Chapter 91
“All right, orientation is over~! From here on out, I’ll take attendance and start class!”
I clapped both hands- and declared.
“Eeeee~”
Then
sounds of betrayed expectations drifted out from here and there in the classroom.
There was a reason the freshmen reacted that way.
In most classes in other departments or general-education courses, professors usually gave only a brief orientation on the first day, didn’t cover any material, and let everyone go home early.
In other words, the first day was for getting familiar with one another’s faces and warming up lightly.
That was the modest expectation of freshmen who had just entered university.
But.
I’d anticipated those expectations long ago.
I wagged my index finger- as if they were being ridiculous.
“You unnis and oppas are mathematics majors at Seoul National University, renowned for your overwhelming workload!”
I said it in the strictest, most solemn, serious voice I could manage.
Of course, from an objective perspective, it probably sounded adorable and cutesy.
“The high-school math level was already lowered considerably by the 2022 revised curriculum. If you unnis and oppas, who are lacking in mathematical knowledge, don’t plow- ahead with the material starting now, it’ll get even~ more difficult later!”
I planted my hands on my hips atop the footstool.
And those words.
They struck the freshmen who had only just shed their high-schooler appearance rather cruelly.
—You don’t know very much. I’ll plow through the material and fill in what you’re lacking.
If you thought about it, that was another way to interpret what I’d said.
The unnis and oppas who heard it right away might feel a tiny bit indignant.
But what could they do?
It was an irrefutable fact.
It was an obvious fact that the standard of high-school education was declining with each passing year.
In fact, I’d heard that when students in the same year got together,
they lamented self-deprecatingly, saying things like, “We’re the squeezed-in generation,” and “We’re the Ministry of Education’s lab rats.”
So, in a way, this was.
A kind gesture from the great Professor Yoo A-yeon, who was going to ease their burden by making them shoulder, little by little even now, the murderous amount of studying they would normally spread out over four years of university!
Therefore, it was actually something they ought to thank me for.
I explained this opinion in a gentle voice.
“All right, everyone understands, right?”
The unnis and oppas reluctantly nodded at my opinion.
Well, it wasn’t as if there was much they could do even if the undergraduates disagreed.
If the professor said she was teaching class, then class was happening!
“Then I’ll take attendance!”
I held the printed attendance sheet in my hands.
Just then.
As if I’d thought of something, I paused and looked all around the classroom.
“Oh, right! I hear these days you can take attendance all at once by simply pressing one button in an app?”
“That’s right, Professor. It’s convenient.”
One oppa answered.
“Heheh, but I won’t be using that function!”
When I said that, everyone looked puzzled.
“Taking attendance with a click- in an app is so impersonal, isn’t it? Besides, I’ve only had a smartphone for three days, so I’m not part of the generation that’s used to digital technology either.”
“Gasp, three days…?”
“Yep! I’m a totally brand-new phone user.”
I held up the smartphone with its cute rabbit case.
An aura of pristine newness radiated from it.
“Besides, I have a really good memory, you know? So I can memorize all your faces and names today! I’ll just read the attendance sheet and call your names!”
When I declared that so confidently.
The unnis and oppas looked slightly surprised.
Come to think of it.
Professors of a certain age might find it gradually difficult to memorize their students’ faces and names.
They probably saw dozens of students in a single semester. How could remembering them all be easy?
But I was a seven-year-old child whose brain cells were fresh and sprouting!
If you rely only on digital devices, which are efficient for everything, your brain activation actually slows down.
At times like this, you have to use an analog mindset, look at people with your own eyes, call them by name, and make your brain work diligently.
I held the attendance sheet in both hands and began calling out each name clearly.
“Kang Chae-young unni?”
“Yes!”
“Kim Do-hyeon oppa?”
“Here.”
“Kim Seo-jun oppa?”
“Yep.”
Every time I called a name.
I etched the face of each unni or oppa who answered into my eyes.
Kang Chae-young had long bangs, Kim Do-hyeon wore round glasses, and Kim Seo-jun had a deep voice.
Pairing each name neatly with a distinguishing feature made memorizing them much easier.
“Park So-young unni?”
“…Yes.”
Along with the quiet answer, an unni raised her hand.
But.
Huh?
Park So-young unni was a little different from the other unnis and oppas.
With her arms firmly folded.
She was glaring straight at me with a very challenging look in her eyes.
It was the kind of look that seemed to say, “Let’s see how well you do.”
