When I woke up, I was in Selene's arms. As I repositioned myself in bed and stared at the ceiling, no thought came to me except for my parents, my sisters, my life on Earth, or my university degree. But all of that is in the past, and if God has allowed me to live in this world, there must be a reason. The Bible says a man must leave his parents and unite with his wife. If my wife is in this world, I have no choice.
I got out of bed quietly, trying not to wake Selene. I covered her well because it was very cold despite it being spring. Winters here must be even harsher than in the forest. Walking on tiptoe, I left without making a sound so she could continue sleeping, as dawn hadn't come yet. I went straight to the dining room, and with a finger I signaled to the guards to keep quiet, preparing a quick breakfast for them and myself. I told them almost in a whisper that I would be going to the company, that one of them should come with me, and the other should wait for Levi at his house. The boys agreed.
When I arrived at the company, the first rays of light were just beginning to appear. The guards, already looking tired, opened the door for me, and since they were to be relieved in an hour, I sent them home.
■― Go calmly, I'll take care of everything. ―
― Ibán ― If you say so. ―
■― Will you be ready by gym time? ―
― Ibán ― Yes, sir. I just need a little rest and I'll be fresh. ―
■― Go rest. ―
The two boys left, and about half an hour later, Levi showed up, still a little sleepy. I was still preparing for today's experiments, so I was in the yard. He said nothing, just sat in his chair and started shivering.
■― It's really cold today. ―
― Levi ― You're telling me. ―
■― Do you want me to make a fire? I'll warm up quickly, but if you stay sitting there, you'll catch a cold. ―
― Levi ― I'd appreciate it. ―
■― Then let's go, help me. ―
We decided to make only one fire, saving wood, since I needed to light one for the experiments anyway. While I checked the condition of the half-filled test tubes submerged in vinegar and bleach, Pudiente appeared, looking nervous and angry with me.
― Pudiente ― I hate you. You're the worst person in the world. ―
■― What's wrong now? What have I done this time? ―
― Pudiente ― I got this letter at my house yesterday. ― He handed me the letter, already with the Marchioness's seal broken. ―
■― It's from the Marchioness. How bad could it be? ―
The letter was from Doina, the Marchioness's daughter, who wanted to have a "date" with Pudiente. In it, she said she would like to have a math competition, and if he won, she would let him invite her as his partner in next week's festival.
― Pudiente ― Do you know what kind of problem you've gotten me into? ―
■― Do you have a girlfriend or something? ―
― Pudiente ― No, but how… ―
■― I don't see any problem. You manage math on your own, right? ―
― Pudiente ― Second-order equations sometimes get tricky for me. ―
■― Then you have to use a secret formula. ―
― Pudiente ― What formula is that? ― I gestured that I had nowhere to write, and he ran off to get paper, a pen, and ink.
― Pudiente ― Just a small thing… Do you know if Neo is coming today? ―
■― No idea. ―
― Pudiente ― Just curious. ―
■― You have to use this formula, Bhaskara. It's like this: (-b ± √(b² – 4·a·c)) / (2·a), with a·x² + b·x + c = 0. It's simple: you can get either two real roots or two conjugate complex roots. ―
― Pudiente ― I don't understand that part. ― He pointed to the complex part.
■― What's the square root of –2? ―
― Pudiente ― I don't understand what a square root is. ―
■― You don't know what a square root is… wait, maybe you know it by another name. ― I drew the root symbol with the number four inside. ― This asks which number multiplied by itself gives 4? ―
― Levi ― Two. ―
■― Thanks, Levi. ― He almost instantly apologized. ― So, what if I have this equation? x² – 1 = 0. Come on, Pudiente, what's x? ―
― Pudiente ― That's easy. We add 1 to both sides of the equal sign, then I have to find a number multiplied by itself that gives 1, which in this case is 1. ―
■― I don't know what you're worrying about. Now, what's x in this equation? x² + 1 = 0 ―
The poor guy thought for a long while. I let him mull it over and turned my attention to the arbum sap.
As I suspected, mixing 25-75% is more useful, since it's somewhat elastic and requires force to stretch—but not too much. Just enough to hold the lid on the jar firmly.
First, I wanted to test its behavior under high temperatures, so with Levi's permission, I used magic to create a layer of eterana over the cauldron, forming a container that increases pressure and therefore temperature, because trapping the water vapor prevents the liquid from changing phase, so it needs higher temperature to escape the dome in the T-s diagram or to greatly increase its entropy.
