H3 Mathematics Resource Page

H3 Mathematics is the pinnacle of the Junior College Mathematics syllabus in Singapore. It contains a glimpse of actual Math that Mathematicians do, and it requires true mathematical understanding and technique to do well. (H1/H2 math requires a lot of practice, but not true understanding. It is quite common for students to “apply the method” and get the correct answer without having any idea of what they are actually doing.)

Topics in H3 Mathematics include Functions, Sequence and Series, Combinatorics, and even Number Theory. Certain schools also include topics like Linear Algebra and Differential Equations. Certainly, the H3 Math questions have a Math Olympiad style to them.

Here are some practice questions for H3 Math (more will be added in the future), with some hints. Questions are adapted from actual H3 prelim papers.

Functions

Q1) The function $latex f$ is such that $latex f(x+2)=af(x+1)-f(x)$, for all real $latex x$ and…

View original post 379 more words

Unknown German Retiree Proved The “Gaussian Correlation Inequality” Conjecture

https://www.wired.com/2017/04/elusive-math-proof-found-almost-lost

$\boxed {P (a + b) \geq P (a) \times P (b)}$

Case “=” : if (a, b) independent
Case “>” : if (a, b) dependent

Thomas Royen used only high-school math (function, derivative) in his proof in 2014. He then published it in arxiv.org website – like Perelman did with the “Poincaré Conjecture”.

Fun Math Without Math

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See clearer if change person to taxi car, bun to passenger.

9 taxi cars send 9 passengers will take the SAME timing as 3 taxi cars send 3 passengers.

[INMO 1993]

QS University Ranking 2017 by Math Subject

This ranking has criteria which are quite subjective, eg. Employer reputation (depends on individual country, local employment for Math graduates, job opportunities), also Citations refer predominantly to English publications which are not in favor of non-English speaking countries like China, Japan, France, etc.

If by real substance of Math education for Mathematicians, the 32th Ecole Normale Supérieure (Paris) – the alma mater of Evariste Galois – should be ranked among the Top 3 in the World, having produced 1/3 of the Fields Medals (Nobel Prize equivalent in Math) by a single country and a single university.

Top 5:

1. MIT
2. Harvard
3. Stanford
4. Oxford
5. Cambridge

18. University of Tokyo

20. Peking University 北京大学

22. Ecole Polytechnique (France)

26. TsingHua University 清华大学

28. Hong Kong University

32. Ecole Normale Supérieure (Paris)

苏联老师Arnold 如何 教中小学 抽象”群”

How this Russian math prof taught secondary school kids the advanced math in Group & Field Theory without the boring axiomatic approach which tortures most math students.

The Map of Mathematics

Show to your schooling children why they need to study Maths – the Queen of all Sciences – which pushes the frontier of human evolution in last 3,000 years. Maths is always invented few centuries or decades before it becomes useful. For examples:  Complex numbers invented accidentally by the 16th century Italian Mathematicians for solving polynomial equation of 3rd degree, became useful in Physics Electrical and Magnetic Fields (19 CE) ; Invention of Analytic Geometry (17 CE) allowed Newton to trace the earth-sun orbit; Calculus propelled Physics and Physical Chemistry; Leibniz’s Binary Math (18 CE) discovery applied in Computing (20 CE)…

Latest Examples

1. Topology was invented in 1900 by French PolyMath Henri Poincaré, today applied in Big Data, AI…

2. His PhD student invented “Derivatives” Partial Differentiation, today applied in Commodity Trading, Stock Trading, Financial Derivatives… with Black-Sholes formula. 1998 USA Sub-Prime Crisis due to the misuse and lack of understanding of its limitation (“fat tail” ).

3. Mathematician SS Chern 陈省身and Nobel Physicist Yang Zhen-Ning 杨振宁were working independently in the USA for 40 years, Chern on Differential Geometry, Yang on Yang-Mill Equation (one the 7 unsolved Math Problems in 21st century). Through a common friend the hedge fund billionaire James Simons – Chern’s former PhD Math student and university colleague of Yang – they realised that the Math “Fiber Bundles” (纤维丛) invented by Chern 30 years earlier could apply in Yang’s Physics (Gauge Theory).

