Hong Kong Joint School Mathematics Society HKJSMS Newsletter
Issue 2 12-13
March 2013
From the Editor Dear readers, entering 2013 and the Year of Snake, the HKJSMS newsletter is taking a great leap forward. We wish a fruitful year. Thanks to the president and the publication team for supporting the new ‘corners’ in this publication. Please enjoy reading :-) Eileen Tam & Jeff Siu
Interview with a university student majoring in Mathematics P.2 // Fun with Mathematics P.4 - 11 Beauty of Mathematics P.5 // Get known to Mathematician P.6 // Introduction of Theorem P.7 Proof without Words P.9 // Relaxing Time! P.11
ALGEBRA
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Interview with a university student majoring in Mathematics Patrick Wong Year 2 Student majoring in Mathematics in the Chinese University of Hong Kong By Jeff Siu J: W: J:
Jeff Siu Patrick Wong
As I know you are studying Year 2 of Mathematics in the Chinese University. Why did you choose to major in Mathematics when you applied into university?
W: Actually when I was in lower forms, I didn’t feel much interest in Mathematics. At that time I thought Mathematics was not meaningful at all as we used to focus on the result we got
J:
from a question rather than the calculation process. However, when I was studying pure Mathematics in A level, it paid much more focus on the calculation logic and the thinking
W: Of course not! Actually I am also now studying two minor courses. They are Economics and
strategies. I really enjoyed the calculation process. Therefore I found Mathematics meaningful and wanted to have further studies on it.
Dutch. I believe that Mathematics is not the only focus when you enter university. We should also study other branches of knowledge. Never-
I want to know more about your life in the university. How many lessons do you have per week?
theless, Mathematics is a great tool to help you to understand the concepts in those subjects. For example, differentiation is a good tool for you to understand and apply the concept of “optimum price” in economics. Even for psychology you also need to apply the concepts of what you learnt about statistics in Mathematics
W: I have 20 lessons per week and a lesson lasts for 45 minutes. J:
Wow! What are the lessons about?
W: There are lots to study and you can choose what you would like to learn yourself. For example, I studied analytical mathematics, calculus, and linear algebra last year. This year I studied abstract algebra, differential equations and the basics of the application of Mathematics in daily life in the first semester. I will study game theory and mathematical models in the next term. J:
lessons. J:
It seems that university has great freedom in choosing subjects to study. However, do universities require students to study some ‘Core Subjects’, like what we did in secondary schools?
W: Of course! We also need to study Liberal Studies, English and Chinese. For Liberal Studies, there are a few branches for us to study. They
Are the lessons only about Mathematics?
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are Science, Chinese culture, humanities, etc. J:
Do you have any professors that you admire a lot?
W: Actually there is one. He is an Assistant Professor called Dr. Lui Lok Ming (雷樂銘助理教授). He teaches me linear algebra and Mathematics applications. His teaching is in detail and rich in content. He explains everything in a clear way. Moreover, he does not only teach, but also tell us why we need to learn those concepts. This led me to know that it is meaningful to learn those mathematical concepts. Now I am follow-
J:
ll````the methods of solving problems. Actually ll````they don’t help much in learning Mathematics. ll````When you enter university, you will find ll````that doing past papers in university does not ll````help you much in examinations. The variety ll````of questions varies so much each year that ll````you cannot catch the trend. What you really ll````need to do is to really understand the ll````concepts behind each new problem. The key ll````to success is that to ‘Really use your brain to ll````think’. This actually relies much on your ll````Mathematical ability. Moreover, you should
ing him to do a project, about how to use Mathematics to handle graphics.
ll````not only know the theorem in the textbooks, ll````but have to really understand and carefully ll````digest what it wants to express behind. This is
What will you do beyond lessons?
ll````the only way to master Mathematics in university.
W: We will not only do Mathematics in leisure time. I love to cook a lot. Every time after I have lessons I will go back to the dorm to cook. Moreover, I will play basketball and help organizing college activities, such as the
J:
W: They definitely help a lot! The textbooks in universities are written by professors studied Mathematics. You can see their thinking logics
“Thousand people dinner” (千人宴) days ago. J:
J:
in the textbook when dealing with problems. Notes also do help. The quality of notes set by professors is high in quality. You can grasp lots
How do you comment on your classmates?
