A genius? I just love studying.

Chapter 66: We are studying the same textbook

Tiewu Middle School,

eight pm,

Deng Leyan rubbed his sore neck, stood up and paced slowly in the room. After studying for several hours in a row, even he felt a little exhausted.

But his brain did not rest and continued to think about what he had just learned.

He is a boarding student, but the school has prepared a study room specifically for him so that he can devote himself to studying without being disturbed.

He doesn't even have to go to class on weekdays.

Facts have proved that if you study well, you can do whatever you want.

Li Bin walked out of the room and handed Deng Leyan a piece of A4 paper filled with text. "Since you're taking a break, I've prepared some interesting proofs in classical geometry for you. Try to prove them using your knowledge of abstract algebra."

He was originally a teacher in the competition class of Tiewu Middle School, but after the talented Deng Leyan emerged from Tiewu Middle School, he had almost become Deng Leyan's personal coach, giving one-on-one lessons most of the time.

If there is a problem he cannot solve, other teachers will help to solve it.

Tiewu Middle School can be said to have formed a think tank to serve Deng Leyan.

Deng Leyan took the draft paper, sat down, and began to read attentively.

There are four difficult problems in ruler and compass construction:
1. Can any angle be trisected?

2. Given the volume V of a cube, is it possible to construct a line segment of length 2V?

3. Given the area A of a circle, is it possible to make a square so that its area is also A?
4. Can a regular N-gon be constructed using a ruler and compass?

Deng Leyan frowned. Seeing this question made his mind go blank for a moment and he had no idea at all.

Li Bin smiled slightly. Sometimes it was a very satisfying thing to see a genius being humiliated.

Especially since this question has actually been proven a long time ago, but the other party is young and has not been exposed to enough mathematical knowledge.

"Group theory itself is just a tool, like equations and calculus, used to solve certain mathematical problems."

Li Bin spoke eloquently, "So after learning group theory, you may feel that it has no practical application, but if you master group theory, you can unlock its powerful capabilities."

"For example, these proofs in front of us once troubled ancient Greek mathematicians for their entire lives!"

Seeing Deng Leyan frowning, Li Bin also reminded him in time, telling him that these questions were not that simple, and he did not want to destroy the genius's self-confidence.

Time passed slowly, and Li Bin had already shut up, allowing Deng Leyan to think quietly.

Soon, half an hour passed.

During this period, Deng Leyan opened the textbook several times to check, but in the end, he still could not write down even a single line of useful answers on the draft paper.

"Okay, here's the answer. Take a look first and ask me if you don't understand."

Li Bin took out another stack of A4 paper and handed it to Deng Leyan.

……

Chengdu No.2 Middle School,

At 9:30 pm,
The glass windows of the study room were covered with a thin layer of mist. Summer had already begun in Chengdu in late April, but the temperature difference between day and night was nearly 10 degrees. The room was as warm as spring, causing the water vapor outside to adhere to the glass windows and condense into mist.

In the classroom, Chen Hui and Liang Peixuan were each concentrating on their study materials, immersed in the ocean of knowledge.

At this time, Zhang Anguo walked in.

He did not disturb the two people who were studying, but came behind Chen Hui and looked over his shoulder at the textbooks on his desk. It seemed that all teachers have this habit.

Chen Hui closed the textbook and turned to look at Zhang Anguo.

He was not disturbed by Zhang Anguo, but just finished reading the textbook "Abstract Algebra".

"I've signed you up for the preliminaries on June 3rd. 48 hours of unlimited questions."

"Yeah." Chen Hui nodded.

Zhang Anguo looked at the closed textbook and asked, "How are you doing with abstract algebra?"

“We have already learned all the content in the textbook.”

Chen Hui said as accurately as possible.

After studying abstract algebra, he discovered that university mathematics was very different from what he had imagined.

It was much more interesting than he had imagined.

Of course, it is also much more difficult.

Although he fully understood the contents of the textbook, he did not dare to say that he had learned abstract algebra. On the contrary, the more he learned, the more he didn't know.

"lesson complete?"

Zhang Anguo asked in surprise.

He was the one who found the textbooks for Chen Hui and planned the learning path for Chen Hui.

Of course he knew that Chen Hui had only studied linear algebra and abstract algebra for just one week combined.

Finished two courses in one week?

Even though he had already studied them once, he couldn't say that he could pick up two courses in just one week, let alone Chen Hui who was starting from scratch.

"Ah."

Chen Hui still just hummed lightly.

First impressions are often so important. Even though the two are now in a cooperative relationship, Chen Hui still cannot feel a sense of closeness towards Zhang Anguo.

Zhang Anguo didn't mind. After a moment's thought, he said, "Don't rush into the next course. I have a few questions here. Try them to consolidate your knowledge."

If someone else said this to him, he would definitely not bother to pay attention to the other person, but if this person was Chen Hui, he would be willing to believe it for the time being no matter how outrageous it was.

As he spoke, he picked up Chen Hui's pen and started writing on the draft paper.

There are four difficult problems in ruler and compass construction:
1. Can any angle be trisected?

2. Given the volume V of a cube, is it possible to construct a line segment of length 2V?

3. Given the area A of a circle, is it possible to make a square so that its area is also A?
4. Can a regular N-gon be constructed using a ruler and compass?

The questions were simple, and Zhang Anguo finished them quickly. "These are some difficult problems in classical geometry. Try to prove them using group theory."

He has not been idle these days. While making a study plan for Chen Hui, he is also reviewing hard. These questions are exactly the exercises he used when reviewing abstract algebra, and now they are just right for testing Chen Hui.

"Drawing with a ruler and compass?"

The moment Chen Hui saw the question, he had already reconstructed the question in his mind.

Ruler and compass construction is nothing more than making points, lines and circles that meet certain conditions. Since any straight line is determined by two points on it, and any circle is determined by the center of the circle and a point on the circle, the ruler and compass construction problem can be reduced to: making points that meet certain specific conditions!
So given two points O and A on the plane, O is the origin and A is (0, 1), knowing which points can be made based on O and A using a ruler and compass is equivalent to knowing which real numbers r∈R satisfy that a point P can be made using a ruler and compass such that |OP|=|r|. The real numbers that satisfy this condition become constructible.

Chen Hui's thoughts flow like a spring and his writing moves like flying.

Zhang Anguo was stunned.

Although Chen Hui hasn't finished writing it yet, the idea is obviously correct.

It's not that he had never thought that Chen Hui could solve this problem, but he had never thought that he could do it so quickly.

Can you really learn it just by taking a look?

Zhang Anguo had to admit that sometimes the gap between people is really bigger than that between people and dogs!
He clearly studied this abstract algebra book back then!

(End of this chapter)