Trump’s Speaking Math Formula

The lower in the score the better : Trump (4.1) beats Hillary (7.7) who beats Sanders (10.1)

Trump defied most expectation from the world to win the 2017 President of the USA. His victory over the much highly educated Ivy-league Yale lawyer-trained Hilary Clinton who speaks sophisticated English is “SIMPLE English“: seldom more than 2-syllable words.

1-syllable words mostly: eg.dead, die, point, harm,…

2-syllable words to emphasize: eg. pro-blem, ser-vice, bed-lam (疯人院), root cause, …

3-syllable words to repeat (seldom): eg. tre-men-dous

His speech is of Grade-4 level, reaching out to most lower-class blue-collar workers who can resonate with him. That is a powerful political skill of reaching to the mass. Hilary Clinton’s strength of posh English is her ‘fatal’ weakness vis-a-vis connecting to the mass.

In election time, it is common to see candidates who win the heart of voters by using the local dialects of the mass, never mind they are discouraged in schools or TV: Hokkien, Teochew, etc.

Singapore Math (PSLE)

This is the typical Singapore PSLE Math for 12-year-old school students sitting for Primary School Leaving Exams (PSLE).

The Primary 6 kids have not learnt Algebra, which will be taught only 1 year later in Secondary schools.

Singapore is proud of her unique “Singapore Math“, characterised by Polya-style Problem-Solving Process , aided with visual Modelling and Guesstimation techniques.

It is Singapore’s secret of being the World’s 2nd position in PISA Test (Math), after China (Shanghai) which beats us with Algebra teaching at this age.

See if you can solve this problem without Algebra.

If stuck … (Answer below) …
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数学名词妙翻译

Homology: 同调 (江泽涵 教授)
Homotopy : 同伦 (江泽涵 教授)
Manifold : 流形 (源于文天祥的《正气歌》: 天地有正气，杂然赋流形)
Geometry: 几何 (明朝 徐启光 / Italy Jesuit 利马窦 )
Topology: 拓朴 (杜甫诗)

Truth = Knowledge + Understanding + Believe

K = Knowledge (comes from Proving)
U = Understanding
B = Believe

U is the mediator between K & B

Truth = KUB
Eg. Mathematics:
1 + 1 = 2

KB = Know & believe but do not understand.
Eg. Diseases like cancer, Chinese Accupuncture, …

B: Believe but do not understand or know.
Eg. Axioms, Religion

K : Know but do not understand or believe.
Eg.

KU : Know, Understand but do not believe.
Eg.

U: Understand but do not know or believe.
Eg. ?

UB: Understand, believe but do not know.
Eg. [Impossible]

3rd Isomorphism Theorem

This 3rd Isomorphism Theorem can be intuitively understood as:

G partitioned by a bigger normal subgroup H
is isomorphic to:
{G partitioned by a smaller normal subgroup K (which is a subgroup of H)}
partitioned by
{H partitioned by a smaller normal subgroup K}

or, by ‘abuse of arithmetic’: divide G & H by a common factor K.

$(G / H ) = (G / K ) / (H / K )$

Analogy:
$100 /$50 = 2 (two $50 notes makes$100)
is same (isomorphic) as
$100 /$10 = 10, (ten $10 notes makes$100)
$50/$10 = 5, (five $10 notes makes$50)
then 10/5 = 2 (ten notes split into five is two )

Singapore PSLE 2015 Math

PSLE is “Primary School Leaving Exams” for 11~12 year-old children sitting at the end of 6-year primary education. The result is used as selection criteria to enter the secondary school of choice.

Hint: Without seeing or feeling the weight of the $1 coin, you still can guess the answer. This is the essence of “Singapore Math” — using “Guesstimation“. Answer (below): Try before you scroll down. If wrong answer, please go back to primary school 🙂 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . Answer: By elimination using ‘guesstimation’ method: A) 6g too light, D) 6 kg too heavy. Left B & C… C) 600 g > half kg for 8 coins ($1), not possible.

Ans: B = 60 g.

Life Algebra

How to solve this ‘Life’ Algebra ?