I went to fetch a thermometer, and when I returned, Pudiente was about to explode. But I left him alone, tied the thermometer with a string, and put it in the water. It was calibrated from 0 to 100ºC, so I couldn't know the exact temperature, but by extrapolating the marks, I could estimate the water's temperature, which obviously exceeded 100ºC (due to the pressure).
I poured the same halves of the 25-75 mixed test tubes and waited to see what happened. But I couldn't ignore Pudiente, who looked ready to tear his hair out.
■― How's it going? ―
― Pudiente ― I'm almost there. ―
■― No matter how hard you try, that equation has no solution. ―
― Pudiente ― You're… damn it… ―
■― Hey, I haven't insulted you. ―
― Pudiente ― Sorry, but how can this be so weird? ―
■― Look, you probably did this: x² + 1 = 0, then moved it to x² = -1, and then wrote x = √-1, right? ―
― Pudiente ― That's exactly what I did, but there's no number that multiplied by itself gives -1. ―
■― There is, but it's not a real number. ―
― Pudiente ― What do you mean, not a real number? ―
■― I don't know how you learned it, but the numbers we can count on our hands are the natural numbers, like 1, 5, 8, 9. Then we have the integers, like -8, -5, 0, 1, 5. After that come rationals and irrationals, like 1/2, 1/3, Pi, or e… Those are the real numbers. Complex numbers are written as a + b·i, where i is √-1, meaning i² = -1. ―
― Pudiente ― So, if you have √-4, how do you solve it? ―
■― It's simple: you have √-4, which by the properties of roots you can write as √4·(i²), right? ― Pudiente nodded. ― That leaves us with i·√4, which is 2i. ―
― Pudiente ― What would a second-degree equation with complex roots look like? ―
■― I can't think of one right now. ― I thought for a moment. ― I think this one will work: x² – 2·x + 2 = 0. Try solving it like before. ―
― Pudiente ― I'll try. ―
The workers started arriving, each at their own pace. Meanwhile, the children passed through the yard greeting the other artisans. Today we had a surprise for them: we bought new clothes for everyone. They're average quality, not custom-made or standardized, but I think they'll love the gesture.
I left Pudiente with his tasks and went straight to continue the experiments. Today the children had 2 hours of language, 1 hour of math, and 1 hour of physical education. Four hours in total—not much, but enough to make a difference.
Regarding the experiments, I gathered all the artisans to start making the glass jars with their respective lids and my new inventions: the O-ring, made from a material similar to silicone derived from white arbum sap, and the fixing rubber, a 25-75% mix of gray and black arbum sap.
It took a while to explain how we would proceed with the new inventions and how everyone needed to stay focused and attentive.
First, we were going to help Emiliano, for which the blacksmiths would be more useful than the carpenters, who would be in the group producing rubber and other items. The children from the general group would be distributed between the two teams.
Last night, I reflected a lot amidst my scattered thoughts. I found a moment to think about the jars, and releasing all types to the market at once would be madness. So I decided to produce and sell only one type: the 1.5-liter jar, large enough to preserve almost any food.
Before starting production, the artisans made steel molds: 3 molds for the glass jars with lids, 5 molds for the O-rings, and 10 for the elastic rubbers. Then, the general group would be at Emiliano's disposal to prepare all the sand mixtures for making the glass. When I finished, I noticed Pudiente was waiting for me.
■― Do you have it ready? ―
― Pudiente ― I think so. ― I checked his calculations, which took him a while, and not that I'm saying anything, but the paper was full of operations.
■― I see you put a lot of effort into it, but it's not entirely correct. ―
― Pudiente ― What's wrong? ―
■― Nothing, I just wanted to mess with you, hahahaha. ―
― Pudiente ― Can you help me, please? I need to review all this, and Doina is coming around 4:00 PM. ―
■― She's coming to the company? No, no, no, absolutely not, I don't want her to know anything about the glass jars. ―
― Pudiente ― Don't worry. The letter says the three of us are going to the university with her to have a math competition. ―
■― The three of us? ―
― Pudiente ― I don't know what that means, but you deserve it for writing those letters to the Marchioness. ―
The artisans were watching us, and I told them they could start working. While Marte was giving the language lesson, I dedicated myself to giving Pudiente a solid math review.
Before my teaching hour, I left Pudiente with a good list of exercises involving second-degree and higher-order equations. Compared to my students, Pudiente is at a higher level; it's clear he's the son of a merchant and received an education from a young age. Today I continued with additions and subtractions.
