Article: 'What to study Math for?' by a leading security expert Google caused a social network storm

With the sharing from a believer in Mathematics like Mr. Duong Ngoc Thai will help people understand somewhat the value of Mathematics in life.

Mr. Duong Ngoc Thai, Google's security engineer shared an article on his personal blog entitled "What to Do Math for" with content that brings the beauty of Mathematics to everyone. In just a short time his blog post attracted the attention of many people.

The following is the article on the blog VNHacker of Mr. Duong Ngoc Thai.

I don't understand what to do and no one is sad to explain.

I like to study mathematics from a young age, but I'm not good at math. I studied mathematics at level 2, and at grade 9, I failed to do grade 3 math. In the 10th grade, I still tried to study by myself, borrow your notebooks to study math and still work hard on articles for Math and Age magazines. Young. If anyone has a good and beautiful solution, they will be named. I still remember you Tran Vinh Hung, whose name also has a name. Recently I learned that Hung was a math professor in America. I was named once, not in a magazine, but a math teacher named in front of the school as a mirror . hard to post to the magazine, even though it was never named.

In the middle of 11th grade I had a computer and started to spend all my time on the Internet. I was at school more and more ignorant, because I didn't want to go to school. So sometimes I was given a "team" at risk of graduation. Until the second half of the 12th grade, I went to school more seriously for fear of dropping college. I was no longer interested in math, and studying was just to take high scores.

Incredible college, the first two-year general program taught a lot of math, but I did not study. Six-and-a-half morning, putting rice in the ribs is not known, but how to put a lot of matrices into the rim, school, and group. I don't understand what to do and no one is sad to explain. I skipped school, stayed home to learn how to hack and go to work to earn money.

What I just said is nothing new, many people have experienced similar stories. Yet after 20 years, I am still learning math with an indescribable joy. What took place next for me is true color.

I do security at a bank. The first few years of good work, lots of things to do and they paid well. I manage a small group, gradually people do all the work, I have nothing to do, so I plan to take a break. I told my boss, but the boss called me to stay, my legs in the outer legs, no need to take a break.

I opened the company, then closed it without selling any product. I realized that I lacked business knowledge so I bought a bunch of books to read, then tried everything, but none succeeded. I did not feel discouraged, but I did not want to waste time. The question of what to do is still not answered.

I read on Professor Ngo Quang Hung's blog about the vicious circle of career, something like what I like I will do a lot, I will be good and good, I like it, like to do more, like that repeat. Finding and jumping into this vicious circle is the key to success.

What is my vicious circle? Thinking back, I found that I was the best and liked the most. 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 making security, "to see how far I can go."

It was a great time. Just like the person who walked in the tunnel for a long time, he thought it would be forever, but then he saw the light gleaming at the end of the road, I spent all the strength of youth to run toward the light, the more everything ran, the more clear , the clearer the faster you run! Within 3 years, from far away, I caught up and went with the world.

In a world where so many people are afraid of math, like learning math to create a competitive advantage is not small .

I found out that learning is like climbing a staircase that every stair has an endless length. The person standing at a staircase 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 because of me, luckily, it is heaven.

My biggest luck is probably not afraid of math. In a world with so many people afraid of math, like learning math creates a small competitive advantage, making me have a distinct identity, different from many others.

It is because of not accounting, I have the opportunity to come to the code, the main tool of the security industry. I discovered that the world has very few security engineers who know the code. I think there must be about 20 new security engineers who have 1 person who knows the password. Simply because you want to learn cryptography, you first need to study math, but mentioning math is that many people have dizziness, dizziness, no energy to learn. I'm the opposite. When I realized the math that I used to be used to protect the Internet, I rushed to learn to forget to eat and sleep.

I don't have any special talent, I have to work 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. Only thing, I never wanted to stop. I was like the one who conquered new lands, the more difficult it was, the happier it was.

What is interesting is that my math knowledge is nowhere near and the code that I have learned does not need the advanced math knowledge. Just use a few maths that level 2 math students can understand, cryptography can protect the Internet. Just knowing a little bit of the math that the code used, I had a competitive advantage over many people and became an expert in my field. Imagine if I knew a little more, I would What else can be done?

But that's because your job needs to use math, and my work and my daily life don't need math?

Thank you for asking. I also asked that.

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 to the brain, not adding the width will also add vertical. The fact that the natural world is dominated by math and math is a good way to understand more about the world in which we live. The famous scientist Richard Feynman says that God's language is 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 of learning what to do as a question of what a child played in the tenth year to do. Learning for fun, is that not enough? In fact, people who have a lot of pleasures do not benefit. Admire a picture, read a detective story, watch a movie, listen to a piece of music, etc. I like it, I do it, I don't need a reason. What would you think if I said that math can also bring joy, math is also very entertaining?

