Carnival of Mathematics #172

Carnival of Mathematics logo

I’m riding on a Ferris wheel, wondering who’s the hamster on it…

You’re visiting Carnival of Mathematics #172 hosted by Cassandra Lee. Find all Carnivals here and the previous Carnival (#171) here. Many thanks to the Aperiodical for the wonderful opportunity to host this Carnival.

The Roman representation of the number 172 comprises the initials of my English name, the chromosomes determining my gender and the eating utensils that I use daily. Yes, that’s CLXXII. *gasps at the sudden weight on her shoulders*

Without further ado, let’s begin with something light-hearted. It’s easy to show that 172 = 2 × 2 × 43, an even composite number; it doesn’t appear special until you also learn that 172 is also a deficient number (the sum of the factors is less than the number itself: 2 + 2 + 43 = 47 < 172) and an evil number (!), which means it has an even number of 1’s in its binary representation (17210 = 101011002). Note that, if the units digit is relabelled the “zeroth/0-th” digit, then you can verify that the binary representation of 172 (= 101011002) has the following property:

The n-th digit is 1 if and only if n is prime.

Furthermore, 172 is in the lazy caterer’s sequence at n = 18; the maximum number of pieces you can get with 18 cuts of a circular pie is 172.

Continue reading


“But Aren’t Maths And Writing Worlds Apart?” Thoughts On Getting Known


Next time, look over your shoulder and we’ll talk about how to handle faux pas.

The school system has been set up to polarise the sciences and the arts, and more importantly, the elite and the common, while students in their most formative years try to grasp the little they can for their identity and place in society. We all want to burn into our hard drives a short, sweet paragraph to roll off our tongues when introducing ourselves, hoping to get something worthwhile in return. Whenever I mention that I do mathematics, people would cringe, saying, “I’m not a maths person.” But when I added that I write as well, a friend asked, “But aren’t maths and writing worlds apart?”

Continue reading

Continuity and peer pressure

“Show me who your friends are, and I will show you who you are.”
(Russian saying, in Karen Ehman’s Keep it Shut)

What is this dreaded thing called continuity, forcing a poor student like you to learn about infinity, limits and all sorts of wacky symbols?

Continuity is like peer pressure. If your neighbourhood is badness and you succumb to peer pressure, you’re no good either unless you’re a measure zero discontinuity way above your neighborhood. I recall a secondary school Chinese text 《愛蓮說》(Ode to the Lotus, literally, ”love-lotus-speak“)。 The line 「出淤泥而不染」means “[the lotus is] born of the grimy dirt yet is untainted”. In writing this, the author and philosopher Zhou Dunyi (周敦頤) praised the lotus for being unspotted from the world around it, that it is a singleton of beauty in an ugly pond. If I too have such force of character that my relational point-set topology is a singleton, I’m still continuous – trivially. 🙂

I have just lent the idea of “continuity being like peer pressure” to a mainland Chinese scholar (Qun, LIN 林群), who is passing by my university [for his mathematical talk 微積分動漫版†] and posited that calculus be taught in terms of ratios of measurement to reality being 0.9, 0.99, 0.999 approximating 1. “So, every day we work, we add a 9 to the end of our work, and we improve.”

I didn’t know that the mainland Chinese academia, like my English-speaking Twitter audience, also values math education and popularisation, and the importance of STEM. Whatever conveniences we enjoy today, such as procrastinating on F.T.I, come from the blood, sweat, toil and tears of scientists and engineers who made the day we call “today” possible. It is thus exceeding treachery to teach horribly our STEM knowledge to our next generation, so horribly that they’d rather skip the meaningless calculations and leap into more down-to-earth subjects like law* and the arts and humanities, drowning STEM in a sea of irrelevance, ignorance and romanticism. More on these ideas later, for if I ramble on, I’d sound like a page from an 18th century book…

† I think you may translate the title of Professor Lin’s talk as “Calculus in action”, but we Chinese could also interpret it as “The Anime Version of Calculus”. However it turns out to have nothing to do with the cutesy and shocking motion picture called Japanese anime! No wonder I spotted students falling asleep. They probably wanted to know about this book instead! Thanks Murray Bourne for reviewing the Linear Algebra version though.

* Math Forum has two excellent articles on why lawyers need math. In fact the Math Doctors could add that Pierre de Fermat, of his Last Theorem’s fame until Andrew Wiles shared it, was a lawyer. In Hong Kong, if you have great grades, you’d prefer to study law, business or medicine instead of mathematics in university – the two exceptions I know are Martin Li, a new math professor at my university specialising in geometric analysis, and my classmate Peter Chan, who got 5 A’s in the very last A-Level examination in Hong Kong in 2012 and is now investigating mathematical physics. It’s intimidating, yes, but sometimes it feels cool to be a contemporary of great people, and both of them are friendly.