After my class, it was time for physical education, and since the children knew they had to go to the yard, they went without hesitation, where Ibán was waiting for them. But since I needed the yard today to make the rubber and other materials, I told Ibán to hold the Defense class in the usual classroom.
The cooks gathered in a corner of the yard, and from the smell, I could guess what we were going to eat today: a typical stew of this world, Corva.
Between preparing the food and handling the arbum sap, the yard looked like a smokehouse. Emiliano was about to prepare the fire to start glassblowing, but Neo arrived, looking tired, and with one thing and another, we didn't get a chance to start the process. By his nature, Neo started blaming the molds—both the jar molds and the rubber molds. While everyone was eating, he made grooves or ridges for the excess arbum sap, which could later be easily cut with a knife.
After lunch and with the children and their mothers gone, Neo drank the stew and got to work. (I know drinking stew isn't healthy, but I don't think it was the first time he did it; it seemed too easy… probably the same thing he did while working on his parents' farm.) Then he went to the glassblowing group, and I stayed with the rubber group.
In the little over an hour we had before Doina arrived, we managed to make more than 20 O-rings and about 44 fixing rubbers. We were running low on raw materials in both my group and Neo's, so we told Pudiente, who sent his right-hand man to shop while we waited for Doina. To our surprise, she arrived exactly on time, not a minute early or late. And that's when everything got complicated.
As soon as Doina arrived, Luca, one of the guards, approached to ask how he could help. But one of her knights pushed him, knocking him to the ground.
― Doina's Knight ― You are not worthy to be in the presence of Miss Doina. ―
Doina didn't react, as if it were normal, as if she had every right in the world to treat Luca that way. Neo, Pudiente, and I were in the yard, still working, and saw the whole scene. Neo didn't hesitate—he dropped what he was doing and went straight over.
― Doina ― How are you, Neo? ―
Neo is something else entirely; without even looking at Doina, he extended a hand to Luca, helping him up. Then he motioned for Luca to enter, and immediately signaled me to go to the door.
#●― I'm fed up with the nobility. Do you know what this one wants? ―#
#■― Do you remember "Pudiente's" love letter? Well, now she's coming for him to have a math competition. ―#
#●― Doesn't she have a home or what? Coming to bother him now. ―#
■― Hello, Miss Doina, what brings you here today? ―
― Doina ― I've come to fetch the owner of the company, Pudiente, and while I'm at it, you two as well. I want to have a math competition with the three of you. ―
#●― But this girl… I don't feel like doing calculations right now. ―#
#■― I guess we have no choice. ―#
After leaving the work half-finished, and with the workers confused, Pudiente gave orders to go home. Meanwhile, Neo and I got into the car, where we waited for Pudiente, who had closed the company.
As Pudiente and Doina got into the car, we started talking about math and problem-solving, even if some of us didn't really want to go anywhere. Upon arriving at the university, five professors and the director greeted us.
― Miriam ― What a pleasure to see you, Doina, it's been so long. ― said that harpy, as Doina got out of the car. But when she saw Neo, her tone changed. ― What are you doing here? ―
●― Calm down, I'm not here to see her for pleasure; I prefer to stay far from you. ―
― Miriam ― I see you're just as insolent as always. ― Neo made a face, showing he didn't care about the director's opinion. (And he doesn't, which is the worst part.)
― Doina ― Don't be like that with him, Director. This boy, along with the other two, will be facing the five university math teachers. ―
We had no choice but to go with them. The university was quite large, the classrooms resembled the large auditoriums of our Earth university, though in comparison, they had a combat arena and several glass domes full of plants.
Winding through the corridors, we arrived at the largest auditorium, where many students were waiting. They were talking, but when the director entered, everyone stood up in unison to greet her. And without letting her start, Doina began speaking. The director glared at her, but couldn't do anything.
― Doina ― You are the smartest students in the university. Today you will witness a battle of intellect. Who is better, the professors or some plebeians? ―
The students began shouting, "The professors!", "Get out, plebeians!", "You smell like monster crap!", and other insults.
Doina calmed the students' jeers and introduced us as "The Plebeians," explaining the rules of the contest. She would write math exercises on the board, and we had to solve them. She divided the board in two: on the left were us, and on the right were the professors. Three against five, but oh well.
