We mentioned Augustin Louis Cauchy in the tragic stories of Galois and Abel. Had Cauchy been more generous and kind enough to submit the two young mathematicians’ papers to the French Academy of Sciences, their fates would have been different and they would not have died so young.

Cauchy was excellent in language. He was the 2nd most prolific writer (of Math papers) after Euler in history. When he was a math prodigy, his neighbor — the great French mathematician and scientist Pierre-Simon Laplace — advised Cauchy’s father to focus the boy on language before touching mathematics. (Teachers / Parents take note of the importance of language in Math education.)

Cauchy’s language education made him very rigorous in micro-details. This was the man who developed the most rigorous epsilon-delta Advanced Calculus (called Analysis) after Newton / Lebniz had invented the non-rigorous Calculus (why?).

Rigorous epsilon-delta…

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Real Life Applications of Algebraic Topology (Big Data)

Big Data: A Revolution That Will Transform How We Live, Work, and Think

What is Algebraic Topology:

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. (Wikpedia)

What is Big Data:

Big data is a broad term for data sets so large or complex that traditional data processing applications are inadequate. Challenges include analysis, capture, data curation, search, sharing, storage, transfer, visualization, and information privacy. The term often refers simply to the use of predictive analytics or other certain advanced methods to extract value from data, and seldom to a particular size of data set. Accuracy in big data may lead to more confident decision making. And better decisions can mean greater operational efficiency, cost reductions and reduced risk. (Wikipedia)

Big Data is said to be the next biggest scientific advance since the internet. Algebraic Topology is one branch of Mathematics that is directly related to Big Data.

Topological data analysis (TDA) is a new area of study aimed at having applications in areas such as data mining and computer vision. The main problems are:

how one infers high-dimensional structure from low-dimensional representations; and
how one assembles discrete points into global structure.

The human brain can easily extract global structure from representations in a strictly lower dimension, e.g. we infer a 3D environment from a 2D image from each eye. The inference of global structure also occurs when converting discrete data into continuous images, e.g. dot-matrix printers and televisions communicate images via arrays of discrete points.

The main method used by topological data analysis is:

Replace a set of data points with a family of simplicial complexes, indexed by a proximity parameter.
Analyse these topological complexes via algebraic topology — specifically, via the theory of persistent homology.^[1]
Encode the persistent homology of a data set in the form of a parameterized version of a Betti number which is called a persistence diagram or barcode.^[1]

Source: Wikipedia

Very interesting!

A nonnegative function f in M(X,X) is the limit of a monotone increasing sequence in M(X,X)

We will elaborate on a lemma in the book The Elements of Integration and Lebesgue Measure.

Lemma: If f is a nonnegative function in M(X,X), then there exists a sequence ( $\phi_n$ ) in M(X,X) such that:

(a) $0\leq \phi_n (x) \leq \phi_{n+1} (x)$ for $x\in X, n\in\mathbb{N}$ .

(b) $f(x) =\lim \phi_n (x)$ for each $x\in X$ .

Proof:

Let n be a fixed natural number. If k=0, 1, 2, …, $n 2^n -1$ , let $E_{kn}$ be the set

$E_{kn}=\{ x\in X: k2^{-n} \leq f(x)<(k+1)2^{-n}\}$ .

If $k=n2^n$ , let $E_{kn}=\{x\in X: f(x) \geq n\}$ .

We note that the sets $\{E_{kn}: k=0, 1,\ldots, n2^n\}$ are disjoint.

The sets also belong to X, and have union equal to X.

Thus, if we define $\phi_n= k2^{-n}$ on $E_{kn}$ , then $\phi_n$ belongs to M(X,X).

We can see that the properties (a), (b), (c) hold.

(a): $0\leq k2^{-n}\leq k2^{-n-1}$ is true.

(I just noticed there is some typo in Bartle’s book, as the above inequality does not hold. I think n is supposed to be fixed, while k is increased instead.)

(b): As n tends to infinity, on $k2^{-n} \leq f(x) <(k+1)2^{-n}$ , i.e. $\phi_n (x) \leq f(x) < \phi_n (x)+2^{-n}$ , thus $f(x)=\lim \phi_n (x)$ for each $x\in X$ .

(c): Clearly true!

Source:

Error-Detecting and Error-Correcting Codes

Today’s post is a bit about coding theory, and how a good code can detect and even correct errors from transmission.

Theorem 1

A code is k-error-detecting if and only if the minimum Hamming distance between code words is at least k+1.

Theorem 2

A code is k-error-correcting if and only if the minimum Hamming distance between code words is at least 2k+1.

Hamming distance is the number of bits in which the code words differ. For instance, the Hamming distance between 1000 and 1001 is 1 since they only defer in the last bit.

Proof of Theorem 1

Lets assume a code is k-error-detecting. Suppose to the contrary the there is a pair of code words c_1 and c_2 with Hamming distance k or less. Then, given the code word c_2, we don’t know if it is a valid code word, or it arose from c_1 due to errors in k-bits. This contradicts the fact that the code is k-error-detecting.

If the minimum Hamming distance is at least k+1, then given a code that differs from a code word by k bits, we know that it is not a valid code word, and hence we have detected a error!

Proof of Theorem 2

Assume the code is k-error-correcting. Suppose to the contrary there is a pair of code words c_1 and c_2 whose Hamming distance is 2k or less. Then if there is a code word whose Hamming distance is k from c_1 and c_2, then it is equally likely to have arose from c_1 or c_2, hence we can’t correct the error!

If the Hamming distance is 2k+1 or more, then any code word with Hamming distance of k (or less) will be closer to one of the code words, and hence has higher probability of having arose from that code word.

The Imitation Game

During the winter of 1952, British authorities entered the home of mathematician, cryptanalyst and war hero Alan Turing (Benedict Cumberbatch) to investigate a reported burglary. They instead ended up arresting Turing himself on charges of ‘gross indecency’, an accusation that would lead to his devastating conviction for the criminal offense of homosexuality – little did officials know, they were actually incriminating the pioneer of modern-day computing. Famously leading a motley group of scholars, linguists, chess champions and intelligence officers, he was credited with cracking the so-called unbreakable codes of Germany’s World War II Enigma machine. An intense and haunting portrayal of a brilliant, complicated man, The Imitation Game a genius who under nail-biting pressure helped to shorten the war and, in turn, save thousands of lives.

Measurability of product fg

In the previous chapters, Bartle showed that that if f is in M(X,X), then the functions $cf, f^2, |f|, f^+, f^-$ are also in M(X,X).

The case of the measurability of the product fg when f, g belong to M(X,X) is a little bit more tricky. If $n\in\mathbb{N}$ , let $f_n$ be the “truncation of f” defined by $f_n (x)=\begin{cases}f(x), &\text{if }|f(x)|\leq n, \\ n, &\text{if } f(x)>n,\\ -n, &\text{if }f(x)<-n\end{cases}$

Let $g_m$ be defined similarly. We will work out the proof that $f_n$ and $g_m$ are measurable (Bartle left it as Exercise 2.K).

Proof:

Each $f_n$ is a function on $X$ to $\mathbb{R}$ .

