Chapter 27: Chapter 26: Four Solutions
A very strange scene was unfolding in Class (2) of the third-years at Jiangcheng Seventh Middle School.
A student stood at the teacher’s podium, telling his classmates below that there were multiple ways to solve the problem.
Meanwhile, the teacher stood to one side, showing no signs of surprise.
...
Li Dong wrote "Solution 1" on the blackboard.
"I think most of you used this method: multiplying both the numerator and the denominator by the conjugate of the complex number in 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 quickly on the blackboard.
"But this is the clumsy way, just brute-forcing the calculation. It’s easy to make mistakes, but the logic is the simplest."
Right after, he wrote "Solution 2" on the blackboard.
"We can also use the properties of the modulus. As you all know, the modulus of a quotient of complex numbers is the quotient of their moduli."
|z| = |1+√3i| / |1+i| = √(1^2 + (√3)^2) / √(1^2 + 1^2) = √4 / √2 = 2/√2 = √2
"You can even do this one in your head."
The students below all nodded. This method was indeed much faster.
However, Li Dong suddenly thought that if he were the one solving it, he’d want to explore other angles, so he continued.
"But what if the question wasn’t asking for the modulus, but for the principal value of the argument?"
He drew a complex plane on the blackboard and wrote, "Solution 3: Geometric Rotation."
"The vector for 1+√3i has an angle of 60°, and the vector for 1+i has an angle of 45°."
"Geometrically, when you divide complex numbers, you divide their moduli and subtract their angles."
"So the angle of z is just 60° - 45° = 15°."
"This is extremely useful when dealing with rotation problems."
By this point, the eyes of more than half the students in the class had glazed over in confusion.
This was the kind of solution an expert would see in a flash, but it was beyond what an average student would ever think of.
But he wasn’t done.
Li Dong seemed to have completely forgotten that he wasn’t teaching Lu Yao, but rather his dear third-year classmates.
"Of course, if you want to understand complex numbers from a higher-dimensional perspective..."
He wrote down "Solution 4: Matrix Representation."
"A complex number a+bi is isomorphic to a 2x2 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 just a linear transformation..."
As strings of matrix symbols appeared on the blackboard, a profound clarity dawned in the eyes of the students below.
Li Dong, however, was speaking with gusto, completely immersed in his ocean of knowledge.
"...So, no matter how you approach it, the answer is A. Understand?"
Li Dong turned around, looking at everyone expectantly.
Below the podium, fifty-odd mouths hung slightly agape.
The girl who had asked the question now had tears welling up in her eyes.
’Am I... really that hopeless? Why can’t I understand any of this...?’
’This... is this the world of a straight-A student?’
The students who’d initially thought Li Dong was just reading off the answer felt like they’d been slapped across the face.
Could someone just reading an answer key come up with matrices? Could they come up with four different solutions?
In the second row, Jiang Yizhou stared intently at the blackboard.
’I know the first method... I could’ve thought of the second one, too... The third one...’
Before he knew it, beads of sweat had formed on his forehead.
He could barely follow the third method.
As for the fourth...
’Matrices? Isn’t that university-level stuff? This is so far beyond the curriculum it’s in another dimension!’
Jiang Yizhou frantically scribbled down the steps he didn’t understand on his scratch paper, muttering under his breath.
"This is beyond the curriculum... The college entrance exam doesn’t even test on matrices..."
But a sliver of fear began to creep into his heart. He wasn’t afraid of Li Dong; he was afraid of himself! Afraid that even he was starting to believe he couldn’t compete with Li Dong!
In the face of Li Dong’s four solutions, his pride-and-joy score of 132 now seemed utterly pathetic.
Misha, on the other hand, had eyes that were sparkling with excitement.
She quickly jotted down the third and fourth solutions. The feeling of looking down on the problem from a higher-dimensional perspective gave her a thrilling sense of enlightenment.
’So this is another way to understand complex numbers...’
She looked up at the boy on the podium, who radiated an aura of confidence, and the vague suspicion in her mind finally solidified into fact.
’He’s really not on the same level as us anymore...’
「Meanwhile, in the hallway.」
Li Dong was receiving a "talking-to" from Old Yang.
"I asked you to explain it to the class clearly, to show them how to lock in their points," Old Yang said, shaking his head helplessly. "There was no need to go into that last solution, was there? It’s completely beyond the curriculum. It served no purpose other than to intimidate them."
Li Dong nodded in complete agreement. He had gotten carried away in the heat of the moment and had almost brought up the mapping between the complex plane and the Riemann sphere that Gauss had once mentioned in their group chat.
"Yes, you’re right, sir. I’ll be more careful next time,"
Li Dong said with a sincere expression.
"Next time I’ll use a different method, like expanding it with Euler’s formula. That should be a little easier to understand."
Old Yang: "..."
He opened his mouth, but all that came out was a long sigh.
"Never mind, you kid... Let’s go. Time for Class (1)."
They soon arrived at the door of the third-year Class (1).
Class (1) was the top class in the third year of Seventh Middle School, the advanced science track.
If anyone at Seventh Middle School was a shoo-in for a top-tier university this year, it was definitely the president of Class (1)—Chen Yue.
It was currently the break between classes. Chen Yue was sitting in the front row, turned around to chat and laugh with a girl in the row behind him.
As Li Dong’s roommate, he knew Li Dong had been studying hard lately, but since they weren’t in the same class, he had no firsthand knowledge of Li Dong’s recent "miracles."
"Hey, wanna duo queue after evening study hall? My Yang Yuhuan is insane, my healing output is off the charts."
Chen Yue looked completely relaxed.
The girl politely declined.
"Nah, I’d rather play with a god-tier jungler on Ma Chao. Your Yang Yuhuan is just a freeloader."
Just as a disgruntled Chen Yue was about to argue his case, he noticed the light from the classroom doorway suddenly darken.
He instinctively turned his head and saw Li Dong standing in the doorway, hands clasped behind his back, with their math teacher, Old Yang, beside him.
Just then, the head of the physics department, Zheng Hua, also hurried over.
The teaching assignments for the third year at Seventh Middle School were rather interesting. For mathematics, Old Yang shouldered the entire load, teaching both the regular and the advanced classes.
For physics, however, Class (1) was taught by Zheng Hua himself, while Jiang Yang taught Class (2).
As soon as Zheng Hua arrived, he went straight to Old Yang and asked,
"So? How did the kid do in your class? Was his explanation any good?"
Old Yang didn’t say a word. He just gave Li Dong a meaningful look, then shot him a thumbs-up and nodded.
The message was clear: ’He’s ridiculously good.’
Zheng Hua understood instantly.
"Haha, excellent, excellent!"
Then, he clapped Li Dong firmly on the shoulder.
"Li Dong, next class is all yours."
Sitting in the front row, close to the door, Chen Yue heard every word of their exchange.
A series of question marks popped into his head.
’What’s all yours? What are they handing over to Li Dong?’
However, he would soon find out.
When the bell rang for class to begin, Zheng Hua didn’t walk up to the podium. Instead, he pulled up a chair and sat down at the back of the room.
The one who walked up to the podium was Li Dong, dressed in the exact same school uniform as everyone else.