The simultaneous inequality equations with 3 unknowns {e, t, m}, resp. for energy, time, money.

Retire after 55 before 60, then you get optimized {e, t, m} — still have good energy (e) with plenty of time (t) and sufficient pension money (m) in CPF & investment saving.

Beyond 60 if continuing to work, the solution of {e, t, m} -> {0, 0, 0}.

Note: After death (e = t = 0), money is not yours (m = 0).

René Descartes

René Descartes (31 March1596 – 11 Feb1650), the 17th century French mathematician who invented XYZ Cartesian Analytical Geometry.

He also invented the ‘Methodology’. His scientific philosophy ‘Cogito ergo sum’ (Je pense donc je suis / I think therefore I am / 我思故我在) influenced Issac Newton later.

Goldbach Conjecture & Theorem Chen

It said: “(The above) Prof Yang was a long time colleague of Chen Jingrun (陈景润). He revealed that Chen was, by nature, not a Math genius, his success in climbing the Math Summit was through sheer determined hardwork — especially during the decade long cruel Cultural Revolution when the intellectuals were persecuted — sustained by a very strong interest in Math.”

“Math journey is not a short 100-meter sprint but a 20-year marathon run.”

“He critised the crazy rush of Math Olympiad training in whole China, not only adding extra school burden on kids, it also kills their interest in Math.”

Note: Recently China government bans IMO training in schools.

Chen Jingrun (陈景润), a student of Hua Luogeng (华罗庚), proved in 196x during The Cultural Revolution: (1+2) Goldbach Conjecture.

Goldbach Conjecture:

Any Even Integer = 1 prime + 1 prime

eg. 12 = 5+7 = “1+1”

Thinking in Philosophy:
Any prime (odd numbers except 2) together with another prime make an Even number:

Two single odd (prime) persons together make a couple (even).

This is the biggest secret in Nature (Natural Number). To prove it is extremely impossible, unless guidance from God.

Theorem Chen:

Chen’s proof is the world’s best approximate one (so far):
Any Even Integer = 1 prime + (1 prime x 1 prime) = “1+2”

Note: (1 prime x 1 prime) = semi-prime

1 单(男) + 2 单(妻x妾) = 对偶
。还是非天意 🙂

IMO 2015 USA beat China after 20 Years

The result is not surprising to China but to USA:
♢Recently China government bans IMO training in schools.
♢Obama was surprised that the USA IMO team consists of predominantly Chinese American students.

IMO Math is like ‘Acrobatics’ to real ‘Kung-fu’, it is not real Math education, but special ‘cute’ techniques to solve tough ‘known’ solution problems. Real Math is long R&D solving problems with UNKNOWN solution (eg. Fermat’s Last Theorem, Riemann Conjecture,…)

2 types of Math: Algorithmic or Deductive (演绎). Chinese long traditional ‘abacus’ mindset, procedural computational Math is Algorithmic, applied to special cases (eg. astronomy, calendar, agriculture, architecture, commerce,…). European Greek’s Euclid deductive, step-by-step axiom-based proofing, is theoretical, generalized in all cases (Geometry, Abstract Algebra,…)

Look at the Fields Medal (aka ‘Nobel Prize’ of Math) super-power – France – which has produced 1/3 of the Fields Medalists, but performing so-so in IMO. In contrast, China has ZERO Fields Medalists, albeit dominating IMO championship for more than 2 decades!

IMO 2015:
https://www.imo-official.org/results.aspx
USA 1st,
China 2nd,
South Korea 3th,
North Korea 4th,
Vietnam 5th,
Australia 6th
Iran 7th
Russia 8th
Singapore 10th [2012 Individual World’s Champion ]
Ukraine 11th
Thailand 12th
Romania 13th
France 14th

United Kingdom 22th

Why doctors need Maths in Big Data Era

In Big Data era, doctors would need to learn in Medical schools the following skills for data-driven analysis:
– Mathematics
– Statistics
– Probability
– Data Analysis and Tools.