W: They love Mathematics very much. They like to discuss Mathematics at any time, even when they are having meals. Now let’s switch the focus on how you see about studying Mathematics. In secondary schools we are told that the key to success in Mathematics is to do a lot of practices. However, is there any difference when studying Mathematics in university compared to secondary schools? W: The method of studying Mathematics totally lll```different in university. I think the reason of ll````doing past papers and practices in second– ll````ary school level is that the question types ll````are similar to the real exams. Therefore you ll````can easily get good results as you can recite
Do you think Mathematics textbook helps you much in studying Mathematics?
of useful information when you read them, such as their thinking logic as I mentioned above. J:
So what types of person do you think is suitable for studying Mathematics in university?
W: Actually not only those who can calculate fast or solve problems quickly is suitable for studying Mathematics in university. If you have a logical mind and knows how to do Mathematical thinking is already suitable. In Hong Kong, a lot of students give up Mathematics as they are not able to get high marks in examinations. However, it is not totally correct. For me, my result in Mathematics was also not very good when studying secondary school. I even failed in Mathematics tests when I was studying pure Mathematics. However, I never gave up because of the
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morial Prize in Economic Sciences, although he is a Mathematician. He works on game theory,
unsatisfactory results. And at last, I entered Mathematics department in university. J:
which is highly applicable in daily life, for example, economics as I mentioned. Even politics and biology are related to that. This proves that
I also heard that some students got A s in Mathematics in public examinations, but cannot study well in Mathematics in univer-
Mathematics is actually highly related to our daily lives and I admire him a lot.
sities. How do you explain that? W: As I have mentioned before, each year the questions in public examinations are similar.
J:
J:
How do you see about the role of Mathematics in career life?
Therefore students do a lot of practices to recite how to solve those problems. Although they are able to get good results, when they enter university, they find the world is totally
W: A lot of people may have a wrong perception that students from Mathematics department can only be a Mathematics teacher. Actually af-
different. The question types vary so much that they cannot catch up. Even you grasp all the knowledge in the textbook, when you deal with
ter studying Mathematics, you may go into any career, such as finance, computer programming, engineering, Mathematics investigations
questions, you will feel puzzled. What approach I should use to solve the problems? You will always ask yourself about this and you may
and so on. Moreover, Mathematics is a training of logical thinking and it is a type of fundamental knowledge. It is useful in any kinds of career,
have no results even thinking the question for a whole day. Hence, those students may easily feel frustrated and cannot do well.
no matter it is directly related to Mathematics or not. Problem solving skills and the way of how you think that you learnt from Mathemat-
I want to know how you see Mathematics too.
W: I think Mathematics is an abstract and interesting subject. If you are not afraid of this kind of language and pay efforts to understand the concept actually lots of people can understand Mathematics. Moreover, finding answers is ac-
ics are good examples. J: Thanks.
FUN with Mathematics
tually not the focus of Mathematics. What’s more important is to appreciate what Mathematics really is. For example, the geometrical meaning of equations in 3 unknowns is actually the intersection of 3 planes. You should not just only focus on finding what is the value of x, y, z. If you can really understand that actually you will feel that Mathematics is interesting and you will have a sense of success. J:
Do you have Mathematicians that you like very much?
W: Actually a lot, I like John Nash partially among them. He is the winner of the 1994 Nobel Me-
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Beauty of Algebra By Andrew Ying “Philosophy is written in that great book which ever lies before our eyes - I mean the universe - but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is
pens with the selling price. Although we can calculate the selling price $x manually by $20X (120%) = $16, we can obviously also write this as x=M (1-d). More importantly, the way of solving problems with algebra is something regarded as more beautiful and elegant than simply using numbers. Notably, the use of algebra in these questions also allows use to manipulate questions by using techniques like solving simultaneous equations.
written in the mathematical language … without which one wanders in vain through a dark labyrinth.” – Galileo Galilei
Algebra is exactly a very important ‘language’ that is described in the above quotation by Galileo. It plays an extremely important role in Mathematics, and helps bring about the beauty of method particularly in Mathematics.
