Four Solutions

At Jiangcheng No. 7 High School's Senior Three Class 2, an extremely strange scene unfolded.

A student stood at the podium, telling his classmates below that there were multiple ways to solve the problem.

Meanwhile, the teacher stood to one side without the slightest look of surprise.

Li Dong wrote Solution One on the blackboard.

"I think most of you used this method: multiply both the numerator and denominator by the conjugate of the denominator."

z =[(1+√3i)(1-i)]/[(1+i)(1-i)]=[(1+√3)+(√3-1)i]/ 2

|z|=√{[(1+√3)/2]^2 +[(√3-1)/2]^2 }=...=√2

Li Dong wrote rapidly across the blackboard.

"But this is the clumsy method. You brute-force the calculations, and it's easy to make mistakes, though the logic is the simplest."

Then he wrote Solution Two on the blackboard.

"We can use the properties of moduli. Everyone knows that the modulus of complex-number division equals the quotient of the moduli."

|z|=|1+√3i|/|1+i|=√(1^2+(√3)^2)/√(1^2+1^2)=√4 /√2 = 2/√2 =√2

"You can work that out mentally."

The students below nodded one after another. This method really was fast.

However, Li Dong suddenly remembered that if it were him, he would probably draw further inferences as well, so he continued.

"What if this question asked not for the modulus, but for the principal value of the argument?"

He drew a coordinate system for the complex plane on the blackboard and wrote Solution Three: the geometric rotation method.

"The vector corresponding to 1+√3i has an angle of 60°, while 1+i has an angle of 45°."

"Geometrically, dividing complex numbers means dividing their moduli and subtracting their angles."

"So the angle of z is 60°-45°=15°."

"This is very useful when dealing with rotation problems."

By this point, more than half the class was looking somewhat lost.

This solution was the kind that experts could grasp at a glance, but ordinary students would never think of.

But it still was not over.

Li Dong seemed to have completely forgotten that the people sitting beneath him were not Lu Yao, but his dear Senior Three classmates.

"Of course, if you want to understand complex numbers from a higher-dimensional perspective..."

He wrote Solution Four: matrix representation.

"A complex number a+bi is isomorphic to a two-by-two matrix [[a,-b],[b, a]]."

"The modulus is the square root of this matrix's determinant."

|A|= a^2 + b^2

"Using matrix multiplication and inverse-matrix operations, this problem is essentially a linear transformation..."

As strings of matrix symbols appeared on the blackboard, the students' eyes below became utterly clear.

Li Dong himself lectured with great enthusiasm, completely immersed in the ocean of knowledge.

"...So, no matter which angle you approach it from, the answer is A. Do you understand?"

Li Dong turned around and looked at everyone expectantly.

More than fifty mouths beneath the podium hung slightly open.

The girl who had asked the question earlier already had tears in her eyes.

"I... am I really that hopeless? Why can't I understand any of this...?"

Is... is this the world of top students?

The people who had initially thought Li Dong was merely reciting the answer now felt like their faces had been slapped swollen.

Could reciting an answer produce matrices? Could it produce four different solutions?

Jiang Yizhou, seated in the second row, stared fixedly at the blackboard.

"I know the first one... I could think of the second one too... the third one..."

Without realizing it, sweat had already begun to seep from his forehead.

He could barely understand the third method.

As for the fourth...

Matrices? Wasn't that university material? This was so far beyond the syllabus it was ridiculous!

Jiang Yizhou frantically copied down the processes he did not understand onto scratch paper, muttering under his breath all the while.

"This is beyond the syllabus... matrices aren't on the college entrance exam..."

But he was beginning to feel afraid. He was not afraid of Li Dong; he was afraid of himself! Afraid that even he himself was starting to feel that he might not measure up to Li Dong!

The 132 points he had always been proud of looked so pale and powerless under the blow of Li Dong's four solutions.

Meanwhile, Mi Xia's eyes were shining.

She quickly recorded the third and fourth methods. The sensation of looking down on a problem from a higher-dimensional perspective gave her an exhilarating feeling of sudden enlightenment.

So complex numbers can be understood like this too...

She looked up at the boy on the podium, radiating confidence. The vague suspicion in her heart finally became reality.

He really is no longer at the same level as the rest of us...

Meanwhile, in the hallway.

Li Dong was receiving Old Yang's "guidance."

"I asked you to explain things clearly to these students, to show them how to secure their points." Old Yang shook his head helplessly. "There was no need to explain that last method, was there? It's completely beyond the syllabus. Other than scaring them, it serves no purpose."

Li Dong nodded in wholehearted agreement. He had gotten carried away just now and had nearly started talking about the mapping between the complex plane and the Riemann sphere that Gauss had mentioned in their old group chat.

"Mm, you're right, sir. I'll pay attention next time."

Li Dong spoke with a sincere expression.

"Next time, I'll use a different method, like expanding it with the Euler formula. That should be easier to understand."

Old Yang: "..."

He opened his mouth, but in the end, all he could manage was a long sigh.

"Forget it, you brat... Come on, let's go to Class 1."

They soon arrived outside the classroom of Senior Three Class 1.

Class 1 was the best class in No. 7 High School's senior year and also the key science class.

If there was anyone at No. 7 High School guaranteed to make it past the first-tier university cutoff this year, it was definitely Class 1's class monitor—Chen Yue.

During the break, Chen Yue was sitting in the front row, turned around as he chatted and laughed with a girl behind him.

As Li Dong's roommate, he knew that Li Dong had been working hard recently. However, since they were not in the same class, he did not have any direct understanding of Li Dong's so-called "miracles."

"Hey, want to duo queue after evening self-study? I'm amazing at Yang Yuhuan, and my healing is insane."

Chen Yue looked completely relaxed.

The girl politely declined.

"I'll pass. I like jungle kings who play Ma Chao. Your Yang Yuhuan is way too deadweight."

Just as Chen Yue was preparing to argue indignantly, he suddenly noticed that the light at the classroom door had dimmed.

He instinctively turned his head and saw Li Dong standing in the doorway with his hands behind his back, while Old Yang, the math teacher, stood beside him.

On the other side, Zheng Hua, the head of the physics department, had also hurried over.

The faculty arrangement for No. 7 High School's senior year was rather interesting. Old Yang carried the entire math department on his own, teaching both regular and key classes.

But for physics, Zheng Hua personally taught Class 1, while Class 2 was taught by Jiang Yang.

As soon as Zheng Hua arrived, he immediately asked Old Yang.

"How was it? Did this kid do all right teaching your class?"

Old Yang said nothing. He merely gave Li Dong a meaningful glance, then raised a thumb and nodded.

The meaning was obvious: absurdly strong.

Zheng Hua understood instantly.

"Haha, good, good!"

Then he gave Li Dong a heavy pat on the shoulder.

"Li Dong, the next class is yours."

Because Chen Yue sat in the front row near the door, he heard every word clearly.

A string of question marks popped up in his mind.

"What exactly is being handed over to Li Dong?"

However, he soon found out.

When the class bell rang, Zheng Hua did not step onto the podium. Instead, he pulled over a chair and sat at the back of the classroom.

The one who stepped onto the podium was Li Dong, wearing the same school uniform as everyone else.