Mathematics comes from very specific problems in everyday life. Greeks do math because they want to measure objects. But then, very quickly, math became a game, the question is no longer to do but to play this game or not! How many of the best minds of mankind have played math games, learning math is a way for ordinary people like us to join them. What is more interesting than being able to meet acquaintances with Fermat, Euler or Abel?

Math has the ultimate beauty, a cold and austere beauty

The arts or sports that we enjoy, after all, are just games, so we can temporarily escape the tediousness 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. Betrand Russel, the eminent mathematician of the early 20th century, wrote that (translated by Google Translate, a technology cannot exist without math):

Math, looking right, not only the truth, but also the ultimate beauty, a cold and austere beauty, like sculpture, does not depend on our weak nature, not based into the great traps of painting or music, but great purity and strict perfection as only the greatest art can show.

You don't need to be good at math, don't need to be a mathematician to feel the beauty Mr. Russel describes. I think having knowledge will help somewhat, 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 median, intersect at one point. Isn't it beautiful? Three straight lines that have nothing to do with each other have one thing in common. Why does the triangle have this property? Please save for homework for readers.

Next, draw a circle. My first mathematical memory was to sit up straight, put my hands on the table and read aloud to 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 r again, we will have a circle area, ie A = π * r * r.

Is 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 right here on earth, right on the Moon, Mars and right everywhere in this 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, over the past two thousand years (yes, he himself jumped out of the bath and ran and la Eureka - I just wondered if he was then cursed by his wife or not?) Came up with a very unique way of calculating.

 

 

First (figure 2) he draws a hexagon with a 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 deduces 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 "easily" calculates the perimeter of each polygon. "Easy" in quotation marks because Archimedes had no money to buy a Casio computer, so it had to be calculated by a square root. In this way, Archimedes estimated that 3 + 10/71 <π

As the author of Infinite Powers wrote, the number π between the two numbers is almost the same as, the only difference is that the first number has a denominator of 71, the next number is 70. The following number is shortened to 22/7, a very famous approximation of π that every student will learn and some people still mistakenly believe that is the true value of π.

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 xỉ. Now it is known that there is no way to calculate the accuracy π, but it is still possible to calculate. Accurately. I don't know exactly how much, but how much I can count accurately, is it interesting?

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

 

The question is how to divide the three corners with the ruler and the compass? From the split into three, it is easy to say, who is more difficult to go to heaven. Two problems at first glance are very similar, but in fact, they belong to two very different worlds. The Greeks began to seek answers from BC, but there was no answer until the end of the 19th century. A very courageous answer: there is no way.

How to prove a problem without a solution? With an extremely ingenious argument as follows. To divide three corners, it is necessary to solve a quadratic equation. By the 19th century, it was known that the quadratic equation solution was represented by the square root. But with rulers and compasses we can only calculate the square root, infer what must be proved!

My heart always has room for proof of impossibility like this. They are the embodiment of the ultimate beauty that Mr. Russel speaks of. No matter who you are, no matter where you come from, there's no need to know me the next day, but I won't be able to divide three corners with a ruler and a compass! The existence of proofs can no longer give me hope. Whenever I can't solve some problem, I often sneak out hoping that this is a wrong topic or that this article can't be solved!

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

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. Number 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 obvious question is how the prime number is distributed in the natural sequence. The visual degree can see that the longer the number sequence, the more primes the number appears and can prove that there are any number of lengths in which there is not any prime number.

Every time I remember these results, I imagine I am walking in the vast desert, not on shore. 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.

My elemental oasis appeared sooner or later, it turned out no one could guess! In other words, for any sequence of numbers, no one has found the formula to predict what the next prime number is. A few thousand years of research on human primes have not yet solved this problem. If you solve it, your name will forever be associated with one of mankind's greatest inventions, on par with Newton or Einstein.

If the distribution of primes is too difficult to understand, this is a problem no one has solved but is much easier to understand: whether all even numbers greater than 2 are the sum of two primes? This question, named the Goldbach conjecture, had the answer to all numbers smaller than 400,000,000,000. Given an even number greater than 2 and less than 400,000,000,000, we always find two prime numbers that are equal to that even number. But no one has seen in the last hundred years whether it is still true if we try even larger numbers!

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

I write this article for no other purpose than to share the pleasure of the person who perceives the beauty of mathematics. Naturally or badly depends very much on the perspective. Maybe you don't see what I'm writing here is interesting, 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 this is a common problem, not only in Vietnam but all over the world. Mathematics is taught in a hard way, so it is learned mechanically. No problem, be optimistic, school is not the only place we can learn. I have never had a college degree, if I can study math, anyone can learn it. The only thing we need is an open mind, daring to try new things, leaving the rest to worry!

Good luck and hope you will also see the proofs and say, oh so beautiful!

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