Algebra also helped the studies of other subjects, in which Physics is an excellent example. All students having studied Mechanics in Physics shall be extremely familiar with the equation F=ma (note that F is force, m is mass, and a is acceleration) as appeared in Newton’s Second Law of Motion. Imagine you need to find the force of an object of 1kg mass of 1ms-2 acceleration. This
Consider a very simple mathematical problem, with a product in a shop in a marked price of $20 and having a discount of 20% off. Obviously we are able to work out the discount by $20 x 20% = $20 x 0.2 = $4. However, this also means that if the marked price changes in any way, for instant from $20 to $30, we would need to do the calculation again. Algebra is there to solve the problem. Simply let $M be the marked price and d be the percentage off. We then would know that the discount $y would be $Md (∵y=Md). The same thing hap-
doesn’t seem difficult at all. But about finding the mass of an object when being applied with a net force of 5N having a 2.5ms-2 of acceleration? This could also be extremely easy in our sense, but how about if there’s nothing called algebra? It is time now to recall your memory that you studied how to rearrange terms in your algebra lessons, not when you are doing calculations involving purely numbers. Algebra evidently makes your life easier and simpler which is an important definition of being more beautiful. You may ask, what would be a life of not having algebra be like? Now, back to the example of Mechanics. Consider the following word equation:
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Final velocity squared is equal to initial velocity squared plus two times the acceleration times the displacement This might be a little redundant and you might not be actually able to recognize this. But how about this:
v2 = u2 +2as I believe you would not object that using algebra is much, much better, but as you can also see, without algebra, our world would have to be filled to a lot of words, and our calculations in various subjects, not limiting to Mathematics and Physics, could take a much more longer time! So, please appreciate the beauty of algebra!
FUN with Mathematics An astronomer, a physicist and a mathematician were holidaying in Scotland. They observed a black sheep in the middle of a field. “How interesting,” observed the astronomer, ‘all Scottish sheep are black!’
Get known to Arthur Cayley By Kwan Chun Hin Algebra, the basic aspect of mathematics, is quite common among people. While we work on it in almost every mathematics lesson, have you ever heard of some unconventional terms, like abstract algebra and linear algeArthur Cayley bra? In fact, these terms (1821 - 1895) and related theories symbolize the development of algebra and are contributed by numerous mathematicians all over the world. Here, I will introduce a special one to you. His name is Arthur Cayley. At first glance, you may think that Arthur Cayley is an ordinary man with a really dull face. However, this is definitely not the case. Born in 1821, Arthur Cayley was a British man who led an unusual life. Due to his diverse interests, he had become a lawyer, a mathematician and a professor. And two things we cannot deny are his aptitude for mathematics and curiosity over knowledge. When he was young, he made an arduous effort in solving algebraic problems. At seventeen, he even entered in Cambridge Trinity College and won the Smith’s prize after years of study, thereby turned into a scholar and a tutor. Later, he was invited to join the Lincoln’s Inn to study law, and was called to the bar, where he met a mathematician, James Joseph Sylvester. When they got to know each
The physicist responded, ‘No, no! Some Scottish sheep are black!’ The mathematician intoned, ‘In Scotland there ex-
ists at least one field, containing at least one sheep, at least one side of which is black.’ James Joseph Sylvester (1814—1897)
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other, James inadvertently discovered that Arthur, as a student of the School of Law, also showed great passion towards mathematics. Since then, they began to research on algebra, producing a myriad of mathematical papers with brand-new and historic ideas. One example would be the theorem that denominated by his name and was called Cayley’s Theorem. The theorem stated that every group of order n is isomorphic to a subgroup of Sn (Please don’t ask me why, I don’t know it either but it sounds cool!). The theorem motivated mathematicians to study them in order to solve combinutoric questions.