Hmm. Is she a tsundere?
I tilted my head slightly, but moved on to the next name for now.
“Park Jae-hoon oppa?”
“Yes!”
And then I got to call a name that made me happy.
“Seo Eun-chae unni?”
“Yeees, Professor!”
That familiar voice!
I quickly raised my head and looked in that direction.
Just as I thought.
In the back row, Eun-chae unni was waving both hands.
“Hehehe, Professor Yoo A-yeon!”
Eun-chae unni whispered with a smile in her eyes.
It finally felt real that Eun-chae unni had become my student.
It felt like only yesterday that I’d cheered her on by handing her hot chocolate outside the CSAT test center.
Of course, class was in session now, so I had to keep public and private matters separate.
“Seo Eun-chae unni, your attendance is confirmed!”
I forced myself to clear my throat and moved on.
But I couldn’t stop the corners of my mouth from slowly rising.
“Shin Yu-jin unni?”
“Yes.”
“Lee Jun-hyeok oppa?”
“Yeeep!”
And so I called out every single name among the roughly forty unnis and oppas.
And their faces and voices.
I stored them one by one- in my mental database.
University students are often afraid of “the professor knowing my name.”
It makes them feel pressured, as if they have been singled out and the professor sees through everything, including whether they attended and submitted assignments.
But what would happen if the professor knew every student’s name?
Some naughty students, thinking the professor wouldn’t know who they were,
supposedly dash off as soon as they check in, or call a friend to take attendance for them before going off to get a good nap.
That might work with other professors, but it absolutely won’t work in front of me!
“Nihihi. I memorized all your faces~ already, so if any of you skip out after checking in or have someone take attendance for you, I won’t let it slide. I’ll mark you absent without hesitation! I’ll let it go once, but if I catch you twice or more, it’s an automatic F!”
When I declared that.
Everyone shuddered~ with looks of horror.
The freshmen had realized why it was frightening for a professor to know their names.
✒️✒️✒️
When I’m explaining something to non-majors, say, Dad or Arin.
Instead of this kind of rigorous exposition, I’d give only easy examples, skip all the intermediate steps with a hand-wave like “just do it like this~,” and tell them only the result.
But things were different now.
The people taking my class were none other than mathematics majors.
They needed to get used to rigorous exposition.
That was why, rather than giving them a friendly explanation that let them understand things roughly,
I planned to implant in their brains solid logic with no rigorous gaps or leaps.
Undergraduate unnis and oppas, please don’t resent me too much.
You’re going to suffer through it for all four years of undergrad anyway, so think of this as taking one hit early.
“Then let’s start class! I’ll plow through the basics first!”
I stepped onto the footstool and firmly readjusted my grip on the white chalk.
Tap-tap-tap-.
[The Rigorous Definition of a Limit]
“All right, unnis and oppas. You all learned limits in high school, right?”
“Yes-.”
“You probably learned it as: when x gets infinitely close to a, f(x) gets infinitely close to L—then lim(x→a) f(x) = L.”
[x approaches a] (?)
↓
[f(x) approaches L] (?)
↓
lim(x→a) f(x) = L
I wrote it carefully on the blackboard with chalk.
“But here’s a question. What exactly does this ‘approaches’ mean? Can ‘approaches’ be described mathematically?”
“Um… uh…”
The classroom fell silent.
Everyone wore an ambiguous expression that seemed to say, “It just… gets closer, doesn’t it?”
“Right? You understand it in words, but it’s vague when you try to explain it precisely, isn’t it? In high school, they just hushed-hushed past this part.”
This is where the trap lies, you see.
“So today. We’re going to learn how to define this ambigu~ous limit very rigorously.”
Tap-tap-tap-.
I moved the footstool around and neatly wrote equations down on the blackboard.
[Definition of the limit of a function]
For every ε > 0, there exists some δ > 0 such that,
if 0 < |x − a| < δ, then |f(x) − L| < ε,
we define lim(x→a) f(x) = L.
“Uh…”
The expressions of the unnis and oppas looking at the blackboard stiffened slightly.
They seemed frightened by the sudden appearance of so many Greek letters.
But the Department of Mathematical Sciences is a department where you don’t play with numbers for four years of undergrad—you fight strange Greek letters.
Don’t be afraid! Stand and fight!
“To explain, epsilon ε represents the tolerance for how close the function value f(x) must be to the limit value L. Delta δ represents the distance to which the input value x needs to be brought close to a.
