Reiner opened the paper and saw Sirius' handwriting.
Upright and meticulous, at least it can be seen that the author of this paper takes the paper extremely seriously.
The content of the paper, as stated in the title, is to explore whether various forms of motion can be integrated into one equation. At the beginning of the paper, he first listed all the currently known forms of motion equations and some that have been integrated by predecessors. content.
For example, linear motion, whether it is uniform linear motion or variable speed linear motion, can be explained by an equation, but this equation is not applicable to curved motion.
The paper takes this as a starting point and begins to study whether curved motion can be integrated.
Sirius first calculated the equations of motion on a concave curved surface, and then calculated the equations of motion on a convex curved surface, integrating them into a similar form. He found that these two equations could be transformed into The same form, and when one of the eigenvalues is zero, this equation becomes an equation of linear motion!
This seems like a surprising discovery, but it raises questions.
The two surface equations only have a sign difference in one place. One of them has a positive sign and the other has a negative sign. In reality, this situation is actually easy to explain. After all, the two motions look like completely opposite mirror surfaces. sports.
But this negative sign appears in the root sign.
This means that in order for the formula to hold true, a negative number must have the root sign, which is unprecedented in mathematical rules.
Even an ordinary magic apprentice can tell that there is no way to take the square root of a negative number. This formula is obviously wrong.
Many mages in the past may have reached this point in their derivation. Seeing the mathematical irrationality, they terminated their exploration and believed that the unification of kinematic equations was impossible.
But Sirius's stubborn mind did not give up. He thought hard and came up with a concept in order to continue the interpretation.
Since there is no way to take the square root of a negative number, then design a number whose square is a negative number!
Sirius defined a number i, i^2=-1, that is, the square of i is -1, and the square root of -1 is i.
He named this number an imaginary number, which corresponds to the number that actually exists. It is a number that is assumed to exist.
After obtaining the concept of imaginary numbers, Sirius's subsequent derivation became smooth and smooth. He integrated curve equations and straight line equations, as well as circular motion and simple harmonic motion. During the derivation process, Sirius discovered that trigonometric functions are in a certain In this sense, imaginary numbers can be converted into exponential form.
Sirius spent a lot of space and exhausted all means, and finally got a formula.
Reiner turned over a page. After the long paragraphs of proof on the previous page, the content on this page was extremely concise.
There is only one formula.
e^πi+1=0.
This formula contains the engineering base, pi, 1 and 0, the plus and equal signs, and the imaginary number i.
This looks so simple and elegant, as if all mathematics is contained within it.
Reiner knew that this formula is called Euler's formula on earth, and is also known as God's formula. It can be said to be one of the most important formulas in mathematics.
But there is no doubt that the concept of imaginary numbers is extremely impactful to normal people.
People can clearly realize that one apple and two apples are natural numbers, and the negative numbers derived from them are also easy to understand. As for irrational numbers, they can also be accurately expressed on the coordinate axis.
But the imaginary numbers are different.
No one can tell what i is and how to express it. People are completely unable to understand what the meaning of this number is.
These seemed to be numbers simply created to explain Sirius's formulas.
For the mages of this world, this is too difficult to understand.
Reiner already roughly knows why Vice President Portordo commented that Sirius's paper is "meaningless", because even without imaginary numbers, the spell model can be constructed smoothly, and it is just a little more troublesome at most. And if imaginary numbers are introduced, So many things that were conventional in the past need to be changed, and the extra theory about imaginary numbers has no impact on the real world at all.
Because imaginary numbers themselves are a system that can exist independently.
Reiner sighed and turned the page.
After establishing the entire system of imaginary numbers, Sirius continued to explore in depth. When he was studying simple harmonic vibration, he discovered that any periodic motion can be regarded as a superposition of sine waves with different amplitudes and different phases, just like a piano. Different keys on the keyboard, when combined, become different chords.
Through calculation, he established a set of mathematical methods for decomposing periodic motion into the sum of countless sine waves. In order to explain this method, Sirius used a lot of discussion to explain this method. He named it the Sirius transformation. It can transform a continuous periodic function in time into a discrete function in the frequency domain, and the series expanded under a certain eigenvalue is called the Sirius series.
In this discussion, Sirius has tried his best to explore the application of imaginary numbers in the real world, but apart from this mathematical transformation method, he has found nothing.
Reiner knows that although imaginary numbers are extremely important, in the current world of mathematics, they are far ahead of the times. Even the simplest system of equations that can apply imaginary numbers to describe electromagnetic fields was only discovered this year. At that time, ten years ago, there was no theory that could make imaginary numbers useful.
Not to mention the important role that imaginary numbers play in microscopic research involving group theory, probability theory, series expansion, complex variable functions, wave equations, quantum mechanics, etc. that have not yet formed a system in mathematics.
As for the Sirius transformation, it may only be possible to apply it in the more distant future when mages have thoroughly studied electromagnetic waves. By then, some people will probably exclaim about this epoch-making theory.
Sirius Oldman's research surpassed the times, but it was evaluated as "meaningless".
What an irony.
At the end of the paper, Siris repeatedly emphasized the correctness of his proof. At the same time, he believed that although these theories may not seem to have any effect now, perhaps in the future, new discoveries will verify their value.
Even if the formula and the theory behind it fail to find any value until the end, Sirius writes that the exploration of mathematics itself is its meaning.
Reiner put down the paper, with mixed feelings in his heart. At this time, Granny Hedwig's hand slowly held Reiner's hand.
"Grandma Hedwig, your son's thesis is correct."
Reiner said, feeling very sad. If it weren't for this old man who didn't even know the words and kept it properly for his son, then this paper and the ideas contained in it might not appear until many years later.
Grandma Hedwig was stunned for a long time when she heard Reiner's words. She seemed to have many things she wanted to say, but she couldn't say a word. Thousands of words were tossed and turned in her heart, and finally she turned into a short response.
"I knew you were right, Sirius."
The sun was setting in the west, and the afterglow of the setting sun shone through the open window on the side of Granny Hedwig's face, leaving a golden glow.
Brilliant and dazzling.