She started with something easy:
1: x² = 9
2: x³ = 8
3: 2x + 5 = 11
4: x² + x = 6
5: 1/x = 2
6: (x + 1)² = 16
7: 2^x = 16
She indicated that we could start and that we had all the time we wanted. The professors copied each exercise to quickly complete the first five and leave the last two for later.
For the first exercise, x² = 9, a professor drew a large square on the board and divided it into smaller squares with sides of 1, 2, and 3. Seeing that a square with side 3 had the correct area, he marked his answer.
For the second exercise, x³ = 8, they solved it by constructing a cube with small unit cubes. By stacking them, they formed a complete cube of 8 cubic units and concluded that each side measured 2.
For 2x + 5 = 11, they drew a long line and placed segments representing x, doubling them and adding an extra segment of 5 units. Adjusting the segments until the total length reached 11, they deduced that each x measured 3.
In x² + x = 6, they drew a square of side x and an adjacent rectangle of width 1 and length x. Adjusting the dimensions until the combined area totaled 6, they concluded that x = 2.
For 1/x = 2, they divided a segment representing one unit into equal parts. By visualizing the length of each part, they determined x = 1/2.
In x² − 2 = 0, they drew a large square and marked an area equivalent to 2 units. Adjusting the side length until the area matched the total, they found x² = 2.
The exercise (x + 1)² = 16 was solved by dividing a large square into a square of side x, two rectangles of x·1, and a small square of 1·1. Adding the areas and comparing them with 16, they determined x = 3.
Finally, for 2^x = 16, they represented a series of doublings: 1, 2, 4, 8, 16. Counting each stage until reaching 16, they deduced x = 4.
Although their methods seemed rudimentary to us, each professor solved the problems consistently, using drawings, comparisons, and geometric reasoning. It was a very different approach from ours, but effective within their time and knowledge.
While the professors were solving the exercises, we were doing them too—though in our case, only Neo. As the professors started, Neo approached Doina and asked if he could do the calculations by himself. Doina didn't refuse, laughing at him as if doubting he could manage it on his own.
Neo first defined the root in a corner of the board, using the symbol √, as the inverse operation of powers: one seeks a number (called the root) which, when multiplied by itself a certain number of times (defined by the index), gives another number (called the radicand). He wrote the definition so everyone could see it.
He then began solving the exercises, writing the steps clearly on the board:
1: x² = 9 -> √(x²) = √9 -> x = 3
2: x³ = 8 -> ³√(x³) = ³√8 -> x = 2
3: 2·x + 5 = 11 -> 2·x + 5 - 5 = 11 – 5 -> 2·x = 6 -> x = 6 / 2 -> x = 3
4: x² + x = 6 -> x² + x - 6 = 0 -> (x + 3)(x - 2) = 0 -> x = -3 or x = 2
5: 1 / x = 2 -> x = 1 / 2 -> x² - 2 = 0 -> x² = 2 -> √(x²) = √2 -> x = √2
6: (x + 1)² = 16 -> x² + 2·x + 1 = 16 -> x² + 2·x + 1 - 16 = 0 -> x² + 2·x - 15 = 0 ->
x = (-2 + √64) / 2 = 3
x = (-2 - √64) / 2 = -5
7: 2^x = 16 -> log base 2 of 2^x = log base 2 of 16 -> x = 4
As expected, Neo finished first, without even filling half of our section of the board. He sat calmly, leaving his work complete.
The students remained silent, trying to understand the logic Neo had applied, but no one seemed to fully grasp it. Meanwhile, the director and Doina were talking, and it was clear that Doina was enjoying the competition.
After the professors finished, each group had to explain their calculations, and then we were allowed to debate them. They explained their calculations, which were clearly more logical than mathematical. Then it was my turn to explain Neo's calculations.