$\{x\in X:f_n (x) >\alpha\}=\begin{cases}\{x \in X: f(x)>\alpha\}, &\text{if }-n<\alpha <n,\\ \emptyset, &\text{if }\alpha\geq n,\\X, &\text{if }\alpha\leq -n \end{cases}$

All of the above sets are in X.

Thus, we may use an earlier Lemma 2.6 to show that the product $f_n g_m$ is measurable.

We also have $f(x)g_m (x)=\lim_n f_n (x)g_m (x)$ , and using an earlier corollary that says that if a sequence $(f_n)$ is in M(X,X) converges to f on X, then f is also in M(X,X), we have that $f(x)g_m (x)$ belongs to M(X,X).

Finally, (fg)(x)=f(x)g(x)= $\lim_m f(x)g_m (x)$ , and hence fg also belongs to M(X,X).

This is a very powerful result of Lebesgue integration, since we can see that the theory includes extended real-valued functions, and prepares us to integrate functions that can reach infinite values!

Source: The Elements of Integration and Lebesgue Measure

Borel Measurable

This is a continuation of the study of the book The Elements of Integration and Lebesgue Measure by Bartle, listing a few examples of functions that are measurable. Bartle is a very good author, he tries his very best to make this difficult subject accessible to undergraduates.

Example:

If X is the set R of real numbers, and X is the Borel algebra B, then any monotone function is Borel measurable.

Proof:

Suppose that f is monotone increasing, i.e. $x\leq x'$ implies $f(x)\leq f(x')$ .

Then, $\{x\in\mathbb{R}:f(x)>\alpha\}$ consists of a half-line which is either of the form $\{x\in\mathbb{R}:x>a\}$ or the form $\{x\in\mathbb{R}:x\geq a\}$ . (We will show later that both cases can occur.) Thus, the set will belong to the Borel algebra B which is the $\sigma$ -algebra generated by all open intervals (a,b) in R.

Both cases can indeed occur. For example, if f(x)=x, then the set will be of the form $\{x\in\mathbb{R}:x>a\}$ . More interestingly, if the set is the step function $f(x)=\begin{cases}-1, &\text{if }x<0\\1, &\text{if }x\geq 0\end{cases}$ , then when $\alpha=0$ , the set will be $\{x\in\mathbb{R}:x\geq 0\}$ .

Lemma: An extended real-valued function f is measurable if and only if the sets $A=\{x\in X:f(x)=+\infty\}$ , $B=\{x\in X:f(x)=-\infty\}$ belong to X and the real-valued function $f_1$ defined by $f_1 (x)= \begin{cases} f(x), &\text{if }x\notin A\cup B,\\ 0, &\text{if }x\in A\cup B,\end{cases}$ is measurable.

This lemma is often useful when dealing with extended real-valued functions.

Proof: If f is in M(X,X), it is proven earlier in the book by Bartle that A and B belong to X. Let $\alpha\in\mathbb{R}$ and $\alpha\geq 0$ , then we have that $\{ x\in X:f_1 (x)>\alpha\}=\{ x\in X:f(x)>\alpha\}\setminus A$ which is in X since it is the complement of the union of A and $X\setminus \{x\in X:f(x)>\alpha\}$ .

If $\alpha<0$ , then $\{ x\in X:f_1 (x)>\alpha \}=\{ x\in X:f(x)>\alpha \}\cup B$ , which is a union of two sets in X and hence also in X.

Hence, $f_1$ is measurable.

Conversely, if $A, B\in \mathbf{X}$ and $f_1$ is measurable, then $\{x\in X:f(x)>\alpha\}=\{ x\in X: f_1 (x) >\alpha \}\cup A$ when $\alpha \geq 0$ , and $\{x\in X:f(x)>\alpha\}=\{x \in X:f_1 (x)>\alpha\}\setminus B$ when $\alpha <0$ , due to a similar reason as above. Therefore f is measurable!

Definitions for measurable functions

I am currently proceeding on a self-guided study of the book The Elements of Integration and Lebesgue Measure, will post some updates and elaborations of the proofs in the book. Every book is constrained by the number of pages the publisher allows, hence some authors will write rather terse and concise proofs, the worst example of which is simply “Proof: Trivial”. Bartle is a very good author, he does provide details of proofs 90% of the time.

Definition: A function f on X to R is said to be X-measurable (or simply measurable) if for every real number $\alpha$ the set $\{x\in X: f(x)>\alpha\}$ belongs to X.

This definition of measurability is not unique, there are other possible forms which are discussed in the lemma below.

Lemma: The following statements are equivalent for a function f on X to R:

((X,X) is a measurable space where X is a set and X is a $\sigma$ -algebra of subsets of X.)

(a) For every $\alpha\in\mathbb{R}$ , the set $A_\alpha = \{x\in X: f(x)>\alpha \}$ belongs to X,

(b) For every $\alpha\in\mathbb{R}$ , the set $B_\alpha = \{x\in X: f(x)\leq\alpha \}$ belongs to X,

(d) For every $\alpha\in\mathbb{R}$ , the set $D_\alpha = \{x\in X: f(x)<\alpha \}$ belongs to X,

Proof:

Note that $B_\alpha$ and $A_\alpha$ are complements of each other, hence statement (a) is equivalent to statement (b). This is due to one of the properties of $\sigma$ -algebra, namely that if A belongs to X, then the complement X\A also belongs to X.

Similarly, statements (c) and (d) are equivalent. We will prove that (a) is equivalent to (c).

Assume (a) holds, we can say that $A_{\alpha-1/n}$ belongs to X for each n.

And since $C_\alpha=\cap_{n=1}^{\infty}{A_{\alpha-1/n}}$ , it follows that $C_\alpha\in\mathbf{ X}$ . Thus, (a) implies (c).

We also have $A_\alpha =\cup_{n=1}^{\infty} C_{\alpha+1/n}$ , and hence if (c) is true, each $C_{\alpha +1/n}$ is in X, and the union of them is also in X (definition for $\sigma$ -algebra). It thus follows that (c) implies (a).

What this lemma says is that there is nothing special about the “>” in the definition of measurability. It could very well have been “ $\leq$ , or even “<” and nothing would change!

‘Beautiful Mind’ mathematician John Nash killed in US car crash

Very sad news…. Rest in peace, Professor John Nash.

Source: https://sg.news.yahoo.com/beautiful-mind-mathematician-john-nash-killed-us-police-143603056.html

Nobel Prize-winning US mathematician John Nash, who inspired the film “A Beautiful Mind,” was killed with his wife in a New Jersey car crash.

Nash, 86, and his 82-year-old wife Alicia were riding in a taxi on Saturday when the accident took place, State Police Sergeant Gregory Williams told AFP.

“The taxi passengers were ejected,” Williams said, adding that they were both killed.

The Princeton University and Massachusetts Institute of Technology (MIT) mathematician is best known for his contribution to game theory — the study of decision-making — which won him the Nobel economics prize in 1994.

His life story formed the basis of the Oscar-winning 2001 film “A Beautiful Mind” in which actor Russell Crowe played the genius, who struggled with mental illness.

“Stunned… my heart goes out to John & Alicia & family. An amazing partnership. Beautiful minds, beautiful hearts,” Crowe said on Twitter.

