On studying mathematics

(I used Google Translate to translate my own article from Vietnamese. It works surprisingly well. Another reason to learn math!)

A wall in my office is used as a board. On it are always chattering math symbols. The problems written there are not complicated, freshman students can understand, but sometimes the solution affects the safety of hundreds of millions of Internet users. In this room, no one asked why studying math, because everyone understood that one cannot do much without math.

I like to study math from a young age, but I'm not good at math. I was deemed “gifted in mathematics” in secondary school, but in grade 9, I failed the entrance test to a specialized math class in high school.

In the 10th grade, I still tried to study math by myself, borrowing notebooks from friends and other people who attend the aforementioned class. There’s a famous math magazine in my country called Mathematics and Youth of which I was a regular reader. In each issue the magazine published a bunch of math problems, and encouraged its readers to send in solutions. The most beautiful solution of each problem is published in the next issue, together with the name of its author.

I worked very hard on these problems, I rarely succeeded, but when I did, I always mailed in my solutions, and on the day of the new issue, rushed to the bookstore to get a copy to check if my name was there. I still remember a person, who I don’t know but I consider a friend because his math was so beautiful. His name, Tran Vinh Hung, was in every issue. Recently I learned that Hung has become a math professor in America. I was named only once, not in the magazine, but a math teacher called my name in front of the whole school as a role model of perseverance, because “He always submits solutions, even when his were never chosen”.

In the middle of 11th grade I had a computer and started to spend all my time on the Internet. My study became worse and worse. Unlike my friends I didn’t want to take private tutoring classes. At some point I was even considered at risk of graduation failure. Not until the second half of the 12th grade, I started to study more seriously, for fear of failing the college entrance exam. I was no longer interested in math, and all I aimed for was just to get high scores.

At college, the first two-year general program included a lot of math, but I did not find them interesting at all. Classes always started at six-and-a-half in the morning. At that time of the day I don’t even know if I can enjoy my favourite meal broken rice with ribs, let alone matrices, rings, groups or fields! I skipped school, stayed at home to learn hacking, and later went to work full time.

What I just shared is nothing new, many people, especially Vietnamese, have experienced similar stories. Yet after 20 years, I am still studying math with an indescribable joy. What took place next for me is simply magical.

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I did security for a bank. The first few years were good, there’s lots of things to do and they paid well. Eventually I was promoted to manage a small group, gradually my team did all the work, leaving me with nothing to do. I was bored and planned to quit. I told my boss, but he called me to stay, saying that I could keep my job while opening my own business.

So I opened a new company, but shut it down within a few years without acquiring a single customer. I realized that I lacked business knowledge so I bought a bunch of books, and learned a lot about the topics that I never thought that I would be interested in such as marketing or sales. I then tried to build many other ideas, but none succeeded. I did not feel discouraged, but I did not want to waste time. The question of what I should do is still not answered.

I read on professor Ngo Quang Hung's blog about the vicious circle of career, something like what one likes one will do a lot, what one does a lot one will become good at it, and what one is good at one will like it even more, and repeat. Finding and jumping into this vicious circle is the key to success.

What is my circle? In retrospect it should be obvious, but it took me a lot of time to realize that I’ve always liked and am quite good at hacking. I kept searching for a long time but not realizing my greatest competitive advantage was the job I was doing. So I went back to hacking, "to see how far I can go." 

It was a such great time. Just like the person who walked in the tunnel for a long time, and thought it would be forever, but suddenly saw the light gleaming at the end of the road, I spent all the strength of my youth to run toward the light, the faster I ran the clearer everything became, the clearer everything the faster I could run! Within 3 years, from far behind, I caught up and have been sailing together with the world ever since.

I found out that learning is like climbing a staircase that every step has an endless length. The person standing at a step cannot know how long it takes to cross to reach the next step. Many people will give up halfway, only a few lucky and persistent to reach the next rung, climb up, and then continue sideways. Perseverance is the only thing that I can control, luck is fortune bestowed by heaven.

My biggest luck is that I’m not afraid of math. In a world with so many people afraid of math, being able to enjoy studying math creates a not so small competitive advantage, giving me a distinct identity, different from many others. 

It is because of not afraid of math, I had the opportunity to study cryptography, the main tool of the security field. I discovered that the world had very few security engineers who know cryptography. This is because in order to study cryptography one has to study math, but mentioning math is enough to make so many people feeling dizzy or nausea, leaving them with no energy to learn. I'm just the opposite. When I realized the math that I used to love is also used to protect the Internet, I rushed to learn at full speed, neglecting to eat or sleep. 

I don't have any special talent, I have to work very hard to learn things that for many people are just basic math. There are books I have read for 10 years, but I still don't understand. But I’ve never wanted to stop. I am like a pirate who sails to the high sea to conquer new lands, the more difficult it is, the happier I am. 