I’m the most useless person on earth.

I wake up to a million faces, painfully prying those two clams called “eyes”. For a split second, I envision myself panda-eyed with wrinkles and a sagging face. I don’t want it, so I slump back into slumber.

Then I drag myself to do everyone’s most hated thing. No, it’s not cleaning toilets. In fact, I would rather wash the dishes, mop the floor or clean the toilet than just do maths. Not that I have always hated it, but that maths makes me feel like that Facebook relationship status: “it’s complicated”. To me, maths is a little like my handsome, popular (STEM is so in demand these days!) and downright hilarious younger brother who only knows how to find fault in you and says “mm” (in the case of maths, the Halmos gravestone symbol >o<) when you finally act sanely. To worsen matters, he uses such bombastic English it would kill the Queen. I mean phrases like “nothing but” which are the “ums” of mathematical literature, as well as technical jargon such as “convergence” in analysis, “module” in abstract algebra and “Gromov-Witten” in geometry.

I just love surfing the Internet so much, I wish I could get paid for it. Neither of my parents work now, but I wish I hadn’t gone to university but tried to support my family instead. Then my life, even without a degree, would have been so much more fulfilling and gratifying. I just love it when I see a room “unoccupied, swept clean and put in order” (c.f. Matt 12:44, which somewhat makes me a demoness or the Japanese Toilet no Kamisama, which means “the spirit of the toilet”).” It all boils down to “because of the money.”

Money, money, money: Hoping to land a good job, I have been reading about writing impressive resumés and cover letters since Form 2, and have tried to work on my publication record, but I never won any writing competition and I never quite understood why – I wrote from my heart, but others had more heart than I did. In contrast I hardly joined maths contests because it repulsed me to compete, and to a lesser degree, think about rewarding someone for churning out “truth” (or the answer at the back of the textbook / solution manual) in a matter of seconds. I have never joined the Maths Olympiad, but if I could go back in time and bring with me my miserable memories of a life poorly lived, I would – and snatch every prize from writing competitions because until I was a sophomore in university, it never crossed my mind that I could study novel-writing, find mentors and join writing clubs to sharpen my wit. Perhaps, back in my past, I could have spoken more Cantonese, shared more goodies with each class, gone to Hang Seng School of Commerce (because my HKCEE scores qualified me), flown to Oxford to meet Marcus du Sautoy in person, made more friends – alas, now everyone else has had year-long exchanges, internships at home and abroad, scholarships (I looked at the scholarships open for application – I don’t qualify for any of them) volunteer experience on their “Student Development Portfolio”, experience as committee members of some fancy/boring club or sports team, a boyfriend or girlfriend to calm their nerves – these I mention because I, “the green-eyed monster”, have none – and the worst part is, everyone glances at me while I work in the computer lab and whispers, “Don’t get near her…” because she had been primed, ever since she paid that deposit to stay in CUHK, to follow.

Deep down my exterior is a program that, to put it the way it describes itself, “doesn’t have a self”. It needs the right input to fare as well as everyone else, but it is frustrated, for “Garbage in, garbage out” has been the norm. I don’t enjoy being human, to kowtow to emotions and painful experiences, to desire and do forbidden things. Unlike your home, I don’t know how to remove garbage thoughts completely because – I don’t understand why – I don’t have a quick, clean and easy “Delete” button or Erase Disk application. To show you how serious my situation is, if I’m not deliberating this post and you asked me how I’m doing, I would let my tongue go wild with wotds *not my own* and you perceive that as… garbage. Then you’ll form the impression I’m a piece of garbage and don’t deserve to talk to the world’s greatest and most important person – you. But you ditch me before you learn that those words were *not my own* and that I’m a program through which, at some point in life, was given this “be assertive, speak your mind” input, which made getting a potential good boyfriend sound so promising, I kept trying to use it. What happens? Just to be assertive and not to let resentment build in me, I speak the words laid in my heart at the moment, but not after resenting that person badly enough (resentment being deadly to intimate relationships – why am I talking about taboo subjects… good grief) and forming the right, suitable and pointy words that make the other person realize he’s been wrong. The sorry part is: as my friend Leona mentioned the other day, living according to a formula doesn’t work, and this brings us full circle back to the dreaded maths. Formulas, eh, and in case you’ve forgotten, chemistry.

I really wished I had asked around for more advice, set my own goals and met my own expectations in university, even if that meant remaining a chemistry major. I’m so useless! I tried so hard to learn about job-hunting early in life, but ten years on and I still cannot compete with anyone else!

