Reiner opened the paper and saw Siris's handwriting.

Dignified and meticulous, at least it can be seen that the author of this paper takes the paper extremely seriously.

As the title suggests, the paper explores the possibility of integrating various forms of motion into a single equation, and at the beginning of the paper, he lists all known forms of equations of motion and some of the things that have been integrated by predecessors.

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 curvilinear motion.

The paper took this as a starting point and began to investigate whether curvilinear motion can be integrated.

Cialis first calculated the equations of motion on a concave surface, and then on a convex surface, and integrated them into a similar form, and he found that the two equations could be reduced to the same form, and when one of the eigenvalues was zero, the equation became an equation of linear motion!

This may seem like a startling finding, but with it comes problems.

The difference between the two surface equations is only signed in one place, one of which is a positive sign and the other is a negative sign, which is actually very easy to explain in connection with reality, after all, the two motions seem to be diametrically opposed mirror motions.

But this minus sign appears in the open root sign.

This means that in order for a formula to work, a negative number must be rooted, which is unprecedented in mathematical rules.

Even an ordinary magic apprentice can tell that there is no way to open the square root of a negative number, and this formula is obviously wrong.

Many mages in the past may have deduced this point, but seeing that there was a mathematical irrationality, they stopped their exploration, believing that the unification of kinematic equations was impossible.

But Siris's stubborn head did not give up, he pondered and meditated, and in order to continue the interpretation, he came up with a concept instead.

Since there is no way to open the square root of a negative number, then design a number, which is convenient to be a negative number!

Siris defines a number i, i^2=-1, that is, i is squared to -1 and the square root of -1 is i.

He named this number an imaginary number, which corresponds to a number that actually exists, which is a number that exists in hypothetical.

After obtaining the concept of imaginary numbers, Cialis followed by the derivation of curvilinear and linear equations, as well as circular motion and simple harmonic vibrations, and, in the process of derivation, Cialis discovered that trigonometric functions can be transformed into exponential forms in a sense using imaginary numbers.

Ciris spent a lot of space, exhausted all means, and finally came up with a formula.

Reiner flipped a page, and after the long paragraph of proof on the previous page, it was unusually concise.

There is only one formula.

e^πi+1=0。

This formula contains the engineering base, pi, 1 and 0, plus and equal signs, and the imaginary number i.

It seems so simple and elegant, as if the whole math is contained in it.

Reiner knows that this formula is called Euler's formula on earth, also known as God's formula, and 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 for normal people.

One apple and two apples, people can clearly understand that these are natural numbers, and the negative numbers derived from them are also well understood, and as for irrational numbers, they can also be accurately expressed on the coordinate axis.

But imaginary numbers are different.

No one can say what i is and how to express it, and people are completely unable to understand what the meaning of this number really is.

It's as if the numbers were simply created to explain these formulas of Siris.

For the mages of this world, this is too difficult to understand.

Reiner already had a rough idea of why Vice Chancellor Portoldo's paper was "meaningless", because even if there were no imaginary numbers, the spell model could be built smoothly, at most it was just a little more troublesome, and if imaginary numbers were introduced, then many things that had been conventionalized in the past would need to be changed, and the extra theories about imaginary numbers would not have the slightest impact on the real world.

Because the imaginary number itself is a system that can exist independently.

Reiner sighed and flipped the page.

After establishing the whole system of imaginary numbers, Cialis continued to explore further, and when he studied simple harmonic vibrations, he found that any periodic motion can be regarded as the superposition of sine waves of different amplitudes and phases, just like different keys on a piano, which are combined to form different chords.

In order to explain this method, Cialis used a lot of exposition to explain, this method he named the Cialis transform, which can transform a continuous periodic function in time into a discrete function in the frequency domain, and the series that unfolds under a certain eigenvalue is called the Ciris series.

In this passage, Sirius has done his best to explore the use of imaginary numbers in the real world, but has found nothing except this mathematical transformation.

Reiner knew that imaginary numbers, while extremely important, were far ahead of their time in the current world of mathematics, and that even the simplest system of equations to describe electromagnetic fields, which could be used to describe electromagnetic fields, was only proposed this year, and ten years ago, there was no theory that could put imaginary numbers to use.

Not to mention the mathematical group theory, probability theory, series expansion, complex variable function, wave equation, quantum mechanics and other microscopic research, imaginary numbers play an important role.

As for the Siris transform, it may be more distant in the future, when mages study electromagnetic waves thoroughly, it will be possible to apply it, and at that time, some people must be exclaiming this epoch-making theory.

Cialis. Oldman's research, ahead of its time, was evaluated as "meaningless".

What an irony.

At the end of the paper, Sirius repeatedly emphasized the correctness of his proofs, and at the same time, he argued that although these theories may seem useless now, perhaps in the future, new discoveries will verify their worth.

Even in the end, the formula and the theory behind it did not find any value, Cialis writes, and the exploration of mathematics itself is its meaning.

Reiner put down the paper, his mind full of mixed feelings, at this time, Granny Hedwig's hand slowly took Reiner's hand.

"Granny Hedwig, your son's thesis, is correct."

Reiner said, sighing that if it weren't for this old man who didn't even know a word, and kept it for the sake of his son, then this paper, and the ideas contained in it, might not appear many years later.

Granny Hedwig was stunned for a long time when she heard Reiner's words, as if she had a lot to say, but she couldn't say another word, and a thousand words tossed and turned in her chest, and finally turned into a short answer.

"I know, Siris, you're right."

As the sun sets, the afterglow of the setting sun shines through the open window on Granny Hedwig's face, leaving a golden hue.

Brilliant and dazzling.