A Beautiful Mind

Synopsis: “HOW COULD YOU, A MATHEMATICIAN, BELIEVE THAT EXTRATERRESTRIALS WERE SENDING YOU MESSAGES?” the visitor from Harvard asked the West Virginian with the movie-star looks and Olympian manner. “Because the ideas I had about supernatural beings came to me the same way my mathematical ideas did,” came the answer. “So I took them seriously.”

Thus begins the true story of John Nash, the mathematical genius who was a legend by age thirty when he slipped into madness, and who—thanks to the selflessness of a beautiful woman and the loyalty of the mathematics community—emerged after decades of ghostlike existence to win a Nobel Prize for triggering the game theory revolution. The inspiration for an Academy Award–winning movie, Sylvia Nasar’s now-classic biography is a drama about the mystery of the human mind, triumph over adversity, and the healing power of love.

Measure and Integration Recommended Book

I have added a new addition to the Recommended Books for Undergraduate Math, which is one of my most popular posts!

The new book is The Elements of Integration and Lebesgue Measure, an advanced text on the theory of integration. At the high school level, students are exposed to integration, but merely the rules of integration. At university, students learn the Riemann theory of integration (Riemann sums), which is a good theory, but not the best. There are some functions which we would like to integrate, but do not fit nicely into the theory of Riemann Integration.

I am personally reading this book as well, as I didn’t manage to study it in university, but it is a key component for graduate level analysis. Students interested in advanced Probability (see this post on Coursera Probability course) would be needing Lebesgue theory too!

Time Management Tips for Students (What to do if fail JC Test / Promo Exam?)

Do you wish there is a method to improve your grades? How do you improve your grades after failing a Common Test for Secondary School or JC?

The Four Quadrant Method is an ideal method for students (especially higher level students like O Level or A Level students) to plan their study schedule and revision time table.

Many students do ok in primary school, but start to falter and fail in secondary school or JC. This may be due to many factors, some of which can be remedied using effective time management.

According to this model, which comes from the book First Things First by Stephen Covey (Highly recommended to read), there are four types of activities:

Quadrant 1) Important and Urgent (crises, deadline-driven projects)
Quadrant 2) Important, Not Urgent (preparation, prevention, planning, relationships)
Quadrant 3) Urgent, Not Important (interruptions, many pressing matters)
Quadrant 4) Not Urgent, Not Important (trivia, time wasters)

The key to doing well in school and exams is actually Quadrant 2! It is highly related to human psychology. Most people would think Quadrant 1 is more important, but actually Quadrant 2 is the most important type of activity for students.

Quadrant 1 activities (in the Singapore context) are activities like assignment due next day, test next day, exam the next day, and so on. They are important and also urgent. The thing is, these things are usually done by most people since there is a time pressure factor to it. Most students will actually do and complete Quadrant 1 activities. However, as you would know by now, just doing the homework the teacher assigns is not enough to do well for the test / exam under the Singapore syllabus. Firstly, the work that the teacher assigns may be basic material, while in Singapore, the school tests and exams all contain advanced and challenging material.

Quadrant 2 activities are long-ranged planning and strategies, like preparing for a test that is 3 months later, preparing for the Promo Exam that is half a year later. Since these activities are not urgent, most people skip them altogether. However, it is highly important to do Quadrant 2 activities everyday. Stephen R. Covey is a genius for discovering that Quadrant 2 is the secret to time management. Students should set aside some time everyday to do long-ranged preparation, e.g. preparing for a test that is a few months into the future.

Quadrant 3 activities are things that are urgent but not important. Examples are checking Email, checking Whatsapp for class group notifications. Yes, checking email and Whatsapp is compulsory nowadays, but it is not considered an important activity in the grand scheme of things. One should set a minimum amount of them for these activities. CCA may also be classified under this category. This Quadrant is highly deceptive, and a huge time sink, but in the end the activities in Quadrant 3 rank very low in importance.

Quadrant 4 activities are things that are not urgent and not important. Examples are checking Facebook, playing computer games, and so on. These activities should be kept to a bare minimum, and only during scheduled breaks for destressing.

The Four Quadrant technique can be coupled with the Pomodoro Technique which is another good technique for time management.

Hope it helps! This method is for parents to teach their child about Time Management, provided their child is motivated and wishes to improve. For children that are not motivated to study / not interested in learning, parents should check out these Motivational books to motivate students instead.

5-21-1471

Albrecht Durer, the German painter and engraver who studied mathematics and applied it to his art, was born in Nuremberg on this day.

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5-21-1923

Armand Borel born in Switzerland. He worked on Lie groups, algebraic groups, and arithmetic groups, helping transform many areas of mathematics, including algebraic topology, differential geometry, algebraic geometry, and number theory.

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5-21-1953

Set theorist Ernst Zermelo died in Freiburg, Germany. He worked on statistical mechanics and the calculus of variations until becoming intrigued by Cantor’s Continuum Hypothesis, posed as Hilbert’s First Problem in 1900, and switching to set theory.

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Tuition Agency / Chinese Tuition

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There are many excellent tutors from RI, Hwa Chong, etc. at Startutor, teaching various subjects at all levels.
High calibre scholars from NUS/overseas universities are also tutoring at Startutor.

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让孩子读书更厉害的书籍

现在的家长都很注重孩子的学习，但是有时候孩子不专注，或者对学习没兴趣怎么办？

俗话说：你可以把马牵到水边，但你无法强迫它饮水（意指有的事情必需本人自愿，强迫无济于事）；老牛不喝水，不能强按头。

可见，强逼孩子读书是没有用的，反而会造成孩子厌倦学习。最重要是培养孩子读书的兴趣，这样往往事半功倍，孩子的学习成绩突飞猛进。

在此，让我介绍一些帮忙孩子读书厉害的书：

1）我的第一本专注力训练书（专注的孩子更聪明）

现在很多孩子都有多动症，就算没有多动症也很难静下来读书，这是21世纪普遍的问题。因为现在太多引诱，比如电脑，手机，电视。毋庸置疑，专注的孩子更聪明，学习也肯定比较好。《我的第一本专注力训练书》为《看到找不到》系列之精彩合集。精选本系列中最经典和最受欢迎的形象页面，按综合难度由易到难编排，增加了专注能量级的划分和“目标锁定”等小细节，提升孩子寻找之后的成就感，逐步提升专注力、记忆力、观察力三大能力。《我的第一本专注力训练书》足足128页，让孩子一次玩得过瘾、找得开心。

2）学会提问(原书第10版)

学会提问是一门很大的学问。批判性思维领域“圣经”之作！权威大师30年畅销不衰的经典！史上最有内涵的思维训练书！亚马逊思维科学领域no.1！俞敏洪高度推荐美国大学生人手一本！打开心智，提早具备未来创新人才的核心竞争力！

3) 棚车少年(套装共8册)(中英双语)（当孩子遇到挫折，这本书能让他们笑着面对人生）

天下没有100%顺利的事，孩子总有一天会遇到挫折。遇到挫折该怎么办，怎么面对？读了这本书能让孩子笑着面对人生，不如买给孩子看看。这本书是中英双语，还能帮助孩子练习语文。《棚车少年》：亨利、杰西、维莉、班尼四兄妹从小就是孤儿，他们知道自己有一个爷爷在绿野镇，但是他们不喜欢他。为了躲避爷爷，孩子们在一个破旧的棚车里安了家，开始相依为命的生活。他们积极向上，阳光开朗，不惧生活的挫折。不仅如此，他们还相互帮助，一起寻找生活的乐趣，在树林里安家、收留小狗望望、探宝、自由赛……最后爷爷找到了他们，原来爷爷很年轻、慈祥、很爱他们。最后他们跟爷爷一起回家，过上幸福快乐的日子。

WordPress to Sina Weibo 微博 Automatic Posting

I recently discovered a way to post (automatically) from WordPress to Sina Weibo 微博(China’s version of twitter, which has more than half a billion users!)