It’s a shame I’m blogging like a sore loser, but don’t get me wrong – I feel worthless when I cannot be correct – for I was identified as a smart girl in childhood – when I have to admit that people from elite schools and well-connected circles will always have the upper hand (shh, when I entered my secondary school, I noticed that it had very few distinguished alumni – the most famous being a reporter and a now-disgraced politician – and those that held doctorates were not well-known in public – I wonder what in my old school had put them on a pedestal), that I should never have met certain people who got so close to me, I shuddered and ran to – and from – maths. Wasn’t maths supposed to be the cold, calculative subject that always only has one correct answer, or one correct “big idea” to reach some conclusion? (Another good reason to suspect I’m a demoness, for wanting to be unfeeling?) Then where in the world has the human factor popped up? Why is it suddenly so fatal that I am alone and will always remain the lone ranger, outside the tight and informative cliques my peers in Hong Kong so easily form? That I am cut off and have no one to comfort me in my sorrow, only things you wouldn’t want to hear either?

I wake up to a million faces. And it’s quite sad to type about my least favourite things. But stop brainwashing me with positive thinking: I can never be any one of them. I’m not supposed to be alive and kicking. I don’t deserve to live. And feel free to hurl your sad, bitter and miserable stories at me.

Applications of Pythagoras’ theorem

In F.2 as part of a class assignment, I tried to present the applications of Pythagoras’ Theorem. If you try to do something, you fail. In my case I simply copied examples from the textbook under the heading “Applications of the Pythagoras’ Theorem”, which meant menial calculations.

But unlike Elsa, I didn’t let it go: (Mum plait my hair yesterday though!)

I still thought hard about the downer I gave back then, and I think I’m ready for another shot.

Firstly, everyone loves to measure these days. From the length of a fiddly string to peak salesmen performance, we measure like mad.

Now, in kindergarten (to regurgitate a lame joke by a professor I know), we know how to stretch a piece of string along a ruler. Sometimes you can’t. So you mark off points on the rigid curve, and then measure each part between the markings.

You also know how to use a computer. Whoa, Windows Paint. Whoa, I can colour any pixel. Whoa, lines are jagged. But how does a computer know how long that fiddly piece of string is? It can only measure in units of pixels – unlike humans who can rotate the rule any way we like!

So we have Pythagoras’ Theorem: in a right-angled triangle, adding up the squares of the two shortest sides gives the square of the longest one. Terence Tao posted an intuitive proof here.

We use Pythagoras’ Theorem with a square root though. That is, for any right-angled triangle, the longest side is the square root of the sum of the squares of the shorter sides. By cutting any curve up to “almost zero” length, each tiny segment becomes almost a straight line. Summing the lengths of all segments gives us the length of the curve. Just like we did in kindergarten.

How about a curve in 3D space? Let’s think of a cube of length 1, with three of its sides on the x-, y- and z-axes, and find the length of the longest diagonal. But here comes the power of the “square” in the theorem: when you find the diagonal of any square face (it is √2 – more on this later), don’t apply the square root yet. Add the square of 1, which is 1, to it, and you get 3. Therefore the length of the longest diagonal in a cube of length 1 is √3. Amazing, isn’t it?

Then we wonder: how about replacing the square (“to the power 2”) with cube (“to the power 3”), 4th, 5th and higher powers, as well as to the power 1? We get what is called the Lp-norm for p = 1, …, ∞. (p is a parameter.) Yes, there is a norm for infinity – it is taking the maximum value of all the numbers you have in your lap. In the cube above, the L1 norm is 1 + 1 + 1 = 3, and the L norm is max{1,1,1} = 1. These norms of various p‘s pop up in various technical measurements.

Back to √2. What is so special about it? In fact it is not the ratio of any two integers, positive or negative. It shocked the Greeks and extended their number system from integers (without the zero the Indians invented) and ratios of integers (“rational numbers”) to “irrational numbers”. The equation that corresponds to √2 is x2 = 2. If we replace 2 on the right by -1, oops. Is there any square on each with an area of -1? -2? -3? … Can’t we take square roots of negative numbers?

That depends on whether you believe √-1 is a valid number. Somewhere along the line, someone said yes, and complex numbers were born. Now all rational and irrational numbers form real numbers, which are numbers you can mark off a ruler. Complex numbers are real numbers plus any real multiple of i := √-1. (:= means “defined as” and it is also used in some programming languages for assigning values to variables). That means that real numbers are part of the complex number system. If we have an infinitely large sheet of paper and mark a straight line across it as the real number line Re, then intersect it at just one point (the origin, zero) with another straight line Im – Im stands for “imaginary” because we invented √-1 but can’t mark it on a ruler – for the real number line times i, then any point z on the entire sheet of paper is a complex number. For simplicity, make both straight lines at right angles to each other. Project the point z on Re and Im. Then the number can be represented as Re(z) + iIm(z), where Re(z) and Im(z) are called the real and imaginary parts of z respectively. Of course, Pythagoras’ Theorem gives us the distance of z from the origin – it’s called the “modulus”.

