PDLP Chapter 5: The Strongest Godfather in Advanced Mathematics Class!
Chapter 5: The Strongest Godfather in Advanced Mathematics Class!
As the light screen appeared, An Yi experienced for the first time the legendary feeling that "knowledge is entering the brain in a strange way".
An Yi vaguely remembered the golden ratio, as his teacher had mentioned it in high school. However, this concept was rarely tested as a standalone difficult problem; it usually appeared in application-based questions, so his understanding of it remained superficial.
I never expected to need it now.
Mathematics is like this: often you don't know the actual use of the knowledge you've encountered, so you're too lazy to study and memorize it seriously. But at some point, when you need to solve a certain problem, you'll find that you really can't do it without that knowledge.
The golden ratio, also known as the approximation of the golden section, is generally taken as 0.618 and is widely recognized as the proportion that best evokes aesthetic appeal. However, this time An Yi will not use this approximation, but its exact algebraic expression—
φ (golden ratio) = (1 + √5) / 2
This algebraic expression will appear multiple times during the factorization of the integral ∫1/(x^5+1)dx that An Yi needs to perform. However, this expression is irrational, and as the girl said, the computational complexity is simply beyond An Yi's current capabilities. Even with [computational ability +1], it's still impossible!
However, by knowing just one small piece of knowledge, those originally massive and uncalculable complex algebraic expressions can be simplified, and the difficulty is greatly reduced to the point that they can be calculated!
This is precisely the point of knowledge that An Yi had previously missed: a truly remarkable property of the golden ratio.
φ²=φ+1
This property helps An Yi quickly simplify nested square roots in all higher powers and integral results, which greatly simplifies the calculation and optimizes the calculation steps!
In this way, calculations that were originally too large and complex for him to perform can now be solved.
"How is it?"
Teacher Liu waited for a while, and seeing that An Yi was staring at the question without writing anything, she secretly breathed a sigh of relief, thinking that the question she had given should have stumped him. Her smiling expression finally returned.
"I think I can solve this problem a little bit, teacher," An Yi replied.
"It's normal if you can't solve it... wait a minute."
Teacher Liu was only halfway through her sentence when she suddenly realized what An Yi had said. Before the rest of her words could even leave her throat, she swallowed them back down.
"What did you say?!"
"Teacher, a gentleman's word is as good as his bond. You wouldn't want to go back on your word, would you?" An Yi stared at Teacher Liu with some suspicion.
"Impossible! Do I look like the kind of person who can't take a joke? You accumulate points, and if you can accumulate enough, I mean it—everyone will be considered to have successfully checked in today!"
Finally, perhaps finding it hard to believe, and thinking that An Yi was putting on a brave face and pretending to be confident, Teacher Liu added another reward.
"And you'll get full marks for your final exam participation, so it doesn't matter if you don't come to class or hand in assignments afterwards!"
"Then I'll really work hard!" An Yi's eyes lit up, showing a look of eagerness, but he just didn't start.
"Go ahead and accumulate it!"
Seeing that he couldn't get any more benefits out of Teacher Liu, An Yi stopped delaying, picked up the chalk, and began to calculate.
∫1/(x^5+1)dx
注意到，1/(x^5+1)=1/5(x+1)+((√(5)-1)/10x+2/5)/(x^2+(√(5)-1)/2x+1)-((√(5)+1)/10x-2/5)/(x^2-(√(5)+1)/2x+1)……
Points are easy to obtain...
Simplify...
An Yi had no intention of writing out every step on the blackboard. That would have made the writing process too complicated and unnecessary, since few people could understand it anyway. As long as Teacher Liu could understand it and know that what he wrote was correct, that was enough.
Therefore, he omitted some of the calculation steps and only wrote out the most crucial steps.
Shortly after, he wrote down the answer.
That is, we get:
∫1/(x^5+1)dx=1/5ln(|x+1|)-(√(5)+1)/20ln(x^2-(√(5)+1)/2x+1)+(√(5)-1)/20ln(x^2+(√(5)-1)/2x+1)+√(10-2√(5))/10arctan((4x-(√(5)+1)/√(10-2√(5))+√(10+2√(5))/10arctan((4x+(√(5)-1)/√(10+2√(5))+C
Even after omitting many of the complex calculation steps, it still took him almost two blackboards to write out the entire calculation process.
