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Introduction to Machine Learning

Very good introduction to Machine Learning by Google. Google is the developer of Tensorflow (on which the Keras package is built). The other platform for Machine Learning is Pytorch by Facebook.

So far, the best introductory book to Machine Learning seems to be the one by the founder of Keras, called “Deep Learning with Python”. See also Best Machine Learning / Deep Learning Books.

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A Maths Topical Prelim Exam Papers

Additional Mathematics Topical Prelim Free Exam Papers

PDF Exam Paper Questions download: Alpha beta questions

“Alpha beta” questions, also known as quadratic roots questions, were first added into the ‘O’ level Additional Mathematics syllabus in 2014. They remain one of the more challenging questions that take students a long time to grasp. Mathematically, they are the simplest type of Vieta’s formulas.

This is one of the series on free exam papers (topical), which is suitable for Secondary 3 or Secondary 4 students, taking A Maths (new syllabus). (Questions are taken from actual school papers.)

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So, you want to become a money Manager?

Money management is a difficult task for many people. It is common to see people struggling with additional expenses, which pop up especially towards the end of the month. Then they have to wait for the salary to be credited so that the financial crisis is stabilized. It is a common practice to see people taking loans to meet various expenses, when they run out of money. Therefore, the next month onwards your salary automatically goes into repayment, interest, etc. Here comes the need for a money management executive.

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Deduction in salary

Loans are indeed a great help during an immediate financial need. It could be an accident or a family function; good or bad the event, you need money to deal with it. This probably may cut money from your salary. The salary deductions from your account are for the following:

  • Educational loans
  • Mortgages
  • Debts from credit cards
  • Unexpected events in life

If you do not manage money flow, even with so much of debts, you are surely going to become bankrupt soon.

Unmanageable bankruptcy

Financial issues that go out of your control could even coil up and end up in bankruptcy. Money management executive would use only simple techniques, which even you could follow.

Kevin Hawhan, from the online payday loan lender thinks that practice makes perfect when it comes to managing your finances and money.

“You just need to practice this simple process of managing money. Inconsistent behaviour is indeed a curse to money management. No weekend mishaps, or Saturday night cash dumps from the cashpoint.”

“You would be in a safe and comfortable zone of finance, by employing a series of sound tips for money management. This would ensure you comfort on financial aspects in future as well.”

Let’s face the reality

Everyone should develop money management skills in life. MSN money conducted a survey in 2007 about finance and consumer debts management. According to this survey, an American household bears consumer debts accounting to $8,000 on an average. Poor money management is directly linked to high rate of consumer debts. This creates difficult times for the whole family.

Proper budgeting

Personal money management would start by creating a budget for family income. Budgeting is effective in studying cash flow in and out, which helps you to analyse your present financial situation.

Most people are troubled with debts and needs a money management executive to deal with finance. Proper financial education is substantial in understanding the working of consumer debts. National financial advisors as well as credit counsellors could help you in adverse situations.

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How to become a money manager

Money managers or investment managers help you in spending and investing your hard-earned money wisely and help you to set goals that can give you fruitful results in future. A money manager is also expected to keep in touch with his clients on a regular basis and provide updates on their client’s investments and its status. A study conducted by the website found that, on an average, a money manager takes home a salary of $72000 per year. Now, that’s a handsome amount. So, what are the steps to become a daily money manager? In this article, we will discuss in detail on how to be a money manager and the skills you need to acquire for it.

Educational qualification

To take up the career of a money manager, you need to acquire a degree in finance. MBA in the same stream will be an added advantage, as many employers prefer aspirants with an MBA. A degree and MBA in finance will get you acquainted with investment concepts and banking affairs.

Certification programs

Certifications can always boost your career and the certification program of the Certified Financial Planner Board of Standards is considered worthwhile. A bachelor’s degree and a minimum work experience of three years make you eligible to enrol for this program. The certification exam includes topics, which test your risk management ability, financial management process, retirement process, emergency fund reserves and liability. Once you become a certified professional, you become a favourite with the employers as your portfolio management skills are tested and certified.

Attain key skills

Acquiring key skills in analytics is very important to become a successful money manager. Acquiring a degree in finance will not suffice for this. You only learn the basics in investments and money management with the degree program. Industry exposure is the key to develop your skills. So, if possible do your internship in a reputed firm, as this will be an added advantage. You are also expected to improve your communication skills when you deal with your clients. Keep in mind that, most of your clients may be new to stocks and investments field and you should be able to make clear all the doubts that your client may ask.

Hope the tips above helped you in understanding the various aspects of becoming a money manager. A good money manager is also a daily money manager at home. Start spending your money wisely and let this start from your home. Only then, you can successfully manage the money of your valuable clients.

