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Chapter 216: Sirius’s Thesis

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Reiner opened the thesis and saw Sirius’s handwriting.

Neat and meticulous, at least it was evident that the author of this thesis treated it with utmost seriousness.

The content of the thesis, as the title suggested, explored the possibility of integrating various forms of motion into one equation. At the beginning of the thesis, Sirius listed all the known forms of motion equations and some content that his predecessors had already integrated.

For example, linear motion, whether uniform or variable, could be explained by one equation, but this equation wasn’t quite applicable for curved motion.

The thesis used this as a starting point to investigate whether curved motion could be integrated.

Sirius first calculated the motion’s equations on a concave curved surface, then calculated them on a convex curved surface, and integrated them into a similar form. He found that these two equations could actually be reduced to the same form. Moreover, when one of the characteristic values was zero, the equation became a linear motion equation!

The two curved surface equations differed only in one place where the sign was different: one was positive, the other negative. In practical terms, this situation was easy to explain; after all, two motions appeared to be mirror opposites of each other.

But this negative sign appeared inside a square root.

This meant that to make the equation valid, a square root of a negative number was necessary, which was unprecedented in mathematical rules. [1]

Even an ordinary apprentice mage could tell you that a negative number couldn’t have a square root. This formula was obviously incorrect.

Many past mages might have reached this point in their deductions, and upon seeing the mathematical inconsistency, they terminated the exploration, believing that unifying the motion equations was impossible.

But Sirius’s stubborn mind didn’t give up. He pondered deeply and, to continue the derivation, proposed a concept.

Since a negative number couldn’t have a square root, he suggested designing a number whose square would be negative!

Sirius defined a number, “i”, where i2=-1.That is, i = √-1.

He named this number “imaginary,” which compared to real existing numbers, was a hypothetical number.

After obtaining the concept of imaginary numbers, Sirius’s subsequent derivations flowed smoothly. He integrated curved equations with linear equations, circular motion with harmonic oscillation. Furthermore, during the derivation, Sirius found that trigonometric functions could be transformed into exponential form using imaginary numbers in a sense.

Sirius spent a lot of effort and exhausted all means. Eventually, he derived a formula.

Reiner turned the page. After the long proofs on the previous page, the content on this page was exceptionally concise.

There was only one formula.

eπi + 1 = 0.

This formula included the base of natural logarithms, pi, 1, 0, the plus and equal signs, and the imaginary number ‘i’.

It looked so concise and elegant, as if the entirety of mathematics was contained within it.

Reiner knew that on Earth, this formula was called Euler’s Identity, also known as the God’s formula, and it was considered one of the most important formulas in mathematics.

But undoubtedly, the concept of imaginary numbers was profoundly shocking to ordinary people.

One and two apples. People could clearly understand these as natural numbers. Negative numbers derived from these were also understandable. As for irrational numbers, they could also be accurately represented on a coordinate axis.

But imaginary numbers were different.

No one could say what ‘i’ was, or how to represent it. People couldn’t understand what significance this number had.

It was as if this number was created purely to explain Sirius’s formulas.

For the mages of this world, it was too difficult to comprehend.

Reiner already had a rough idea why Vice Principal Bodoro evaluated Sirius’s thesis as “meaningless.” Even without imaginary numbers, magical models could be constructed smoothly: at most, it would be a bit troublesome. But if imaginary numbers were introduced, then many past conventions would need to be changed. Additionally, the extra theoretical content related to imaginary numbers had absolutely no impact on the real world.

Because imaginary numbers themselves formed a system that could exist independently.

Reiner sighed and turned the page.

After establishing the entire system of imaginary numbers, Sirius continued his exploration. When he studied harmonic oscillation, he found that any periodic motion could be regarded as the superposition of sine waves with different amplitudes and phases, just like different keys on a piano combined to form different chords.

He established a mathematical method that decomposed periodic motion into the result of adding countless sine waves with different amplitudes and phases. To explain this method, Sirius used a lot of exposition. He named this method the Sirius Transformation, which could transform continuous periodic functions in the time domain into discrete functions in the frequency domain. The series expanded under a certain characteristic value was called the Sirius Series.

In this exposition, Sirius did his best to explore the application of imaginary numbers in the real world. However, apart from this mathematical transformation method, he gained nothing else.

Reiner knew that although imaginary numbers were extremely important, they far exceeded the era of this level of mathematics. Even the simplest equations that could make use of imaginary numbers, for example, describing systems like electromagnetic fields, were only proposed this year. Ten years ago, there was simply no theory that could make use of imaginary numbers.

Not to mention group theory, probability theory, series expansion, complex variables, and studies like wave equations and quantum mechanics, where imaginary numbers played a crucial role.

As for the Sirius Transformation, perhaps it would only find application in the more distant future, when mages thoroughly researched electromagnetic waves. At that time, surely someone would exclaim over this epoch-making theory.

Sirius Odman’s research transcended the era but received the evaluation of “meaningless.”

What irony.

At the end of the thesis, Sirius repeatedly emphasized the correctness of his proofs. At the same time, he believed that although these theories might seem to have no use now, perhaps in the future, new discoveries would validate their worth.

Even if in the end, this formula and the theories behind it couldn’t find any value, Sirius wrote, the significance lay in the exploration of mathematics itself.

Reiner put down the thesis, feeling a whirlwind of emotions. At this moment, Grandma Hedwig’s hand slowly grasped Reiner’s hand.

“Grandma Hedwig, your son’s thesis is correct,” Reiner exhaled with a sigh. If it weren’t for this old woman, who didn’t even recognize letters but carefully preserved it for her son, then this thesis and the ideas contained within it might not have appeared for many years to come.

Grandma Hedwig listened to Reiner’s words, stunned for a long time, as if she wanted to say something but couldn’t find the words. Thousands of words swirled in her mind, eventually condensed into a brief response.

“I knew it, Sirius, you’re right.”

As the sun set, the evening glow shone through the open window onto Grandma Hedwig’s face, leaving a golden hue.

Brilliant and dazzling.

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T/N:

[1] In the “Real number system”, the number system that most people are familiar with, a negative square root does not exist. But in other number systems like the “complex number system”, it does exist.

I don’t study uni math so please pardon me if I get any of the technical terms wrong.