Even if we set the tolerance ε to 0.01, a corresponding δ must exist, and even if we choose an incredibly tiny value like ε = 0.0000001, a corresponding δ must exist as well—that’s what it means to be able to say that the limit truly exists!”
For the limit of a function to exist, even if you specify an extremely~ small tolerance ε, you must always be able to find a δ that brings it neatly within that range.
As I wrote this explanation all the way through.
Everyone had a dazed— expression, as if they’d been struck by the Empty Void.
But their hands, which hadn’t neglected their studies throughout their school years, seemed unaffected; they were hurriedly taking down the equations I’d written on the blackboard.
Scratch-scratch-.
A moment ago, I’d told them to feel free to ask about anything they didn’t know.
But the unnis and oppas were now in a state where they didn’t even know what they didn’t know in the first place, so they didn’t even know what to ask.
I let out a small sigh, hoo-.
‘Well, I didn’t expect you to understand it after seeing it once. You’ll review it on your own.’
The more difficult a concept is, the more times you have to read it before you realize it. Just like the secret formulas in martial-arts novels.
It’s the same even for a genius like me.
No one in this world can understand something after seeing it only once.
The only difference is whether they absorb it quickly or slowly.
✒️✒️✒️
“All right, then let’s pause here for a moment. A story from the history of mathematics.”
I said, dusting off the chalk with a tap-tap.
“Who do you think came up with this definition?”
“Um… Newton?”
“Wrong! Newton is known as the founder of calculus, but unfortunately, it wasn’t him. The rigorous definition of limits wasn’t established until 200 years after calculus was developed. Pretty late, right?”
I swiftly wrote a name on the blackboard.
[Weierstrass (Weierstrass)]
“This epsilon-delta definition was systematically established in the 1860s by a mathematician named Weierstrass.”
I continued explaining.
“Actually, when Newton and Leibniz first created calculus, they relied on intuition about infinitesimals—that is, ‘infinitely small things,’ right?”
“Infinitely small things?”
“Yep. Something larger than zero but with no size, infinitely small. That’s an infinitesimal. But even after introducing that, differentiation and integration calculations worked amazingly well. Yet they couldn’t rigorously explain, ‘What exactly does infinitely small mean?’ or ‘Why are these calculations always correct?’”
I shrugged.
“It was basically, ‘I don’t really know, but the calculation works when I think of it this way, right?’ A little unsettling, isn’t it? It’s an ambiguous explanation on the level of what you learned in high school: ‘When it gets infinitely close, it converges somehow!’”
“It does seem a bit like that.”
The unnis and oppas nodded.
“But mathematics before 1860 all followed this rough-and-ready style of glossing things over. Then, once this epsilon-delta definition appeared, a huge change took place.”
[ Intuition → Logic ]
“In short, mathematics changed from a mathematics of intuition, where we roughly assumed things would work that way, into mathematics that could be described using rigorous logic.”
I looked all around the classroom.
“From then on, limits, continuity, and differentiation—all of them could be proved with precise logic. No matter how strange and bizarre a function looked, this one definition could handle it all.”
I grinned-.
“So, to sum up. In Newton’s time, they were at the level of saying, ‘If you do it this way, the calculation works somehow!’ and moving on. After Weierstrass, they became able to prove logically ‘why those calculations are always~ correct.’”
I set down the chalk and dusted off my whitened hands, which had become like glutinous rice cakes, tap-tap-.
The falling powder was practically on the level of a Dubai chewy cookie. Thinking about it suddenly made me crave one.
“The field built on such solid logic is the analysis you’ll be learning from here on out!”
Tap-tap-tap-.
While I wrote on the board, the unnis and oppas busily moved their pens and took notes.
Scratch-scratch-.
Their expressions had been frightened at first, but now they were following my explanation carefully, if laboriously.
Good, good.
That’s an excellent attitude!
“All right, who understands up to here?”
When I asked.
“Uh… more or less!”
Answers drifted out from here and there, but none sounded confident.
“More or less is excellent too! But if you don’t keep looking at it over and over, you’ll forget it all and turn into total dummies, okay? Got it? Be sure to review! Promise!”
I hooked my pinky in the air, making a gesture of promise.
A few unnis and oppas hooked their pinkies in the air after me.
At that insignificant little action.
I felt good, as if I had clicked- with the college students twelve years older than me and we were in sync.
“Hoo heheheh.”
“Hoo heheheh.”