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No human being, however great or powerful, was ever so free as a fish

” No human being, however great or powerful, was ever so free as a fish”
– John Ruskin

The full context can be found here:

Romans 6:19-20
I speak after the manner of men because of the infirmity of your flesh…

You hear every day greater numbers of foolish people speaking about liberty, as if it were such an honourable thing; so far from being that, it is, on the whole, and in the broadest sense, dishonourable, and an attribute of the lower creatures. No human being, however great or powerful, was ever so free as a fish. There is always something that he must or must not do; while the fish may do whatever he likes. All the kingdoms of the world put together are not half so large as the sea, and all the railroads and wheels that ever were or will be invented, are not so easy as fins. You will find, on fairly thinking of it, that it is his restraint which is honourable to man, not his liberty; and, what is more, it is restraint which is honourable even in the lower animals. A butterfly is more free than a bee, but you honour the bee more just because it is subject to certain laws which fit it for orderly function in bee society. And throughout the world, of the two abstract things, liberty and restraint, restraint is always the more honourable. It is true, indeed, that in these and all other matters you never can reason finally from the abstraction, for both liberty and restraint are good when they are nobly chosen, and both are bad when they are badly chosen; but of the two, I repeat, it is restraint which characterises the higher creature, and betters the lower creature; and from the ministering of the archangel to the labour of the insect, from the poising of the planets to the gravitation of a grain of dust — the power and glory of all creatures and all matter consist in their obedience, not in their freedom. The sun has no liberty, a dead leaf has much. The dust of which you are formed has no liberty. Its liberty will come — with its corruption.

(J. Ruskin.)

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Recommended China SIM Card

For travelling to China, it is recommended to get the SIM Card from ICC on Qoo10 (see link below). It has many benefits:

  • Biggest benefit is that it can access WhatsApp/Google/Facebook which is otherwise blocked in China. The reason is that the cards are designed for Hong Kong /Macau, but yet can work in mainland China.
  • Cheap and affordable. It should be much cheaper than using data roaming from Singtel/Starhub etc.
  • Unlimited data. It has 3 GB for fast data (LTE speed). After that, it has unlimited data but at a throttled speed.

Do note that there is one disadvantage which is you cannot make or receive phone calls or SMS. But this can be easily overcome by using apps such as WeChat which has voice call and message services.

Official URL:

[S$4.99](▼92%)[ICC]◆ICC◆【China SIM Card 8-30 Days】4GLTE Unlimited Data Call* ❤WhatsApp/Google/FaceBook❤ Hong Kong/Macau


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5 Awesome Career Paths for Mathematics Lovers

5 Awesome Career Paths for Mathematics Lovers

By Sarah Quek

Unbeknownst to most, a career in mathematics goes beyond education.

Mathematics might be one of the toughest subjects at any education level. But a solid understanding of key concepts and mathematical ideas could help your career take flight and expose you to a myriad of industries.

The demand for mathematics experts has boomed in various industries in Singapore. If you have always loved numbers, read on to find out where mathematics can take you!

1. Mathematics and Consultancy.

Consultancy is a diverse sphere in itself. The popularly known and most preferred consulting career lies in the business consultancy. However, there are some other aspects of consultancy that a mathematic-proficient student can venture in.

These include; hospital consultancy, marital consultancy, management of risks consultancy and more. These types of consultancies require extensive knowledge and use of arithmetic in their daily work. The employees apply algebra, use of graphs, tables and excel to analyze data garnered from markets or presented by clients. Some financial management works require mathematical theories whose roots are deeply rooted in calculus, probability and trigonometry. Generally, analytical skill forms every part of consulting. Mathematics defines which analytical skill to use at a specific time and for a specific purpose.

2. Mathematics and Insurance.

Mathematics forms the foundation of any insurance job. An employee is required to manipulate data from different market niches and clients using different mathematical formulas, draw graphs and tables and ensure proper storage of data. Probability is also useful especially in the areas of risk management.

3. Mathematics and Engineering.

This should come as no surprise, but mathematics is a must-have skill in the engineering industry. A deep understanding of probability and 3D mathematics is necessary especially when it comes to engineering design.

To a student who would wish to know how to build a bridge across 10 km lake, stop a skyscraper from falling and power 400-ton aircraft to fly, graduating in a class of applied mathematics is a necessity.

4. Mathematics and digital marketing.

Numbers are applied in every stage of this dynamic career. Search Engine Optimization (SEO) tools and search engines employ different units of variables to find averages, sums, densities and more to rank websites and other digital marketing tools as per the data they harbour. Filtering, segmentations, mail structures and designs are products of various mathematical concepts.

5. Information Technology.

The last in the list is a career in information technology. A number of mathematical principles and formulas are used in gathering, analysis, presentation and analysis of data. These include but are not limited to statistics, algorithm, and probability.