You might be wondering, ‘How could he excel at both law and mathematics?’ The answer is simple: he excelled at time management as well. During the 14 years of his career as a lawyer, he balanced his time spent on the two aspects. He kept himself not being distracted. Not only did he succeed in his research on mathematics, but also developed his eloquence, which could be reflected by his companion’s, James, words. For example, in 1852, there was once James talked about his opinion on Arthur, saying that he ‘habitually discourses pearls and rubies.’ While James and Arthur continued to collaborate together, making great progress on algebraic theory of invariants. In 1863, Arthur Cayley decided to follow his own interest and pursue what he enjoyed the most – mathematics. He gave up his well-paid job as a
lawyer and became a professor of pure mathematics at Cambridge. Due to his excellent language and administrative skills, he was elected the new Sadlerian chair there. Nonetheless, he had never been pleased with his current situation. He kept working on mathematics and finally published his own book, called Treatise on Elliptic Functions in 1876. In the sunset of life, he was still keen on painting, travelling and architecture. His interests made his life fascinating and much more meaningful than the others. In 1895, Arthur Cayley passed away peacefully. Throughout his life, he had been granted numerous academic honors, for instance, the Copley Medal of the Royal Society and the Royal Medal. He had once said ‘As for everything else, so for a mathematical theory, beauty can be perceived but not explained.’ It gives us the message that mathematics, as well as life, has to be experienced by heart, so that the ‘beauty’ can be discovered.
Introduction to the Cauchy-Schwarz inequality By Tsang Chi Cheuk Let me introduce to you one of the most famous and useful inequalities in Mathematics, the Cauchy-Schwarz inequality. This is what it looks like:
You are probably thinking: What madness is this? You can have 3 summations, 2 series, and a root in one inequality? What is the significance of the 3 summation anyways? ... Calm down first and let me introduce to you this inequality step by step. To simplify matters, let us take a look at this inequality at small values of n.
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For n=1, the Cauchy-Schwarz inequality becomes: which is quite trivial con``sidering both sides are equal. For n=2, the Cauchy-Schwarz inequality becomes:
This is not hard to prove at all, in fact, I’ll show to you how to do it here:
However, , so , which if you check the reasoning above, is exactly the Cauchy-Schwarz inequality. By considering 2 points in a 4-D space, 5-D space, etc… one can derive this form of the CauchySchwartz inequality for all values of n, which is:
There, much simpler than the form above, right? Now that the inequality doesn’t seem too threatening, we can try to understand some of its implications.
Since both sides must be positive, taking the square root yields the result to be proven. For n=3, the Cauchy-Schwarz inequality becomes:
This looks pretty complicated, doesn’t it? Let’s try to approach it from another angle: coordinates. Consider 2 points in a 3D coordinate space, X= (x1,x2,x3) and Y=(y1,y2,y3). One can prove, using Pythagoras’ theorem, that the distance of X to the origin is . Similarly, the distance of Y to the origin is . If you know vectors, you will recognize that the dot product of X and Y (for those who don’t have a clue what I’m talking about, just treat it as a name for the particular expression), but one can prove that the dot product of X and Y is the product of length of X, length of Y, and cosine of the angle between X and Y. In mathematical expression, that’s
where | X．`Y | is the dot product of X and Y, ||X|| and ||Y|| is the distance of X and Y to origin respectively, θ is the angle between X and Y.
The triangle inequality, which states that the sum of the lengths of any 2 sides of a triangle cannot be smaller than that of the remaining side, though sounding very trivial, can be proven using the Cauchy-Schwarz inequality, however we shall omit the proof here. You may think this result is quite useless, but if you look at it from another angle, it is of great importance. Up to now, we have been discussing distance between A and B as the length of the shortest line between A and B. How about if we consider another definition for distance: the taxicab metric.
