Chapter 219: Number Line
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“This thesis…”
Lord Percival didn’t finish reading. In fact, the moment he saw the title and the author, he had already guessed much.
He had thoroughly read this thesis by Sirius ten years ago, during a seminar suggested by his mentor, Bodoro.
Percival still remembered that afternoon. The first time he read the thesis, he felt an electrical current running through him. He couldn’t believe that Sirius, someone his own age, had reached such heights.
While their peers were still worrying about dinner and tomorrow’s weather, Sirius had already set his sights on the distant stars.
However, this thesis had a fatal w: Sirius had introduced imaginary numbers.Undoubtedly, imaginary numbers did not exist in the real world. They were purely conceptual, difficult to understand, and had no real-world application.
At that seminar, Bodoro had expressed the same opinion. Despite the thesis’s elegant and concise forms, they were meaningless.
Now, Percival couldn’t believe that this thesis had resurfaced before him.
Another discussion meeting, another young and talented mage presenting it. Percival felt a sense of déjà vu.
As Percival was lost in thought, Roger had already finished reading the thesis.
He put it down and spoke.
“Absurd, absolutely ridiculous!”
Clearly, he also didn’t believe in the existence of imaginary numbers and thought it was pure imagination.
“Imaginary numbers? Square root of -1? What was the author thinking?”
As Roger spoke, the others also finished reading the thesis.
Bel remained silent. His mathematical skills werecking, so he refrained frommenting.
Hannah nced at Percival before speaking.
“This is a purely mathematical thesis. The concepts seem forcibly created to solve problems. Honestly, I don’t see the point of imaginary numbers.”
Igor, after hearing the others’ments, nced at Reiner and spoke slowly.
“Uh, I think this thesis might be better reviewed by a mage specializing in this field. Some of the mathematical concepts are beyond my knowledge.”
Savros shook his head, keeping his thoughts to himself.
Seeing everyone’s reactions, Reiner was prepared.
<i>Ten years ago, no one could ept the existence of imaginary numbers. How could people’s views change so easily ten yearster?</i>
He stood up and spoke,
“Everyone, you must be puzzled by the imaginary numbers in this thesis, thinking they are meaningless in the real world.”
“Aren’t they?” Roger retorted sarcastically. “If I have one apple, that’s normal and reasonable. But if I say I have i apples, what exactly do I have in my hand, Reiner? Can you show me?”
He used a very realistic example, earning a nod from Hannah beside him.
“Of course, I can’t show you, Lord Roger.” Reiner replied calmly to Roger’s near-mocking question.
“A number that cannot exist in reality is meaningless. Even if the theory of imaginary numbers isplete, it’s just a calction on paper. How does it rte to our real lives?”
Roger continued, seeking agreement from others. “The reason not many invest time in pure mathematics is not just theck of real-world feedback, but because pure number games can easily be meaningless theories like imaginary numbers, bing mathematics for mathematics’ sake.”
His words made everyone ponder. Indeed, if a theory couldn’t be applied to practice, no matter how ingenious and sophisticated, it was useless. The development of magic has always been about solving real problems, not exploring knowledge without practical significance.
This was exactly the pragmatism of the mage, a matter of efficiency.
Everyone looked at Reiner, waiting for his exnation.
“Lord Roger, I think I need to educate you on mathematics.” [1]
Reiner walked to the ckboard and picked up a piece of chalk.
“W-What did you say?” Roger was angered by Reiner’s attitude and wanted to stand up, but his reason restrained him.
Reiner smiled and drew a horizontal line on the ckboard, adding an arrow.
“This is a number line we often use in mathematics. Now, I’ll mark three points: -1, 0, and 1. These are integers and negative numbers, the most basic concepts even six year olds know.”
The Thesis Review Committee members watched as Reiner continued to mark the number line, adding 0.5 between 0 and 1.
“Integers can’t fill the entire x-axis because there are also decimals. Between 0 and 1, there are countless decimals. But even then, the number line isn’t filled.”
Reiner drew a square and its diagonals.
“The square root of 2, an irrational number, also exists on the number line. By now, rational and irrational numbers have filled the number line.”
After hearing Reiner, Roger spoke.
“If the number line is full, where do imaginary numbers fit? Aren’t they on this board?”
He intended to mock Reiner, but Reiner unexpectedly nodded.
“Imaginary numbers don’t fit on this line, but they at least exist on this board.”
“What?”
Even Percival questioned, looking at the number line again. Rational and irrational numbers filled itpletely, leaving no ce for imaginary numbers.
“Imaginary numbers exist here.”
Reiner marked a point above 0 on the number line.
“Are you kidding me!?”
Roger stood up, thinking Reiner was mocking him, ready to confront the young mage, but Bel stopped him.
“Let’s hear him out, Roger.”
Bel sensed something and looked at Reiner.
“The imaginary number i is one unit above the origin.”
Reiner’s words seemed strange, but Igor quickly realized it. In his mind, the number line shrank, and the entire ckboard appeared in his eyes.
“Yes, a single number line can’t represent all points on a ne, but adding another axis creates a coordinate system to represent all points.”
Igor realized the Cartesian coordinate system was already widely epted. People had long epted using two numbers to represent a point’s coordinates. The existence of imaginary numbers was akin to adding an additional axis, expanding the world a hundred times!
Such a simple concept, yet none of the mid-level mages present had thought of it before Reiner’s demonstration.
Their imagination limited their knowledge!
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T/N:
[1] Burn