What is interesting is that my math knowledge is nowhere good, and the cryptography that I have learned does not need any advanced math knowledge. With just the math that any secondary students can understand, cryptography can already protect the Internet. And just by studying a little bit of the math that the cryptography relies on, I had a competitive advantage over many people, and became an expert in my field. Imagine if I knew a little more, what else could I do? This is a question that drives me forward.

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But that's because your job needs to use math, and my work and my daily life don't need math at all, why should I study it?

Thank you for asking. I also asked myself this question.

Some people say learning math is like exercise. You can live without exercise, but exercise will help you stay healthy and thus improve the quality of life. An American scientist once said that learning math is like upgrading the firmware of the brain, making it bigger horizontally if not vertically. The natural world is dominated by math, and studying math is a good way to understand more about the world in which we live. The famous scientist Richard Feynman says that God speaks calculus. If these ideas still haven't convinced you, let me tell you the rest of the reason I study math.

I study math because math is so beautiful. Learning math brought me so much joy, making the question why studying math sounds as silly as asking a child why playing hide and seek. I learn for fun, isn’t that enough? As human beings, we have a lot of pleasures or hobbies that do not serve any higher purposes than having fun. Admiring a painting, reading a story, watching a movie, listening to music, etc. we don't need any other reason rather than entertainment to submerge ourselves in these activities. What would you think if I said that math can also bring joy, that math is also very entertaining? 

Mathematics started from very specific problems in everyday life. The Greeks did math because they wanted to measure objects. But then, very quickly, math became a game, the question is no longer what to do with it but whether it is a good game! How many of the best minds of mankind have played these games, learning math is a way for mere mortals like me to join them. What is more interesting than being able to meet and make acquaintances with the likes of Fermat, Euler or Abel? 

The arts or sports that we enjoy, after all, are just games, to help us temporarily escape the tediousness and boredom of everyday life. Mathematics, from this perspective, is the pinnacle of entertainment art, because it takes place entirely in human imagination, not limited by space, time or senses, as described by Bertrand Russell, the eminent mathematician of the early 20th century:

Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.


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You don't need to be good at math, don't need to be a mathematician to enjoy the beauty Mr. Russel describes. Of course it can’t hurt knowing math, but there are many dream-like elementary math results that anyone can see.

Get a piece of paper and a pen. You go get it, I'll wait. Draw a triangle. Next you connect each vertex to the center point of the opposite edge. What do you see? The three lines you just drew, called the medians, intersect at one point. Isn't it beautiful? Three straight lines that have nothing to do with each other end up having one point in common. Why does the triangle have this property? This is your homework today.


Figure 1: taken from Paul Lockhart's Measurement book. A book is worth reading if you want to entertain with math.
Next, draw a circle. My earliest mathematical memory was to sit up straight, put my hands on the table and read aloud together with my classmates: the circumference of the circle is 2 times pi multiplied by a radius, written as C = 2 * π * r. This formula says that the ratio between the circumference and the diameter of a circle, no matter how large or small, is a constant. If we take half of the circumference multiplied by pi again, we arrive at the area of the circle, that is A = π * r * r. 

Isn’t it strange? There is no reason for the ratio between the circumference and the diameter or the ratio of the area and the square of the radius of every circle to be equal, and equal to a number that appears everywhere in mathematics and life. But this is the truth. It is true here on Earth, true on the Moon, Mars and true everywhere in the observable universe. 

How do people calculate pi? Notice that we can easily calculate the area or circumference of a square or rectangle, simply because the straight line is easier to measure than the curve. Based on this observation, Archimedes (yes, the very person who jumped out of a bathtub and shouted Eureka! - I just wonder if he was then cursed by his wife or not?) came up with a very unique way of measuring pi.


Figure 2: A hexagon with each edge equal to the radius. Pictures 2, 3, 4 are taken from Infinite Powers, a book worth reading for those who want to know what calculus is for.

Figure 3: Selecting the median point in each arc to create a 12-edge polygon from the hexagon.

Figure 4: As the number of edges increases, the circumference of the polygon is closer to the circumference of the circle.

First (figure 2) he drew a hexagon with the side length equal to a circle radius. The circumference of the hexagon will be 6 * r. Apparently the circumference of the circle is larger than the circumference of the hexagon, so Archimedes deduced that π must be greater than 3. 

Archimedes then doubled the number of edges of the polygon (Figure 3 and Figure 4). Because the edges are straight lines, Archimedes could "easily" calculate the perimeter of each polygon. "Easily" in quotation marks because Archimedes had no money to buy a Casio computer, and he had to manually calculate many a square root. Eventually, Archimedes estimated that 3 + 10/71 < π < 3 + 10/70.

As the author of Infinite Powers wrote, the unknown number, and forever unknown, π is trapped and squeezed between the two numbers that look almost identical except that the former has a denominator of 71, the latter of 70. The latter result reduces to 22/7, the famous approximation to π that all students still learn today and that some unfortunately mistake for π itself. 