Under the definition of taxicab metric, the distance between points P and Q is the distance of the shortest line, which can only go horizontally and vertically, between P and Q. Imagine walking on streets like a grid paper, you won’t be able to walk diagonally, so naturally you count the distance between the 2 points as the sum of the vertical distance and horizontal distance you took. The triangle inequality may not seem so trivial now, does it? Here’s when the Cauchy-Schwarz inequal-
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ity comes to rescue, for it implies that the sum of lengths of any 2 sides of a triangle must still be longer than the 3rd side. In short, the CauchySchwarz inequality shows that the triangle inequality hold under any definition of ‘distance’ considered as a ‘metric’. Some of you may heard of the Heisenberg uncertainty principle in quantum physics, for those who haven’t, the principle states that the uncertainty in position of a particle, times the uncertainty in momentum of a particle, cannot be smaller than a certain constant. In physics terms, that’s
This principle is arguably one of the foundations of quantum physics as it establishes the fact that quantum particles cannot be located precisely without losing track of its velocity, and vice versa. And guess what, the Cauchy-Schwarz inequality is essential in proving this principle. That means all the advancements in quantum science we have today would not have existed if not for the presence of the Cauchy-Schwarz inequality. Perhaps you will now want to thank the discoverers of this great inequality, but you may face quite a problem here. As the name of the inequality implies, several mathematicians came across this inequality in a short period of time, and it is hard to determine who coined this inequality in the first place. Nowadays, it is widely accepted that Augustin-Louis Cauchy has first discovered the inequality in 1821, then Viktor Bunyakovsky published a similar version of it in 1859, then Hermann Amandus Schwarz re-discovered it in 1888. I would suggest you to thank all 3 mathematicians, for without them, this inequality would not have been so well-known, and our world will be very different. If you are interested in knowing more about the Cauchy-Schwarz inequality, make sure to find out more in the Internet or in books about algebraic analysis!
Challenge Time! 1. (APMO 1991 Q3) Let a1, a2, … , an, b1, b2, … , bn be positive real numbers such that a1+a2+…+an = b1+b2+…+bn. Show that . 2. (USAMO 2002 Q2) such that
Let ABC be a triangle
, where s and r denote its semiperimeter and inradius, respectively. Prove that triangle ABC is similar to a triangle T whose side lengths are all positive integers with no common divisor and determine those integers.
(Answer will be provided in next issue.)
Proof without Words By Ian Kwan
1. Introduction As the saying goes ‘Nothing strengthens authority so much as silence.’ By Leonardo da Vinci, being silent is a quality well appreciated in different cultures, including Buddhism, Medieval Europe, and shockingly, even in mathematics. Being silent while doing math is easy, you may think, but that is not exactly what I mean by ‘silent’. In mathematics, there is a term for it, called ‘To prove without words’, and to accomplish that, you are to prove something without using written or spoken language, and that certainly includes equations, so, in other words, you are only entitled to use diagrams and pictures to prove something. Sounds impossible? Yes, that’s because even if you want to prove something as simple as congruent triangles, you need to use
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words and letters such as ‘given’, ‘SSS’ and ‘ASA’. But it turns out to be possible, and maybe even, graceful and appealing. Now, let me prove how ‘Proving without words’ IS possible, ready to drop your jaw? 2. The Silent Proof for Pythagoras Theorem As far as I know, there are at least 8 ways to prove Pythagoras Theorem. But this one is a quite special one among the 8, so sit back and listen carefully. Oops, I mean watch carefully. If you still survive now, there is a great chance that your mouth is in a wide open state, either you don’t get the picture, or surprised to see that ‘To prove without words’ is really possible and is fascinated by the beauty of the proof.
3.1 The Algebraic proof First, let the sum of the sequence be n, and you will get: 1+1/4+1/16+1/64+1/256+1/1024+…=n Shift ‘1’ to the right hand side and multiply both side by 3: 1/4+1/16+1/64+1/256+1/1024+…=n-1
> 1+1/4+1/16+1/64+1/256+…=4n-4 We will get the equation: 4n-4=n and we will know that n=4/3 3.2 The Silent (Geometric) Proof After convincing you that 1+1/4+1/16+1/64+1/256+1/1024+...=4/3 is actually correct, I can finally show you the much more graceful, simple, and most importantly, silent proof. Don’t blink.