也难怪先前他单凭【计算能力+1】能够解出∫1/(x^6+1)dx，却对∫1/(x^5+1)dx这道题无能为力，因为后者的难度和计算量足足是前者的三四倍有余！
After finishing writing, An Yi let out a soft sigh of relief, put down the chalk stub that was now only a small piece, and couldn't help but shake his hand... Writing on the blackboard with chalk was a relatively rare thing for him, so doing such a large amount of writing work in a short period of time even made his fingers feel a little sore and numb.
The silence in the classroom was beyond description; the phrase "you could hear a pin drop" was insufficient to capture the extreme situation. Everyone in the lecture hall held their breath, staring back and forth between the blackboard, An Yi, and Teacher Liu. A deathly silence permeated the air.
The main problem was that An Yi's very first step, "noticing," successfully put all the students in the audience to their knees. How did he manage to notice that? Everyone else has two eyes, so how come his attention span is so amazing? Are you perhaps a disciple of the great mathematician Ramanujan's school of thought? The kind whose eyes light up!
In short, to put it another way, after An Yi took the first step, all the students in the audience couldn't keep up and were in a state of existential doubt, wondering, "Who am I? Where am I? Is what's written on the blackboard really math and not English?"
Normally, after falling into this state, one would either lose focus or get distracted, or lie down to play games or read novels. However, this time was obviously different. Everyone was focused on waiting for the result, and some mischievous people had already set up their phones and turned on the camera when An Yi started answering the second question.
The public's sense of smell is often very keen. When there is any indication that a big scandal is about to break, they will be able to detect it. Some generous members of the public will record it and share it on the internet.
However, it is obvious that no one is paying attention to this matter at the moment, so naturally no one can imagine what kind of impact it will have if the relevant video is posted online. At this moment, everyone's attention has shifted to Teacher Liu as An Yi shook his hand and turned his head.
Unfortunately, at least for now, in this lecture hall, apart from An Yi who wrote the answer on the stage and Teacher Liu who posed the question, no one else can determine whether this question is correct or not!
The atmosphere seemed to grow heavy with the deathly silence, and all eyes and gazes seemed to exert a tangible pressure, focusing on Teacher Liu.
As a teacher with considerable experience, Mr. Liu even felt uncomfortable with this pressure... mainly because even when the classroom is full, there aren't usually so many people looking at him at the same time!
Teacher Liu took a deep breath, carefully looked An Yi up and down, then smiled with relief, raised her hands, and slowly and earnestly clapped.
He didn't say anything, but his gesture spoke volumes.
The onlookers, who had been holding their breath and waiting for the result, suddenly realized what was happening and joined in the unexpected celebration. The large lecture hall was filled with a deafening roar, almost an overwhelming sense of awe.
An Yi enjoyed the sound waves that rose up for him. In the eighteen years he had lived since childhood, this was the first time he had enjoyed such treatment. The soreness and numbness in his right hand fingers disappeared, replaced by a tingling sensation that came from nowhere throughout his body from head to toe, almost leaving him at a loss.
Almost instinctively, he looked at the girl beside him. Sunlight slanted through the glass, falling on the girl, making her already fair skin appear almost transparent. She was also smiling and clapping, her voice still clearly reaching his ears amidst the noise.
"Well done!"
To be honest, An Yi felt that he was still a long way from winning over the girl, but if the system reversed the calculation and counted the girl's affection for him... then the affection level was already 200/100!
Is this the charm of "Advanced Mathematics"? It's really not difficult to get along with at all! That's what An thought.
"But don't forget what you promised me, okay? You have to keep your word." The girl smiled slightly.
"So... what exactly happened?" An Yi's feverish state subsided slightly, and he asked somewhat nervously.
"Keep it a secret for now~"
The girl kept everyone in suspense, saying, "You'll find out later."
An Yi could only suppress her anxiety for the time being.
Meanwhile, the noise in the large lecture hall finally subsided, and the onlookers got their fill of gossip. An Yi had become the "godfather" of his three roommates and all the students who had skipped class, and had even received a "guaranteed exemption from classes and homework." As for Teacher Liu…
Teacher Liu was on the verge of tears!
No, this can't be right! This is Jingjiang University, just an ordinary non-985/211 university, and this class he's taking isn't even in the math department, but in the aerospace engineering department... This is engineering! How can there be a student so good at math that they can solve this kind of problem?
As expected, he should have successfully defeated his opponents and left behind a record of achievements. Now, however, he has truly "lost both his wife and his army"!