The trick is to use IFTTT.com (If this then that).

Steps:

1) Setup up publicize for WordPress to Twitter. (WordPress.com can do this automatically).

2) Go to IFTTT.com, and set up a recipe from Twitter to Weibo. (There is a premade template for that, takes less than 5 minutes to sign up)

Done!

There may be a way for WordPress –> Weibo direct posting, I am still researching on that. (Update: Yes, there is a recipe for direct WordPress –> Weibo too!) It depends on whether you want a short summary, in which case WordPress –> Twitter –> Weibo may suit you better. If you want a full text, then WordPress –> Weibo is great. Or, you can use both!

Hope it is helpful!

无微不至:微博营销实战指南

《无微不至:微博营销实战指南》内容简介：企业如何利用微博进行营销？如何了解消费者的购买心理？如何把握微博的传播机制，发现用户的行为模式，找到有价值的客户？如何挖掘数据价值，制定营销方案，实现营销的最佳效果？《无微不至:微博营销实战指南》从如何搭建企业微博营销平台、构建微博体系、塑造企业微形象、选择微博营销模式，以及微博营销的技能、微博写作技巧等方面详尽地讲述微博营销的方法、技巧，具有实操性强，案例经典，拿来就能用的特点。在《无微不至:微博营销实战指南》中，读者还会学到以下经典内容：微博营销已不是简单开个账户，发发帖子。微博传播永远是内容为王，无论是重口味，还是小清新，一定要与草根文化血脉相通。写微博和说相声是一样的，要善于抖包袱，要在140字中写出跌宕起伏。10%的人影响了90%的人的购买行为，微博是影响他人购买决策的一个有效工具。社交广告即将或者已经成为最主流的社会化营销解决方案。高质量的内容和互动永远是提高粉丝转发率、留住粉丝的不二法宝。中国移动、中国电信应该如何做微博营销？《独唱团》爆单，快书包如何转危机为商机？如何打造企业官微？1000个真实的粉丝意味着什么？如何用微博编织人脉？微博内容写作十大技巧是什么……

Chinese Math and Science Books

Just to introduce a few books that can simultaneously improve your child’s Math and Science knowledge, and Chinese at the same time!

The latest news is that China is building the Kra Canal, a news that would mark the beginning of the increased dominance of China in Southeast Asia, hence having a good grasp of Mandarin is no longer optional, but 100% compulsory if you want to have a slice of the pie of the jobs and benefits generated by China.

华文数学/科学书本

1）
可怕的科学•经典数学系列(套装共12册)(三度荣获国际科普图书最高奖)

This series of “Horrible Science” Books is translated into Chinese, and is an award winning series of books. Highly recommended!

2)
小学奥数700题详解:3、4、5、6年级

$math olympiad$

700 Practice Problems for Math Olympiad! These books are very useful for GAT / DSA / GEP Preparation.

3)
走进奇妙的数学世界1-3(套装共3册)

$math olympiad$ $world of math$

Into the world of Mathematics! 《走进奇妙的数学世界1-3(套装共3册)》内容简介：数学最让人困惑的是为什么这样和有什么用，很多人即使大学毕业也不明白，这套书完美地阐释了数学的本质，把数学和生活紧密联系在一起。13种基本数学思想，层层深入，完美阐释数学的本质。以两个小矮人贯穿全文，图文并茂，讲故事、出谜题、做游戏，游戏背后蕴藏数学概念让孩子以最简单、最科学的方式走近数学，爱上数学！不仅仅讲算术，更重在启发从不同角度看待事物、解决问题的思考方式，培养孩子的逻辑思维能力，提高综合素质。

Finally, for readers of my blog who are new to Chinese, and wish to learn this 5000 year old language, I would recommend some books to learn Chinese for beginners here：

轻松学中文1(课本)(附CD光盘1张)

Learn Chinese in an easy manner! Easy steps to Chinese. (With CD)

Free USD 50 Amazon voucher (For Business and Engineering majors)

Hi, just to share a way to win a free USD 50 Amazon voucher!

This offer is only open to students from NUS, NTU, SMU; both Business and Engineering Bachelor Degree students, who have just completed their studies in 2015 or completing studies in 2016.

Details:

Greetings!

We, Universum, are currently conducting a research about career expectations of students. For this study, we are currently recruiting undergraduate students to participate in a 3-day online focus group.

If selected, the students will receive a USD 50 Amazon voucher after their active participation during the discussion. The online focus group will be scheduled from now to 14 June 2015, and will be active for 3 days. Selected students are encouraged to log in to participate actively during this period.

Interested students will just need to register here or http://goo.gl/forms/xp1OnFrXlQ. We will contact the students if they are selected. They will then be notified as soon as possible of the periods of the focus group and will receive further details of the interview’s topic.

Nadine Dinh

B2C Marketing and Relations Manager for APAC

3 is everywhere

Math Online Tom Circle

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Why 381,654,729 is awesome

Math Online Tom Circle

Pandigital numbers

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The Problem in Good Will Hunting

Math Online Tom Circle

Movie (Maths Problem Scene):

Homeomorphically Irreducible Trees with 10 dots.

Who was the real Good Will Hunting?

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A Graph Theory Olympiad Question Whose Answer is 1015056

April’s Math Olympiad Question was a particularly tough one, only four people in the world solved it! One from Japan, one from Slovakia, one from Ankara, and one from Singapore!

The question starts off seemingly simple enough:

In a party attended by 2015 guests among any 7 guests at most 12 handshakes had been
exchanged. Determine the maximal possible total number of handshakes.

However, when one starts trying out the questions, one quickly realizes the number of handshakes is very large, possibly even up to millions. This question definitely can’t be solved by trial and error!

This question is ideally modeled by a graph, and has connections to the idea of a Turán graph.

The official solution can be accessed here: http://www.fen.bilkent.edu.tr/~cvmath/Problem/1504a.pdf

To read more about Math Olympiad books, you may check out my earlier post on Recommended Math Olympiad books for self-learning.

神童背诵圆周率秀心算

This 5-year-old boy can remember Pi upto 600 decimal points.

Math Online Tom Circle

5岁儿童背 Pi 600小数位:

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Coursera Probability Course and Recommended Probability Book

Just completed the Coursera Probability Course by UPenn (University of Pennsylvania), lectured by Professor Santosh S. Venkatesh who is the author of the highly recommended book: The Theory of Probability: Explorations and Applications.

