The Good Teacher

A/N: Try this link (https://www.desmos.com/calculator/mexr42hgc8) to follow the mathematics explained in this chapter.

I use LaTeX conventions to define the mathematic formulae here. It may not make sense on your mobile device or computer. You can find the properly formatted derivation at the Discord Server.

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Guy somehow got used to the convenience brought on by The Church, especially in pedagogy. Although the RoK was an insane tool that offered its own benefits, they were generally self-serving in nature. Guy could use it to refer back to his past knowledge and he could use the computer to run simulations or programs, but these functionalities could easily become redundant. With increased strength and cultivation, Guy could memorise and recollect vast quantities of information at a moment's notice. This also extended to running simulations or programs, as it was possible to segment mental operations at higher realms of cultivation.

The only function of the RoK that could scale and have potentially game-breaking ramifications would be the printing functionality. Guy could only print out one copy of books from the RoK that exceeded the similarity index limit. For instance, even if Guy had a copy of some rare cultivation manual in the RoK, he could only produce a singular printed copy of it. However, if he really wanted to make a profit out of that, he could commission a printing company to make infinite copies out of that replica obtained from his RoK, since these repeat copies wouldn't be moderated by Mast's powers. (Only printed replicas of books from Guy's original world are moderated. Only a single hard copy of them can exist outside the RoK - they cannot be reprinted physically).

Guy could do that, but he chose not to. For one, it would be highly immoral and disrespectful to break secrecy especially if it wasn't in his authority to divulge it. Secondly, doing so would inevitably draw the attention of interested parties who may be beyond his level of strength. Finally, Guy just wasn't that ambitious.

In the end, Guy found himself quickly growing out of the RoK. The Church, on the other hand, was becoming more and more relevant to him. It was primarily because its functions and uses extended outwards. As a teacher in a world that was sorely regressed technology-wise, Guy was constantly wanting for tools and materials that could improve his student's level of assimilation. Most of these resources could be reproduced using available resources through Guy's experience teaching in villages and remote regions. However, as Guy began covering harder and more advanced topics with his students, the available resources were quickly becoming obsolete and insufficient.

In this predicament, The Church was to Guy, as a bladder of water was to a thirsting man stranded in the middle of a desert. It was a remote teaching tool, a VR simulator, a student progress tracker, and a self-cultivation tool all wrapped into one convenient, albeit potentially dangerous, package.

At first, Guy had his apprehension and aversion with The Church, given his early experiences accessing it with the use of the mask. But after finding another pathway, and one that did not require him to suppress his rationality for unabashed zeal, he was getting more enthused to it. In fact, in the past months, Guy had used The Church multiple times to augment some of his lessons. Although his level of mastery with The Church's facilities wasn't as superior as the "other guy's", they did the job and netted him a small boost in cultivation along the way.

This time, Guy intended to use The Church to graphically display the effect of applying the Taylor Series approximation to functions.

The group were currently standing in a greyish room, and projected in front of them was a pair of axes representing the Cartesian x-y plane.

"For this exercise, we will approximate the function:"

$$

f(x) = e^x

$$

As Guy voiced out this command, a line started to form on the x-y plane that closely resembled the exponential function.

"We can prove that this line, in fact, represents the exponential function by substituting values for the independent variable."

Guy gazed meaningfully at his students, who nodded affirmatively to indicate that proof wasn't necessary.

"We can choose the point from where the approximation can be initiated, that is $x_0$. To make our lives easier, let's take $x_0=0$. Thus, we can calculate the coefficients for the Taylor Series:"

$$

f'(x_0) = e^0 = 1

$$

$$

f''(x_0) = e^0 = 1

$$

$$

f'''(x_0) = e^0 = 1

$$

"As you already know, the exponential function with the natural base is special in that when you derive it, you get the same function. By now substituting these coefficients into the series, we get:"

$$

f(x) = 1 + x + frac[1][2]x^2 + frac[1][6]x^3 + frac[1][24]x^4 + dots

$$

"If we take each term in this series, and treat them as separate functions, it looks something like this:"

On the plane before them, the line representing the exponential function slowly regressed into the background. In its place, the following functions started to materialise with different colours.

$$

f(x) = 1

$$

$$

f(x) = x

$$

$$

f(x) = frac[1][2]x^2

$$

$$

f(x) = frac[1][6]x^2

$$

$$

f(x) = frac[1][24]x^2

$$

"Now, let us start superposing these functions one at a time and see how they compare against the original function."