The list contains only a handful of career options that one can venture in. Don’t hesitate if you are a lover of math. Dip your shoes in the vast ocean, the world is your oyster!

About Author

Sarah is a digital marketing content producer at First Page Digital. She specializes in SEO, social media marketing as well as content marketing. Find out more about a career in digital marketing by clicking here.

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$1 Dunkin’ Donut Promotion (2 Donuts + 1 Coffee for just 1 dollar)

There is a little known and unbelievably worth it Dunkin’ Donut $1 promotion for 2 donuts and 1 cold brew coffee/ice coffee.

The terms and conditions:

  • You need to be a new Fave user. (This has to be your 1st purchase on Fave)
  • Download the Fave app
  • Search for Dunkin’ Donut promotion (It is displayed as $4.90 for 2 donuts + 1 cold drink)
  • Important: Enter the Promo Code STORMSCAPE when making payment
  • Check that the price drops to $1!

Tip: To be very sure, you may want to ask the staff whether the promotion is still on before making purchase. Technically, it should be until 2 June 2019.

Original Chinese Post:

只要下载Fave app,新用户可以用1新币的价格获得2个甜甜圈+一杯饮料!只需要在结账的时候输入STORMSCAPE,并出示给店员,就可以享受这个劲爆的优惠了~



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Acceleration = d/dx (1/2 v^2)

Acceleration = \frac{d}{dx}(\frac{1}{2}v^2):

\displaystyle \boxed{a=\frac{d^2x}{dt^2}=\frac{dv}{dt}=v\frac{dv}{dx}=\frac{d}{dx}(\frac{1}{2}v^2)},

where x is displacement, v is velocity, and a is acceleration.

\begin{aligned}  a&=\frac{dv}{dt} \text{(this is definition of acceleration)}\\  &=\frac{dv}{dx}\cdot\frac{dx}{dt} \text{(chain rule)}\\  &=v\frac{dv}{dx}.  \end{aligned}

\begin{aligned}  \frac{d}{dx}(\frac{1}{2}v^2)&=v\frac{dv}{dx}. \text{(chain rule)}  \end{aligned}

Therefore, a=\frac{d}{dx}(\frac{1}{2}v^2).

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Part Time Employees Law (Malaysia)


根据 2010 年雇佣(兼职员工)条例,兼职员工是指那些工作时间占全职员工工作时间的30% 至70%的人士。少过全职工作时间的30%属于散工,并不在此条例管辖范围。

在2010 年雇佣(兼职员工)条例下,兼职员工可享有公积金、社险、公假、年假、病假、休息日和超时津贴。
若雇主没有提供福利给兼职员工和不遵守2010 年雇佣(兼职员工)条例,违例者一旦被定罪,每项罪行可被罚款最高一万令吉。

有薪公共假期: 兼职员工有权享有不少于7 天宪报公布的有薪
公共假 期:其中四天必须是:国庆日,元首诞辰,苏丹或州元首 华诞或联邦直辖区日及劳动节.
休息日:兼职员工享有休息日,不过必须符合条件,即必须工作5天或一周工作时间不少过20 小时
超时工作津贴:兼职员工的工作时间只占全职工作时间的70%至100%, 他们是根据每小时1.0计算,若他们工作时间超过全职工作 时间,是以每小时1.5倍计算。
公假工作享有薪水和超时津贴:兼职员工在公共假期工作可享有2天的薪水,在超时津贴方 面,若工作占全职工作时间的70%至100%,可申领2 倍,若超过全职工作时间,可申领3倍。
休息日工作享有薪水和超时津贴:兼职员工在休息日工作可享有2天薪水,在超时津贴方面,若 工作占全职工作时间的70%至100%,可申领1.5倍,超过全职工作时间,可申领2倍.

工作少过两年者, 每十二个月不断 的服务可领取6 天;
工作两年但少过五年者, 每十二个月不断的服务可领 取8天;
工作超过五年者, 每十二个月不断的服务可领取11天

工作少过两年者, 每十二个月不断 的服务可领取10 天;
工作两年但少过五年者, 每十二个月不断的服务可领取13天;
工作超过五年者, 每十二个月不断的服务可领取15天.
*2010年雇佣(兼职员工)条例里并没有提到兼职员工可以 享有产假津贴和住院病假的福利.

1. Employment Act 1955 Employment (Part Time Employees) Regulations 2010
2. Akta Kerja 1955 Peraturan-Peraturan Kerja (Pekerja Separa Masa) 2010





*加入 我们“法律与你同行”的FB 群组:

*Like 我们“法律与你同行”的FB Page:


*Kuek, Ong & Associates
*original link:

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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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