2015 British GCSE Tough Math Question

2015 GCSE O Level Math paper, not so difficult but thousands of Sec. 4 British students complain tough. It is an excellent question combining Probability & Algebra.

http://mashable.com/2015/06/05/math-exam-gcse-question/

The British Math system which Singaporeans inherit at Sec. schools is applied and non-abstract thinking. The fact that it uses abstract ‘n’ (instead of concrete number, say, 10), the students’ brain cannot visualize ‘n’ sweets.

The French Math system, on the opposite, is abstract since Troisième, equivalent to Sec. 3, but not very applied.

It is good to have the best of both worlds: applied and abstract Math from Sec.3 /4.

Quiz: Can You Solve This Sum ?

[Hint]: Think out of the box…

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1 + 13 + 15 = 30 in 9 Based Numeric System.
i. e. 1 + 12 + 14 = 27 in Base10

Explain: In Base 9
${1}_{9} = 1.9 = {1}_{10}$
${13}_{9} = 1.9 + 3 = {12}_{10}$
${15}_{9} = 1.9 + 5 = {14}_{10}$
${30}_{9} = 3.9 + 0 = {27}_{10}$

Excellent MITOpenCourseware

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

Thanks Prof. Gilbert Strang for the unselfish sharing.

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

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

Example:

Matrix Tensor Product

Note the difference from Matrix Multiplication:

Singapore Math for 11-year-old Kids : Modelling

Compare below the Singapore Math versus Algebra Method:

Mrs. Lim 68 (son 36, daughter 32)

Algebra Method: Try yourself !
(It involves solving 2 equations of 2 variables.)

There are certain types of problem easier to solve with Modelling (example: this one and the “Monkeys & Coconuts Problem” with Modelling Method); whereas there are some other types difficult to visualize in model, but straight forward with Algebra Method.

Modelling Math is excellent for lower primary kids (Primary 1 to Primary 4), while upper primary (Primary 5 – Primary 6) should be taught Algebra (1-variable to 2-variable equations).

The reason in PISA Test China (Shanghai) kids (PISA World’s #1) beat Singapore kids (PISA World’s #2) in Math is they learn Algebra in upper primary schools (11-12 year-old).

PSLE Maths

This is Singapore PSLE (Primary School Leaving Exams) Maths for 12- year-old kids.

It is not difficult for adults, but I wonder most kids have such analytical skill ?

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(1)

(2):
R – B = 1 for every suite of 5 cards
If R-B = 36
36×5= 180 cards

Walnut Math

A friend from China gave us a bag of walnuts plucked from their home-grown walnut tree. I decide to count them by applying math:

A stack of walnuts piled in a pyramid, with base layer arranged in a square of 6×6 walnuts, above layers 5×5, 4×4, 3×3, 2×2, and finally top 1 (1×1).

How many walnuts are there in total ? (Answer: 91)

This is simple math but only taught in A-level (with proof by induction).

$\displaystyle \boxed { \sum_{1}^{n} k^2 =\frac { n (n+1)(2n+1)} {6} }$

Note:
If $k^2$ is changed to $\frac {1}{k^2}$, this is Euler’s “Basel Problem” which leads to the unsolved Riemann Hypothesis.

Below is a 400-year-old walnut tree: walnut is called “Wise fruit 聪明果”, it looks like human brain, also has proven nutritious benefits to brain.

Half Factorial

Prove : For any positive integers p, k,
$(p^k)! \text { is divisible by p }$
Proof:
Apply Factorial Formula:
$\boxed {n!=n.(n-1)! }$

$(p^k)! = (p^k). (p^k -1)! = p.(p^{k-1}).(p^k -1)!$
hence divisible by p. [QED]

Why 0!=1

Answer: (don’t scroll down until you have tried)

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Mathematician’s Job: High Pay but Lowest Stress

High pay high stress ?