As the functions were added incrementally, the line started to get closer and closer to the faded exponential function in the background.

"Wow!" Markus exclaimed. "This works with every function?"

"It does," Marie chimed in confidently.

"How do you know?" Markus retorted doubtfully. "We just learned this."

"It's not my fault that you aren't bright", Marie clicked her tongue in annoyance and derided Markus. "I already tried it out mentally, and it works with every function I can think of."

"You did all of that in your head?" Guy commented in surprise.

Marie smiled back sheepishly and answered, "I devised a method to visualise and execute functions mentally. It's what I used back when I tried to observe the waves of fate."

"So Teacher Larks, how does this connect to your previous point about representing functions as a weighted summation of sinusoids?" Marie proposed.

"That is the Fourier Series," Guy clarified.

"What is Fourier?" Marie tried.

"That's the name of the mathematician who developed it," Guy answered.

"He must be a great Mage to be able to create such a method. Just like Euler and Taylor," Marie commented with an awestruck expression.

"Sure..."

"Anyways, unlike with the Taylor Series, the Fourier Series and the subsequent Fourier Transform, which is what will help you in your current predicament, necessitate a lot- A LOT of prerequisites. We will have to barrel through a plethora of topics such as complex numbers, integration..." Guy wore a distressed expression before inquiring, "Are you sure you want to go through with this? We will be rushing through them. It will be intense!"

Marie shrugged nonchalantly and said, "If it is necessary. Besides, I think it will be fun!"

"Of course, you'd think that," Guy chuckled. "That leaves you, Markus-"

"I'm going to sit out of this one," Markus offered immediately with a wry smile. "I can anticipate a roadblock incoming. I'm planning to take it slow and digest what I've just learned."

Guy nodded with a sympathetic smile, "Don't beat yourself up over this. It has a lot to do with how easily people perceive certain truths and concepts. You find it easier to relate to phenomena that can be observed physically, while Marie finds it easier to assimilate abstract theories."

Markus shrugged understandingly. "Don't worry, Master. I'm not affected. Understanding your limitations is the first step to development, after all."

Upon saying that, Markus' ethereal figure turned static and regressed to his seat on the congregation.

"Now," Guy declared to his remaining audience member. "Let's start with complex numbers, then."

Complex numbers are an interesting development in mathematics as they comprise of a real and a rather unfathomable "imaginary" component. They stem from the idea of square-rooting a negative number. As known, it is impossible to obtain a root of a negative number as the converse isn't possible: it is impossible to obtain a negative number by squaring another number. A mathematician in the 1500s conceived this idea of imaginary numbers when he hit a brick wall in trying to solve a cubic equation non-graphically. By introducing the imaginary component 'i' into the fold, his problem could be overcome resulting in, on-paper, a viable method to obtain the solution.

At that time, the use cases for complex numbers were limited. However, De Moivre and Euler in the 1700s realised that complex numbers and the so-called imaginary value 'i' could also assist in solving complicated trigonometric and exponential problems. This brought to attention an interesting conceptualisation of what a complex number could mean. Where before, it was simply a placeholder to ease mathematics, it now had a graphical explanation.

Complex numbers could be described as a transform operation in two-dimensional space. This includes both scaling and rotation. Essentially, a complex number $3+2i$ ('3' is the real, and '2' is the imaginary component) could be described as a scaling operation of magnitude $sqrt[13]$ $(3^2 + 2^2 = 13)$ and rotation of approximately $33.7 degree$ $(tan^[-1]frac[2][3])$.

"Make sure to keep this conceptualisation in mind," Guy highlighted as a projected image in front of him visually presented this explanation. "It will come into use when we finally touch upon the Fourier Series and the Fourier Transform."