"Are you really a student majoring in aircraft propulsion engineering at the Aerospace Engineering University?"
As the applause subsided, Professor Liu persisted, still somewhat unwilling to give up. He now seriously suspected that An Yi was someone from the mathematics department who had taken attendance in place of him.
"Answer honestly, I'm just curious, and I won't change what I promised."
"No, teacher."
"I knew it! Tell me, are you from the math department?" Professor Liu slammed his fist on the table in excitement.
"No, I'm an industrial engineering major from the aerospace engineering program." An Yi scratched his head sheepishly.
"Aerospace engineering has this major?" Teacher Liu was completely confused.
"It's a small major, with only two classes and less than a hundred students. It can't compare to the several hundred students in flight control and manufacturing..." An Yi explained.
Teacher Liu stared at An Yi in disbelief, feeling as if she was about to vomit blood.
He had indeed never known about industrial engineering before, but upon hearing An Yi's description, he understood. In a university like aerospace engineering, a relatively minor major... it's generally just there to fill quotas, and its academic requirements are definitely incomparable to those of flight engineering and control. It was immediately identified as a "watered-down" major.
As a result, in such a "watered-down" major, a freak actually emerged who was able to solve this integral problem that even math students might not be able to solve?
"You...are you usually very interested in points?" After thinking for a while, Teacher Liu could only come up with this explanation.
It might be a wild jaguar. Although it is an extremely rare species, it is not impossible!
"I've only been learning about integrals for a short time, and I only have a basic understanding."
An Yi could only maintain a polite yet awkward smile. "Actually, I was just lucky to solve this problem. Really, teacher, you have to believe me."
This matter cannot be accepted hastily. An Yi is quite self-aware. The reason he was able to solve these two integral problems is twofold: firstly, he was helped by his "computation ability +1", and secondly, he received a gift from "Advanced Mathematics". However, neither of these is a constant occurrence!
It's best to remain humble until he truly internalizes this ability.
"real?"
"Really!" An Yi nodded seriously.
"Do you know any other ways to solve this problem?"
Teacher Liu stared at An Yi suspiciously, then suddenly asked, adding a tempting hint: "Solve it and I'll give you an even bigger reward."
"What reward?" An Yi asked cautiously.
Having radishes is certainly a good thing, but the first reaction upon seeing radishes should be to be wary of the potential danger.
An Yi did indeed know the second way to solve the problem, which was the second item in the "Gifts of Advanced Mathematics" [Knowledge: Palindromic Polynomials]. Using this knowledge, let t = x + 1/x, reduce the fourth-degree polynomial to t² − t - 1 = 0, and solve for t = φ. This gives the same factorization as the first solution method, and the subsequent steps are the same.
The difference lies in the fact that although it uses the property of the golden ratio φ²=φ+1, it doesn't use it directly; rather, it's derived. It can only be said that they arrive at the same conclusion through different paths, and are indeed two different problem-solving methods.
"So you knew all along..."
Teacher Liu was speechless, choked with emotion, and sighed deeply, "Fine, you may leave!"
"A miscalculation."
An Yi slapped his head in annoyance, turned around and walked off the stage, muttering, "Old foxes are indeed shrewder..."
"You've tricked me twice already with your sweet talk. If I can fall into the same trap a third time in such a short time, what kind of teacher am I?" Teacher Liu thought to herself with a hint of smugness.
In his view, An Yi was undoubtedly a big shot, but he didn't know the specifics of his background. So he decided to let An Yi go for now and investigate him thoroughly after class.
Of course, given his status, he wouldn't stoop to arguing with An Yi over such a trivial matter.
It was just that as a teacher, I was curious about a student who suddenly appeared under my tutelage and seemed like a promising talent. I wanted to find out what was going on and see if I could nurture him.
If we can really discover a promising talent, then canceling roll call for one class is nothing!
Meanwhile, in some unknown corner, the fun-loving people are up to something.
If a person who likes to have fun doesn't cause trouble, then they're not a person who likes to have fun. And coincidentally, there really is such a person in the lecture hall at this very moment.
Yu Wenbo appears to be an ordinary freshman, but he is actually a small content creator on Bilibili. He likes to edit and upload all sorts of random things, and some of his videos have even become popular for a while.
When An Yi was called on to answer the question, he sensed the potential for material and immediately filmed the entire process!
After some simple editing, adding captions and emojis, he edited the title and uploaded it.
Title Name——
The strongest godfather in advanced calculus class!