, DevOps,… ) , lawyers with legal cases, doctors with medical questions …

Example 1 is obvious to this guest whose PhD research topic is in Differential Geometry “Chern Class” invented by Prof S. S. Chern (陈省身) :

Example 2 is obvious to this guest unless he/she can’t count the number of sides (8):

Example 3 is to the guest with some background in secondary school organic chemistry but forgotten, definitely not linear…

Example 4 to the hobbyist of Rubik’s cube, but the answer can be guessed easily by elimination: definitely not 2019 (this year), or π.

Answer in Multiple choices(Only 1answer is correct, the other 2 answers are obviously wrong) :

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