Coursera Review

The course isn’t very hard, it is very suitable for undergraduates and even high school students should be able to understand majority of the content. It actually overlaps with the A level syllabus in Singapore, and hence I would say that a 17-18 year old student would be able to grasp most of the concepts in this course.

The lecturer is very good at words, and his lectures are full of imagery and vivid descriptions. The homework is a little tricky, and hence would require some thought, even though the concepts tested are elementary (elementary in the sense that it doesn’t require calculus).

A sample of a tricky question is the “Six Saucer Question”: Six cups and saucers come in pairs: there are two cups and saucers that are red, white, and blue. If the cups are placed randomly onto the saucers (one each), find the probability that no cup is upon a saucer of the same color.

It is very tricky and to get it correct on the first try is a major accomplishment.

Overall, this Coursera Course is highly recommended, and students should try to take it the next time it comes out!

WW2 Enigma Machine

How to break 158,962,555,217,826,360,000 codes ?

Math Online Tom Circle

There are 158,962,555,217,826,360,000 possibilities of codes in the German Enigma Machine:

Flaw cracked by the genius Mathematician Alan Turing (Father of Artificial Intelligence) :

“A key can never be itself” — this is ‘the straw that breaks the camel’s back’, a critical clue to break the 158,962,555,217,826,360,000 possible codes !

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Math is Best Paid Job in 2015

Math Online Tom Circle

These 3 jobs all require Math: Actuary, Statistician and Mathematician.

Others good to have Math : Software Engineer, Systems Analyst, Data Scientist.

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Chinese Remainder Theorem History (韩信点兵)

I have written a guest post on https://chinesetuition88.wordpress.com on the very fascinating Chinese Remainder Theorem and its History (韩信点兵). Do check it out, you will be amazed at the genius of Chinese General Han Xin.

Students who are interested in Chinese Tuition may check out https://chinesetuition88.wordpress.com for more details.

Chinese Tuition Singapore

淮安民间传说着一则故事——“韩信点兵”，其次有成语“韩信点兵，多多益善”。韩信带1500名兵士打仗，战死四五百人，站3人一排，多出2人；站5人一排，多出4人；站7人一排，多出6人。韩信马上说出人数：1049。

Translation:

In Ancient China, there was a General named Han Xin, who led an army of 1500 soldiers in a battle. An estimated 400-500 soldiers died in the battle. When the soldiers stood 3 in a row, there were 2 soldiers left over. When they lined up 5 in a row, there were 4 soldiers left over. When they lined up 7 in a row, there were 6 soldiers left over. Han Xin immediately said, “There are 1049 soldiers.”

Amazing! How did Han Xin do that?

Han Xin was not only a brilliant mathematician and general, he was also a very magnanimous guy full of wisdom.

Once, when he was suffering from hunger, he met a woman who provided him with food. He promised to repay her for her kindness after he had made great achievements in life, but it was rebuffed by her…

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Think Out of The Box

Math Online Tom Circle

Answer (scroll down)

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Terry Tao, Ph.D. Small and Large Gaps Between the Primes

Math Online Tom Circle

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Quiz: Can You Solve This Sum ?

For more logic puzzles, check out:

Puzzle Baron’s Logic Puzzles

Math Online Tom Circle

[Hint]: Think out of the box…

Answer below (scroll down)
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Answer:
1 + 13 +…

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Cheryl’s Birthday Problem

We all know by now Singapore Math is not easy, but here is the viral Singapore Math problem that took the world by storm!

Question:

Albert and Bernard just became friends with Cheryl, and they want to know when her birthday is. Cheryl marks 10 possible dates: May 15, May 16, May 19, June 17, June 18, July 14, July 16, August 14, August 15, or August 17.

Then Cheryl tells Albert the month of her birthday, but not the day. She tells Bernard the day of her birthday, but not the month. Then she asked if they can figure it out.

Albert: I don’t know when Cheryl’s birthday is, but I know Bernard doesn’t know either.

Bernard: At first I didn’t know when Cheryl’s birthday is, but now I know.

Albert: If you know, then I know too!

When is Cheryl’s birthday?

Source: http://www.vox.com/2015/4/15/8420577/cheryls-birthday-singapore-math

There is a nice Numberphile video about it too.

Do give it a try! (The fun is in trying to solve the question)

Also, another fun part is sending this question to your friends!

Also see: Meet the mathematics lecturer behind ‘Cheryl’s birthday’ puzzle – See more at: http://www.straitstimes.com/news/singapore/more-singapore-stories/story/meet-the-mathematics-professor-behind-cheryls-birthday-p#sthash.qKZZtwpk.dpuf

To be honest, though Cheryl’s birthday puzzle is difficult, there are more challenging logic puzzles around. For a good challenge (and good practice), check out Puzzle Baron’s Logic Puzzles. It is a very good practice for children gearing up for Math Olympiad since they love to test logic questions in Math Olympiad.

Egyptian Math Mystery

Translation:

[The world’s most mysterious number is 142857.]

It is found in the ancient Egyptian Pyramids.

142857 x 1=142857

142857 x 2=285714

142857×3=428571

142857×4=571428

142857×5=714285

142857×6=857142

142857×7=999999

Amazing? Each multiple is a cyclic permutation of the original numer 142857.

You may read more about Egyptian mathematics in this wonderful book: Count Like an Egyptian: A Hands-on Introduction to Ancient Mathematics.

$egypt math$

3 of the Top Jobs in America involve Math

Although in Singapore currently doctors and lawyers are the top jobs, the trend is changing, starting with the most technologically advanced country – America. 3 of the top jobs in America are about Math. As the world becomes more dependent on technology (and hence mathematics), Mathematics will play a more prominent role in the global scene. Eventually the change will come to Singapore too, as more and more jobs require mathematical skills.

According to our Law Minister Mr Shanmugam, Singapore is facing a glut (excessively abundant surplus) of lawyers, which means that Singapore may not have so many jobs for lawyers. “The study of law provides an excellent training of the mind, so I don’t want to be seen as discouraging people… but you have to have a realistic understanding of the market, the economy, the total structure,” said Mr Shanmugam – See more at: http://www.straitstimes.com/news/singapore/more-singapore-stories/story/singapore-facing-glut-lawyers-shanmugam-20140817#sthash.BojzeqhX.dpuf

Hence, young students may want to consider a new discipline that is Math related, like Actuary, Math, or Statistics. To read up more about what true Mathematics is (it is very different from high school mathematics, where students just practice differentiation and integration), check out this book How to Think Like a Mathematician: A Companion to Undergraduate Mathematics.

Site: http://www.businessinsider.sg/best-jobs-of-2015-2015-4/#.VTEI7ZPoaKg

Perhaps if you had known that some of the best jobs of 2015 would require mathematical skills, you would’ve paid more attention in your high school algebra class.

Professions like actuary, mathematician, and statistician are three of the top jobs in America right now, according to CareerCast.com, a career guidance website that just released its 27th annual Jobs Rated report.

“Jobs in mathematics rank among the nation’s best jobs because they are financially lucrative, offer abundant opportunities for advancement, and provide the opportunity to do great work in a supportive environment,” says Tony Lee, publisher of CareerCast.com, in a press statement.