Not really true … among the top 17 high-paying jobs (annual earning above US $100,000) in the USA, Mathematician’s job has the lowest stress below 60 (in the scale from 0 no stress to highest stress at 100). Other high-pay-high-stress professional jobs (Annual Salary above US$100,000) :
Orthodontists ($196,270 : 67.0) Computer Hardware Engineer ($106,930 : 67.0)
Astronomers ($110,440 : 62.0) Political Scientists ($100,900 : 60.8)
Law Teachers ($122,280 : 62.8) Actuaries ($107,740 : 63.8)
Physicists ($117,040 : 61.3) Optometrists ($111,640 : 70.3)
IT Managers ($132,570 : 64.3) Geoscientists ($108,420 : 62.5)
Mathematicians (\$103,310 : 57.3)

Shanghai Jiaotong Universities Ranking 2014 (Mathematics)

The unique universities ranking by numbers of Nobel Prizes / Fields Medals and Research citations from faculty members and alumni:

France:

Singapore: need to work harder to win some Nobel / Fields medals !

Calabi-Yau “The Shape of Inner Space”

Do we really live in 10-dimensional Space ? Harvard Prof S.T. Yau (1st Chinese Fields Medalist) talked on the inner space of Geometry and String Theory in Physics:

《演讲原文摘要》:

2006.03.18 人物-数学界的诺贝尔奖获奖者- 丘成桐:

Reference: The best seller book “The Shape of Inner Space” can be loaned at the Singapore National Library branches.

IMO Number Theory

IMO中的数论

Theorem:

U and V co-prime if there exist intergers m, n such that

m.U + n. V = 1

Concrete and Abstract in Modern Math

Preface on “Introduction to Number Theory” by Hua Luogeng (1956).

“Math evolved from concrete to abstract, the former is the source of inspiration of the latter. One cannot just study the abstract definitions and theorems without going back to the source of concrete examples, which has proven valuable applications in Physics and other sciences.”

“Mathematics, in essence, is about the study of Shapes and Numbers. From Shapes give rise to the Geometrical Intuition, from Numbers give the Relationship and Concepts

张益唐谈做数学

2013年7月13日 台大访问笔记摘要 Summary:

http://blog.sina.cn/dpool/blog/s/blog_c24597bf0101ctdp.html

Holistic Approach to Attack Math :

10岁的启蒙书:

Ref: 白居易写给元稹《与元九书》

Shimura Modular Form:

On Riemann Hypothesis:

Applied Math in Medicine

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

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

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

The successful diagnosis (with Math): 95%

Lie Algebras & Lie Groups

Best way to study abstract math is to use concrete examples, visualizable 2- or 3- dimensional mathematical ‘objects’.

Groups:
(Do not need to search too far…) Best example is the Integer Group (Z) with + operation, denoted as {Z, +}. (Note: Easy to verify it satisfies the Group’s 4 properties: CAN I“.

It has infinite elements (infinite group)

It is a Discrete Group, because it jumps from one integer to another (1,2,3…, in digital sense).

The group of rotation of a round table, which consists of all points on a circle, is an infinite group. We can change the angle of rotation continously between 0 and 360 degrees, rotating around a geometrical shape – a circle. Such shapes are called Manifolds (流形).

All points of a manifold forms a Lie group.

Example: The group of rotations of a sphere around a central axis, (eg. The Earth), is a Lie group SO(3) – a special orthogonal (meaning preserve all distances) group.

SO(2): Rotations of a circle in 2-dim space.

SO(n): Rotations of a manifold in n-dim space.

Note: SO(2), SO(3), SO(n) are both infinite groups and manifolds, so they are Lie groups.

Example of infinite dimensional Lie group is a Loop group.

Approximation at a given point
1. For SO(2) – manifold circle – it can be approximated at a given point by the tangent.
2. For SO(3) – manifold Sphere – at a given point by a tangent plane.
(Note: The cross-product operation makes the 3-dimensional space into a Lie algebra.)
3. For Lie group in general – SO(n) – its special point is the identity element of this group. The approximation tangent space at this special point is the Lie algebra.

Why study Lie Algebra instead of Lie Group ?
Unlike a Lie group which is usually curved (like a circle), Lie algebra is a flat space (like a line, plane, etc). This makes Lie Algebras easier to study than Lie groups.

Notes:
1. The Lie algebra of an n-dimensional Lie group is an n-dimensional flat space, also known as a Vector Space.
2. Kac-Moody Algebras are the Lie algebras – which we should think of as the simplified versions – of the loop groups.
3. Quantum groups are certain deformations of Lie groups.

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