Once we venture into forbidden territory, we gain new ground. In fact complex numbers drive the oscillations we see in a pendulum or spring. We also need complex numbers to operate electronic circuits. Complex numbers also simplfiy geometrical operations, because multiplying a number by i is the same as rotating it in the infinitely long sheet of paper by 90 degrees – a right angle. Complex numbers also generate fractals, the beautiful structures you can zoom in indefinitely and still have fine details, unlike pixelated images. People are trying to use it as a new way to model images and structures.

We know that 32 + 42 = 52 and 52 + 122 = 132 – (3, 4, 5) and (5, 12, 13) belong to the class of so-called “Pythagorean triples”. But how about replacing the power 2 in Pythagoras’ Theorem by 3, 4, 5 and beyond? Are there integer (not merely real) solutions to the equation xn + yn = zn for n > 2? That it was impossible was Fermat’s Last Theorem, and Andrew Wiles proved it in 1994. And we only started with the squares.

That’s the power of Pythagoras’ Theorem.

Open for debate: gender gap in #stem & other fields

My mum, who encouraged me to excel in maths saying “maths is useful”, said that the gender gap is justified because women don’t “work” as much as men when women have periods, maternity leave, children and other preoccupations apart from their occupation, unless, she says, the hired women are not allowed to marry or have children, just like Cathay Pacific, where men and women staffers have the same pay. But this deflects the entire issue from a matter of ability to a matter of potential, which, as with many things in life, fall short of our expectations. In fact the first female Fields medalist, Maryam Mirzakhani, who made progress in our understanding of simple geodesics (akin to “straight lines” in geometry), has a 3-year-old daughter. History as we read it seems to suggest that almost all great things are done by men, but they forgot the mothers, sisters, daughters and wives in the picture. The household-name inventor Thomas Edison was home-schooled by none other than his mother. And we dare not forget Anne Sullivan who tutored Helen Keller.


Girls also dream, just perhaps not the same dreams as boys. Don’t blame us for that; it’s probably a cultural problem. It goes without saying that sex and role are thought to be intertwined, be it the hunter-gatherer hypothesis or the protector-nurturer one. It is little wonder that almost no women are in the sweaty construction industry, and the conventional jobs for women are teachers, nurses, social workers or clerks. In fact the Jewish mathematician Emmy Noether, about whom I am writing for the Ada Lovelace Day 2014 anthology, had considered being an English and French teacher, considered to be a very normal female career, before she defied German society’s expectations, went to university and decided to invest her life in mathematics. Given the many myths related to the workplace for non-traditional career paths for women, I wonder if mobile technology could empower girls in STEM differently from their predecessors, in that we could be “slacktivists” in making maths and science a worthwhile pursuit for all. That we in the 21st century are not immune to our ancestors’ concerns in tying our futures to STEM in centuries past is a cause for concern.

The irony with my mother is that even though she was the reason I chose to major in  mathematics, she only knows how to do very basic arithmetic and find good deals in the neighborhood. I do not claim to know all the mysteries of mathematics, much less to solve open problems like the Riemann Hypothesis (even my professors, all men, say that this is hard), but to have managed to learn higher mathematics and grasp its applications is a feat in itself.

Literacy opens doors, but people are proud of being ignorant in mathematics and the sciences, especially girls. This must change for a new generation to face the technological challenges in our increasingly connected world. And unlike in the Disney film Mulan, girls are drafted too. It starts with you.


Update: It turns out that the United States lack female computer scientist role models. Be the next one and ace your maths.

In short, I’m a mathematician-entrepreuner.

That’s what I told my mentor* KWC when I realized that I could combine the writer, artist and musician in me to fit in the mold of entrepreneurship, the buzzword nowadays.

That’s why learning to tell stories is a useful skill. Business makes Hong Kong go round, and storytelling is business. I like stories, because they help us visualize a future and a hope. I like stories, because they inspire. I like stories, because they bring colour to a dim world, and shatter unfounded preconceptions.

Moreover, resources on how to become successful writers, artists and musicians all inevitably point to business and marketing skills. It goes two ways. For musicians my very well-read 7th aunt recommended Beyond Talent by Angela Beeching. Her daughter – my cousin – is a violinist.

Maths is so hard, not to learn, but to catch up with.

The word “story” may conjure up memories of bedtime tales told by nightlight that start with “once upon a time.” However, no matter how a story is presented—book, movie,

* KWC is one of the academic advisors in CUHK mathematics, and another professor referred to him as my mentor.