Here are the 10 best jobs of 2015:

2015 Rank	Job Title	Mid-level Income
1	Actuary	$94,209
2	Audiologist	$71,133
3	Mathematician	$102,182
4	Statistician	$79,191
5	Biomedical Engineer	$89,165

Terence Tao, genius mathematician

费马大定理 Fermat’s Last Theorem

Intriguing review (Chinese) of FLT by a non-mathematician. He aims to convey the beauty of Mathematics to students, who unfortunately treat Math as a tool to pass exams from PSLE, O and A level, university math course, then ditch Math upon graduation. Math is the beauty of the universe.

Math Online Tom Circle

费马大定理 Fermat’s Last Theorem (FLT): 17世纪业余数学家法国大法官费马开的一个”玩笑”, 推动350年来现代数学突飞猛进。

FLT 数学长征英雄人物:

1. Fermat (费马 1601@ Toulouse, France)
2. Galois (伽罗瓦): Group Theory (群论)
3. Gauss (高斯)
4. Cauchy (柯西) Lamé (拉梅) Kummer (库马)
5. Solphie Germain
6. Euler (欧拉)
7. Taniyama (谷山丰), Shimura (志村五郎)

集大成:
8. Andrew Wiles (怀尔斯) (证明@1993 -1995)

1.Elliptic Curve (椭圆曲线)
2. Modular Form (模形式)
3. Fermat Last Theorem (费马大定理)

(1) = (2) = (3)

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Music and math: The genius of Beethoven

Math Online Tom Circle

Music and math: The genius of Beethoven – Natalya St. Clair:

Music = Math

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Math is Forever (Spanish)

With humor and charm, mathematician Eduardo Sáenz de Cabezón answers a question that’s wracked the brains of bored students the world over: What is math for? He shows the beauty of math as the backbone of science — and shows that theorems, not diamonds, are forever. In Spanish, with English subtitles.

Yes, indeed, 1000 years from now, students will still be learning Pythagoras’ Theorem, while other fragments of human knowledge would have faded away.

Check out also this book: Arithmetic and Algebra Again: Leaving Math Anxiety Behind Forever, suitable for students who really need some encouragement and motivation to overcome fear of math! Albert Einstein once said, “You never fail until you stop trying.” Hence, even if you have not done well in math for the past years, there is still hope, don’t give up!

April Fools Video Prank in Math Class

Check out this really funny video on a April Fools Prank during a Math Class!

The teacher played a trick on his math class for April Fool’s Day. In this one, he’s showing a “homework help” video that gets some trigonometry wrong.

Looking for more Math Jokes? Check out the book below!

Math Jokes 4 Mathy Folks

Excellent MITOpenCourseware

Math Online Tom Circle

Strongly recommended free excellent MIT Math for high school, undergrads/grads and any self-study learners.

Thanks Prof. Gilbert Strang for the unselfish sharing.

http://ocw.mit.edu/faculty/gilbert-strang/

I find extremely pleasure when I discovered his brilliant lecture notes in “Generating Function” – a Discrete Math technique for computing sequencing using function, and the application in complex Combinatorics. Download here:

Example:

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Permutation Math Olympiad Question (Challenging)

March’s Problem of the Month was a tough one on permutations. Only six people solved it! (Site: http://www.fen.bilkent.edu.tr/~cvmath/Problem/problem.htm)

The question goes as follows:

In each step one can choose two indices $1\leq k,l\leq 100$ and transform the 100 tuple $(a_1, \cdots, a_k, \cdots, a_l, \cdots, a_{100})$ into the 100 tuple $(a_1, \cdots, \frac{a_k}{2}, \cdots, a_l+\frac{a_k}{2}, \cdots, a_{100})$ if $a_k$ is an even number. We say that a permutation $(a_1, \cdots, a_{100})$ of $(1, 2, \cdots, 100)$ is good if starting from $(1,2,\cdots, 100)$ one can obtain it after finite number of steps. Find the total number of distinct good permutations of $(1, 2, \cdots, 100)$ .

The official solution is beautiful and uses induction.

Personally, I used a more brute force technique to get the same answer using equivalence class theory which I learnt in my first year of undergraduate math! It is not so bad in this question, since n is only 100, but for higher values of n the approach in the official solution would be better.

If you are looking for recommended Math Olympiad books, check out this page. In particular, if you are looking for more Math Olympiad challenges, do check out this book Mathematical Olympiad Challenges. In fact, any book by Titu Andreescu is highly recommended as he is the legendary IMO (International Math Olympiad) coach that led the USA team to a perfect score!

What’s Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success

Recently, Professor Jo Boaler released her new book What’s Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success.

The minute it came out, it became an instant best seller on Amazon. Currently, there are some issues on Math education in the United States, due to the very controversial syllabus called Common Core. Professor Jo Boaler attempts to address these controversies and give suggestions and advice to parents.

I totally agree with Professor Jo’s viewpoint that the first step to engage students in math learning is via practical means and showing them how mathematics is useful and relevant to their lives. Next is to always adopt a “growth mindset”, that no matter how weak or strong a child is in math, it is always possible to improve. Just having this mindset makes a huge difference. I took Prof. Jo Boaler’s online course on “How to Learn Math“, and what she said actually makes perfect sense. Hope a new generation appreciative of math will emerge due to new research on how to best learn Math, which Prof. Jo Boaler (PhD in Math Education) is an expert in.

Without further ado, I will link Prof. Jo Boaler’s introduction to her own book:

Hi Everyone,

I wanted you to be the first to know that my new book: What’s Math Got to Do With It:? How Teachers and Parents Can Transform Mathematics Learning and Inspire Success has just hit the bookstores and of course Amazon and other online outlets.

You can now get a copy here: What’s Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success

The changes from the original book include:

2 new chapters
A focus on mindset
Ideas for the Common Core
An infusion of new research through the book

Why not buy the book for your principal? Or your colleagues? your family? your students’ parents? or others who you think may need to understand the nature of good mathematics teaching? You may need people to know the research evidence behind what you are doing, as well as get some new ideas yourself.

For youcubers in the UK there will be a new edition of The Elephant in the Classroom coming out in the Autumn, we will let you know when, of course.

I also wanted you to know about some book signings that are planned:

Friday April 3 Portsmouth, New Hampshire. The Exeter High School Auditorium 7-8.30pm talk followed by book signing. See:

http://www.seacoastonline.com/article/20150330/NEWS/150339887/101019/NEWS

At NCSM:
Monday April 13th Boston, NCSM Following Jo’s keynote talk

At NCTM:
Thursday April 16th, 11.30 After Jo’s networking session.

We will also be arranging a book signing in the San Francisco bay area soon too.

I hope to see you at one of them. Below is our youcubed team reading the book yesterday 🙂

Viva La Revolution

The Math of Shuffling Cards

Previously, the first YouTube video wasn’t working. I have added a new link to the interesting “Looking at Perfect Shuffles” video. 🙂

Mathtuition88

A magic trick based on the “Perfect Shuffle”. Featuring Professor Federico Ardila. I watched his videos on Hopf Algebras while learning the background material for my honours project on Quantum Groups.

Mathemagician Persi Diaconis discusses which is the best way to shuffle: Overhand shuffle, Riffle Shuffle, or “Smoosh” Shuffle? Watch the video to find out!

Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks is an interesting book by Professor Diaconis, featuring Magic Tricks that have a mathematical background! This book is a great idea for a gift for students, teachers, or friends!

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NUS High Selection Test (DSA)

Official Website: http://www.nushigh.edu.sg/admission-n-outreach/admissions/eligibility-n-admissions-process

The official website, unfortunately, doesn’t tell much about how the NUS High Selection Test / DSA is like, in particular the format of the exam.

However, from online sources from students who took the test, we can have a glimpse of what the NUS High Selection Test (DSA) is like.

Disclaimer: I have not taken the NUS High Selection Test (DSA) before, and I am only listing down suggested format of the tests based on the online sources. I have taken the GEP Selection Test (both round 1 and round 2) though, at Primary 3.

Source 1: http://wwwdontmesswith6a.blogspot.sg/2011/06/nus-high-selection-test.html

This is a highly reliable blog post by the sister of Lim Jeck, a highly skilled Math Olympiad Participant who has achieved perfect score at IMO. From the blog post, we can tell that:

The Math Paper is 1 and a half hours.
Math Paper is “ok” (easier than NMOS) Do take note that the blogger is very good at math, so “easy” is subjective.
Math Paper has 7 pages, inclusive of cover page and last page.
23 Non-MCQ questions, where you have to shade the integer answer. (Do bring a pencil!)
“The first few Math questions are easy, like P6 Math questions. One of the easiest Math questions is, the average of 3 numbers is given, you add another 2 numbers and you get another given average, you have to find the sum of the 2 numbers added. There are varying marks for different questions. I think the harder questions carry 4 marks.”
(Again, easy is subjective, what is easy for a Olympiad Gold Medalist may not be easy at all)
“Total marks for Maths and Science are 55 and 30 respectively.For Maths, max of answer shades is 4, so max answer may be 9999. Maths questions carry 1 mark, 2 marks, 3 marks and 4 marks. Think Q23 (last qn) is a 4-mark question.”
(We can assume that due to the format of this test, all answers are integers!)

Read more at http://wwwdontmesswith6a.blogspot.sg/2011/06/nus-high-selection-test.html to get an idea of the original post and how the Science NUS High DSA (supposedly more difficult than the Math NUS High DSA) is like.

To deal with difficult NUS High DSA problems (last few questions of the Selection Test), most likely the student has to be trained in Math Olympiad. A book like The Art of Problem Solving Volume 1: The Basics AND Basics Solution Manual (2 Volume Set would be ideal in beginning the journey in Math Olympiad. Note that Math Olympiad is nothing like normal school math, and even a fresh university graduate in a math-related major say Engineering/Accounting would have great problems solving a Primary 6 Math Olympiad question, if he doesn’t have the necessary Math Olympiad background!

If you are also interested in preparing for GEP (Primary 3 or Secondary 1 intake), do check out my most popular page on Recommended Books for GEP.

Other blogs with info on the NUS High DSA Selection Test:

Update (2016): Check out this Pattern Recognition (Visual Discrimination) book that is a guided tutorial for training for GEP / DSA Tests!

The Math of Shuffling Cards

Mathemagician Persi Diaconis discusses which is the best way to shuffle: Overhand shuffle, Riffle Shuffle, or “Smoosh” Shuffle? Watch the video to find out!

Mathematicians have prevented a world disaster, behind the scenes

Recently, after taking the Coursera course on Cryptography, I had a better appreciation of mathematics and the role of cryptography in our modern society.

I was pleased to read this article Quantum compute this: Mathematicians build code to take on toughest of cyber attacks, and Washington State University mathematicians have designed an encryption code capable of fending off the phenomenal hacking power of a quantum computer.

The quantum computer, though not yet invented, is widely believed to be available soon in the next few years. In the hands of hackers, the quantum computer would be a formidable weapon as current cryptographic methods are extremely vulnerable to the quantum computer as it can factor numbers extremely quickly, leading to number theoretic codes being broken.

What would happen if a Quantum Computer is built

Quantum computers are near

Quantum computers operate on the subatomic level and theoretically provide processing power that is millions, if not billions of times faster than silicon-based computers. Several companies are in the race to develop quantum computers including Google.

Internet security is no match for a quantum computer, said Nathan Hamlin, instructor and director of the WSU Math Learning Center. That could spell future trouble for online transactions ranging from buying a book on Amazon to simply sending an email.

Hamlin said quantum computers would have no trouble breaking present security codes, which rely on public key encryption to protect the exchanges.

In a nutshell, public key code uses one public “key” for encryption and a second private “key” for decoding. The system is based on the factoring of impossibly large numbers and, so far, has done a good job keeping computers safe from hackers.

Quantum computers, however, can factor these large numbers very quickly, Hamlin said. But problems like the knapsack code slow them down.

Fortunately, many of the large data breaches in recent years are the result of employee carelessness or bribes and not of cracking the public key encryption code, he said.

Hence, when many people say mathematics is useless, they are actually extremely wrong, as mathematics permeates every aspect of life! Even though maths like calculus is not directly used in everyday life, it is part of our phone, computer, and every part of the modern lifestyle.

Kudos to the mathematicians who have averted a world disaster, before quantum computers are even invented!

If you are interested in what a quantum computer is, and what it can do (it is so powerful that whoever has one would hold the keys to the entire internet), check out this book Schrödinger’s Killer App: Race to Build the World’s First Quantum Computer.

Written by a renowned quantum physicist closely involved in the U.S. government’s development of quantum information science, Schrödinger’s Killer App: Race to Build the World’s First Quantum Computer presents an inside look at the government’s quest to build a quantum computer capable of solving complex mathematical problems and hacking the public-key encryption codes used to secure the Internet. The “killer application” refers to Shor’s quantum factoring algorithm, which would unveil the encrypted communications of the entire Internet if a quantum computer could be built to run the algorithm. Schrödinger’s notion of quantum entanglement—and his infamous cat—is at the heart of it all.

Vector Subspace Question (GRE 0568 Q3)

This is an interesting question on vector subspaces (a topic from linear algebra):

Question:
If V and W are 2-dimensional subspaces of $\mathbb{R}^4$ , what are the possible dimensions of the subspace $V\cap W$ ?

(A) 1 only
(B) 2 only
(C) 0 and 1 only
(D) 0, 1, and 2 only
(E) 0, 1, 2, 3, and 4

To begin this question, we would need this theorem on the dimension of sum and intersection of subspaces (for finite dimensional subspaces):

$\dim (M+N)=\dim M+\dim N-\dim (M\cap N)$

Note that this looks familiar to the Inclusion-Exclusion principle, which is indeed used in the proof.

Hence, we have $\dim(M\cap N)=\dim M+\dim N-\dim (M+N)=4-\dim (M+N)$ .

$\dim (M+N)$ , the sum of the subspaces M and N, is at most 4, and at least 2.

Thus, $\dim (M\cap N)$ can take the values of 0, 1, or 2.

Answer: Option D

If you are looking for a lighthearted introduction on linear algebra, do check out Linear Algebra For Dummies. Like all “For Dummies” book, it is not overly abstract, rather it presents Linear Algebra in a fun way that is accessible to anyone with just a high school math background. Linear Algebra is highly useful, and it is the tool that Larry Page and Sergey Brin used to make Google, one of the most successful companies on the planet.