For me, Archimedes' method is no different from a good story that can be read and read over and over again. After Archimedes, many mathematicians have devised many other "stories" to approximate π. Now it is known that nobody can determine the exact value of π, but it is possible to approximate π to any level of accuracy. Doesn’t that sound puzzling? We cannot know it, yet we can approximate it however accurate we want.

Next, go find a compass and a ruler. Draw any angle, then use the compass and the ruler according to the instructions in Figure 5 to bisect that angle. 

Figure 5: Split a corner with a compass and a ruler.

This looks easy, how’s about taking it to another level? Let’s try to trisect the angle. One might think that it should be easy to go from two to three, but it is actually as difficult as reaching heaven from earth. The two problems at first glance look identical, but in fact, they belong to two very different worlds. The Greeks began to seek answers thousands of years ago, but there was no answer until the end of the 19th century. A very courageous answer: it is impossible.

How to prove that a problem cannot be solved? With an extremely ingenious argument as follows. To trisect an angle, it is necessary to solve a cubic equation. By the 19th century, it was known that solutions of cubic equations are represented by the cube root. But with rulers and compasses we can only calculate the square root, QED! 

My heart always has a special room for proof of impossibility like this. They are the embodiment of the supreme beauty that Mr. Russel speaks of. No matter who you are, no matter where you come from, no matter what you do, you won't be able to trisect an angle with a ruler and a compass! Impossibility proofs can also give me hope. Whenever I can't solve any problem, I often thought that maybe it can't be solved at all!

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It would be a glaring omission if talking about the beauty of mathematics without mentioning prime numbers.

A natural number n is called divisible by a natural number m when there is a natural number k such that n = m * k. Primes are natural numbers divisible by 1 and itself. For example, 2, 3, 5, 7, 11, etc. are prime numbers. 6 is not a prime number because it is divisible by 2 and 3. These are concepts anyone can understand, but hidden behind them are extremely interesting and difficult questions. 

From more than two thousand years ago, Euclid with a breathtaking argument proved that there were infinitely many prime numbers. The next obvious question is how prime numbers are distributed. By visualizing the first few thousands of primes, one can see that the longer primes appear less and less frequently. One can also prove that there are number sequences with arbitrary lengths in which there is not a single prime.

Every time I recall these results, I imagine I am walking in a vast desert. The oasis where I want to go is the next prime number. I don't know where it is, but, thanks to Mr. Euclid, I know it must appear, sooner or later, this desert must have a stop. Mathematics gave me such faithful beliefs, which I could not find anywhere else.

It turns out that nobody could predict when my oasis would appear! In other words, for any sequence of numbers, no one has found a formula or a proof to predict what the next prime number is. Human beings have studied prime numbers for thousands of years, yet this remains an open problem. As nicely put in this article, present an argument or formula which (even barely) predicts what the next prime number will be (in any given sequence of numbers), and your name will be forever linked to one of the greatest achievements of the human mind, akin to Newton, Einstein and Gödel.

If the distribution of primes is too difficult to understand, this is a problem no one has solved yet but is much easier to understand: are all even numbers greater than 2 the sum of two primes? This question, called the Goldbach conjecture, had been tested positively to any even numbers smaller than 400,000,000,000. But no one has seen to it whether it is still true if we try test larger numbers!

Is there any other form of artistic entertainment that only within a few minutes, just by imagination and very elementary knowledge, help you arrive at the threshold of knowledge of all mankind? Learning math is like having a miraculous door of Doraemon, it takes only a moment to be able to travel to places where nobody has ever come. What's more fun that that?

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I write this article for no other purpose than to share the pleasure of a person who is lucky enough to perceive the beauty of mathematics. Of course, beauty is in the eye of the beholder. Maybe you don't see what I'm writing here is interesting, and it's okay. I just want to say that math has completely changed my life, and hope that you give math a chance to change the life of you or your children.

I understand that most of us do not have a good experience when studying mathematics in high school or college. I assure you that this is a common problem, not only in Vietnam, but all over the world. Mathematics is taught and learned mechanically, without joy. But don’t worry, and please be optimistic, school is not the only place we can learn. I have never had a college degree, and if I can study math, anyone can learn it. The only thing we need is an open mind, daring to try new things, the rest can be left to math! 


Good luck and I hope that one day you’d find yourself studying a math problem, then, with a happy smile, whispering to yourself, isn’t it beautiful?

Comments

Neovn said…
Mình cứ ngỡ cậu là một nhà văn chứ không phải một kỹ sư của Google.
Cám ơn Thái, bài viết rất hay!
Lê Huy said…
Bên VN mình cũng có một anh best Toán học đó anh, anh Nguyễn Ngọc Trung huy chương vàng Toán học quốc tế tại Kazakhstan.
Dũng said…
"...learning math is like upgrading the firmware of the brain, making it bigger horizontally if not vertically"

really?