3. Proof for 1+1/4+1/16+1/64+1/256+1/1024+…= 4/3
I recently had the chance of setting the paper for some lower form students and this was one of the questions. Of course, I didn’t gave them the answer, and I gasped when I received the answers from them — the answers were in a clear split, one side had the answer ‘infinity’ and the other had the correct answer of 4/3. Let me explain why the answer is not ‘infinity’. As you can see, the values in the series are undergoing a drop in an exponential manner and are becoming more and more insignificant, meaning that they are getting closer and closer to zero. Therefore the answer is not infinity (it will be easier to get the above explanation if you have the concept of ‘limit’).
Once again, I’ve successfully gagged you with the beauty of silent, and of course the beauty of Mathematics too. 4. A Brief Conclusion There are lots and lots of ways to prove something in the world of Mathematics, but still as I’ve mentioned before, ‘Nothing strengthens authority so much as silence.’, to prove without words is the most beautiful and potent way of proving something, at least in my opinion. Other ways of proving includes ‘by Brute Force’ (the exact contrast of proving in silence), ‘Reducto ad Absurdum’ (another name of ‘to
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prove by contradiction’, which may sound more comfortable and familiar for you) and Mathematical Induction etc. But none is as fun as to prove without words, where you will spend minutes to comprehend the diagram and finally scream ‘ohhhhhhhhh!!!’ And there comes the moment of glory, the miraculous moment of finally knowing what on earth is going on.
Relaxing Time! By Irene Kwok
Bored with words and starve for games? Why not come and try all these mathematic puzzles out? 1. Kakuro What is Kakuro?
5. The Final Problem If you have read ‘Sherlock Holmes’ by Sir Conan Doyle before, these three words may ring a bell. However, there is nothing for you to worry about, because the ‘problem’ for you to deal with is not the infamous crime lord Professor Moriarty, it is just a mathematical problem. The Problem: So, prove without words, that xx+yy+zz>xy+yz+xz, where all x, y and z are three distinct, positive
real numbers. (Hint: you are advised to construct rectangles). The answer of ‘The Final Problem' will be found on the back of the newsletter, and, remember, we will appreciate your silence. Shhhhhhhh…
Kakuro is a puzzle game that's a cross between both traditional crosswords and Sudoku. Just like Sudoku, you must fit the digits 1 to 9 into a grid of squares so that no digit is repeated within a defined area. And just like a crossword there's a grid of filled blocks and clues to solve. How to play it? 1. Each row and column (from colored box to colored box) contains numbers from 1 to 9, no numbers can be used repeatedly. 2. The numbers in the pale yellow boxes indicate the sum of each row and column (from colored box to colored box). For each row, refers to the number on the left hand side. For each column, refers to the number on top.
FUN with Mathematics
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2. Ring Irregular Number Place
Solution of the question in ‘Proof without words’
What is Ring Irregular Number Place? It is the variation of Sudoku. The major difference is that Sudoku is a 9x9 square while Ring Irregular Number Place is of irregular shape. How to play it? 1. Each block bounded by a bold line contains all numbers from 1 to 9. 2. Each row and column contains different numbers ranging from 1 to 9. No numbers can be repeated.
Solution of the question in ‘Relaxing Time!’
Upcoming competition 3/3/2012 (Sun) 港澳數學奧林匹克公開賽 16/3/2013 (Sat) 香港初中數學奧林匹克全國青少年數學論壇選拔賽 23/3/2013 (Sat) 第 12 屆培正數學邀請賽 決賽 20/4/2013 (Sat) 第 30 屆香港數學競賽 決賽
Acknowledgement
Mr Patrick Wong, CUHK
Publication Team Andrew Ying // Brian Lui // Eileen Tam // Ian Kwan // Irene Kwok // Jeff Siu // Kwan Chun Hin // Tsang Chi Cheuk
Need your contributions The HKJSMS welcomes contributions to the upcoming issues of our newsletters from teachers and fellow students of our member schools. Please feel free to send your submission, such as event write-ups, articles or jokes related to Mathematics, to jsms.hk@gmail.com. We need your work to make the newsletter more rich!
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