Professor Stewart’s Incredible Numbers

Amazon just informed me of a new book which is the #1 New Release Math Book on Amazon!

The book is titled: Professor Stewarts Incredible Numbers.

At its heart, mathematics is about numbers, our fundamental tools for understanding the world. In Professor Stewart’s Incredible Numbers, Ian Stewart offers a delightful introduction to the numbers that surround us, from the common (Pi and 2) to the uncommon but no less consequential (1.059463 and 43,252,003,274,489,856,000). Along the way, Stewart takes us through prime numbers, cubic equations, the concept of zero, the possible positions on the Rubik’s Cube, the role of numbers in human history, and beyond! An unfailingly genial guide, Stewart brings his characteristic wit and erudition to bear on these incredible numbers, offering an engaging primer on the principles and power of math.

Previously, I read Galois Theory, Third Edition (Chapman Hall/Crc Mathematics), also by Ian Stewart, and I have to say his style is very accessible to the average reader. Not overly technical or abstract, he actually explains Galois Theory in as concrete a way as possible, which is not easy, since Galois Theory is one of the most abstract topics in mathematics.

I read the Third Edition, featured above, but lately there is a newer and better fourth edition: Galois Theory, Fourth Edition.

What is a Tensor?

Most people don’t encounter Tensors (the higher level advanced version of Matrices) until they reach senior undergraduate, or even graduate level.

What is a Tensor?

The best explanation I have ever seen, comes from this video by the author of A Student’s Guide to Vectors and Tensors, Daniel A. Fleisch. Using children’s blocks and laymen language, he explains what is a tensor clearly and succinctly in a way that is unbelievably crystal clear.

This YouTube video is watched over 200,000 times, a very commendable achievement for a math video!

Official Definition by Wikipedia

Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples of such relations include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of numerical values. The order (also degree) of a tensor is the dimensionality of the array needed to represent it, or equivalently, the number of indices needed to label a component of that array. For example, a linear map can be represented by a matrix (a 2-dimensional array) and therefore is a 2nd-order tensor. A vector can be represented as a 1-dimensional array and is a 1st-order tensor. Scalars are single numbers and are thus 0th-order tensors. The dimensionality of the array should not be confused with the dimension of the underlying vector space.

If you have some programming knowledge, you may view tensors as a type of multidimensional array. A more mathematical abstract way can be achieved by defining tensors in terms of elements of tensor products of vector spaces, which in turn are defined through a universal property.

Cool? The word “tensor” really strikes me as a word that is really sophisticated and complicated!

How to find the distance of a plane to the origin

Given the equation of a plane: ax+by+cz=D, or in vector notation $\mathbf{r}\cdot \left( \begin{array}{c} a\\ b\\ c\\ \end{array}\right)=D$ , how do we find the (shortest) distance of a plane to the origin?

(When a question asks for the distance of a plane to the origin, by definition it means the shortest distance.)

One way to derive the formula is this:

Derivation

Let X be the point on the plane nearest to the origin.

$\overrightarrow{OX}$ must be perpendicular to the plane, i.e. parallel to the normal vector $\mathbf{n}=\left(\begin{array}{c}a\\b\\c\\\end{array}\right)$ .

Furthermore, X lies on the plane, hence we have $\boxed{\overrightarrow{OX}\cdot\mathbf{n}=D}$

Using the formula for dot product, we can get $|\overrightarrow{OX}\cdot\mathbf{n}|=|\overrightarrow{OX}||\mathbf{n}|\cos \theta=D$

Since $\overrightarrow{OX}$ is parallel to $\mathbf{n}$ , $\theta$ is either 0 or 180 degrees, hence $\cos \theta$ is either 1 or -1.

Thus, we have $|\overrightarrow{OX}||\mathbf{n}|=|D|$ .

The shortest distance from the point X to the origin is then $\displaystyle|\overrightarrow{OX}|=\frac{|D|}{|\mathbf{n}|}=\frac{|D|}{\sqrt{a^2+b^2+c^2}}$

Ans: Shortest distance from point to plane is $\displaystyle\boxed{\frac{|D|}{\sqrt{a^2+b^2+c^2}}}$

H2 Maths Condensed Notes and Prelim Papers

If you are looking for a short summarized H2 Maths Notes, with Prelim Papers to practice, do check out our Highly Condensed H2 Maths Notes!

GEP Questions: The Gauss Trick

We will continue our series on GEP Questions. To learn more about Recommended Books for GEP, to practice GEP Questions, visit the link here.

Today, we will discuss the quintessential GEP Question: The Gauss Trick. This GEP question illustrates the fact that giftedness can be trained to a large extent.

Question:

Find the sum of 1+3+5+7+…+95+97+99.

Solution will be below after this text, scroll down after you are ready to see the answer!

If a 9 year child can solve this on his/her first attempt (i.e. see the question for the very first time), then the child is actually at the level of Carl Friedrich Gauss, the legendary mathematician! When Gauss was a young kid, his teacher set the class a difficult question, 1+2+3+…+99+100, hoping to keep the children quiet and busy while he could have some time to relax. Little did he expect Gauss to come up with the right answer minutes later.

Most 9 year old kids would not be able to solve this on their first try. I wouldn’t be able to solve it correctly even if given an hour when I was a kid! If the children have seen it before and practiced, that is a different story, as it becomes so easy even for a 7 year old kid. Hence, practicing GEP Questions actually leads to an indirect boost of IQ in this manner! The transition from ignorance to knowledge, leading to increased intelligence, can be accomplished by practice! To practice more of these GEP Questions, check out the Math Olympiad Recommended Books page. As a tutor, I know that this Gauss Trick is a “must-know” question for students aiming high for school Math / GEP Selection Test, since almost every kid knows this nowadays, and it is highly popular in tests.

In fact, this technique (Gauss Trick) commonly tested in GEP Questions can be used all the way to JC and beyond! In JC it is covered under the topic of AP/GP (Arithmetic Progression and Geometric Progression).

Solution:

Now, for the solution. This sum is the famous arithmetic progression, where each term differs from the next by a fixed constant. In this case, the constant is 2.

Some solutions use a number pair matching, which can be problematic to explain when the number of terms is odd, but still works nevertheless. We can use another method of writing the sum backwards.

First we note that there are 50 terms in this series. We can know that either by noting that they are the odd numbers from 1 to 100, and half of the numbers are odd, hence there are 100/2=50 terms. Alternatively, we can note that 1=2(1)-1, 3=2(2)-1, 5=2(3)-1, …, 99=2(50)-1, where the number in the brackets acting like a counter.

1+3+5+7+…+95+97+99
99+97+95+…+5+3+1

Note that each pair, 1+99, 3+97, 5+95 actually add up to the same thing, i.e. 100.

Adding up the two expressions above, we get 100×50=5000.

Dividing that by two (since we double counted), we will get 5000/2=2500.

Ans: 2500

Hope you enjoyed solving this question!

Check out this amazing book on Gauss: The Prince of Mathematics: Carl Friedrich Gauss

Gauss is a true Math genius, and you can read more about his life in this interesting biography! This historical narrative will inspire young readers and even curious adults with its touching story of personal achievement.