― Doina ― I think we have some winners, don't we? What do you think? ―
The students began shouting, "the professors!", "their calculations make no sense!", "they just copied from the teachers"…
― Doina ― Not only did they finish first, but they also solved the problems better. It seems that because you are nobles, you are unable to understand the true nature of numbers, because your teachers only teach you logic from centuries ago. ―
― Jur (Professor) ― But Miss Doina, most of those calculations make no sense. ―
― Doina ― Pudiente, Neo, Hunt, can you explain why your calculations do make sense? ―
■ ― Of course, but I'd like to erase the professors' part so I can explain everything better, without erasing our calculations. May I? ―
― Doina ― Go ahead. ―
■ ― First, I will define the numbers. I'll create groups. The first is called the natural numbers group. This group consists of counting numbers, like 1, 2, 3, 4, 5, 6, 7, 8, 9. ―
― Virtus (Professor) ― Those aren't the numbers. ―
■ ― You're completely right. You use base-4 numbers, but I use base-10 numbers. They're not different; both represent the same quantity. I use a drawing to represent the number 4+1 and draw it as 5. Fine. And for the number 2·4+2 I use 10, and for 2·4+3 I use 11, and so on. I have several parts—the units, which is the last number—and when I reach 9, I have no more numbers, so I add one to the next. That is, 10. ―
― Bia (Professor) ― That makes sense; some countries do that. But in base-4, as you said, your 5 would be written as 40. ―
■ ― Exactly. Continuing, the natural numbers have two properties: addition and multiplication. I won't go into details so we don't spend all night here. If we represent the naturals as this circle… ― I drew a circle with the numbers inside. ― The integers are the following. ― I drew another circle containing the first one. ― To these, we have to include the negative numbers. ―
― Or (Professor) ― That makes no sense; numbers less than nothing don't exist. ―
■ ― Mathematics doesn't always have to have a physical meaning, but in this case, it does. ― I told Neo and Pudiente to take a piece of chalk and stand in front of everyone. ― How many pieces of chalk does each of you have? ―
― Doina ― One. ―
■ ― And if I take Pudiente's chalk, how many pieces does Pudiente have? ―
― Bia ― None. ―
■ ― Exactly, Pudiente has 0 pieces of chalk. ―
― Or ― Okay, represent nonexistence with that drawing. We do that too, but it doesn't make sense that the boy has less than no chalk. ―
■ ― He does. Now let's say Pudiente has to give a piece of chalk to Neo, but since Pudiente has no chalk, he gives Neo a paper that says "I'm going to give you a piece of chalk." And if now I give a piece of chalk to Pudiente, Neo can claim the chalk from Pudiente. ―
― Doina ― That means Pudiente has a piece of chalk, but at the same time he doesn't. ―
■ ― I see you're understanding it. Pudiente, before I gave him the chalk, had -1 chalk. But now, adding 1, he has 0 chalk because -1 + 1 is 0. ―
― Virtus ― Can you give us another example? ―
■ ― Yes, pay attention. If Pudiente represents distance 0, that is, the origin, and we walk toward the door… ― I took large steps toward the door. ― We can see that there are 50 steps to reach the door.
● ― I think no one's lost, but what if I walk in the opposite direction of Hunt? ― Neo took 7 steps in the opposite direction. ― Look, I'm at -7 steps because I need to reach Pudiente to get to 0, and then I need to take 50 more steps to reach my friend. ―
We spent about an hour giving examples so they could understand negative numbers, also explaining the "incoherence" of multiplying something by -1. We explained that it represents a change of direction. Almost naturally, a girl asked what happens if we multiply -1 · -1, and we explained it equals 1. Then we explained why a negative number squared is also the root of a positive number.
Later we explained rational numbers they already knew them, so it wasn't difficult. When we got to irrational numbers, they didn't like representing 1/3 as 0.333…, and the problem of adding 1/3 + 1/3 + 1/3 or 0.333… + 0.333… + 0.333… = 1, but they reluctantly accepted it. Then we moved to real numbers, as the set of all previous groups. Between one thing and another, we spent more than two additional hours explaining.
― Doina ― With all this knowledge, are you able to solve an equation that has no solution? ―
● ― I think so. Which equation is that? ―
Doina stood up from her chair and took a piece of chalk dramatically (how do you take chalk dramatically? By stomping as you pick it up to become the center of attention). Doina wrote the following equation x² + 1 = 0. Neo didn't hesitate to write (i). Everyone, except Pudiente and Levi, was left speechless.
― Doina ― Can you explain why i? ―
■ ― Pudiente is going to do it, he knows more. ― He looked at me with a face that didn't know where to start.
― Pudiente ― Well, this equation has a problem, because we have to find a number multiplied by itself that gives -1. But as they said before, any negative number squared gives a positive number. But if no number exists, who's stopping us from inventing an imaginary number? I can say i² = -1, so √(-4), for example, is √(4·i²), and knowing this, we can determine that this root is 2·i. ―
Pudiente's explanation wasn't perfect, but it was innovative enough to excite the whole class.
The professors started talking among themselves; the students' whispers were not quiet. It took about 20 minutes to restore silence. In those intense moments, I looked at Doina and gestured to Pudiente. She nodded while looking at him, then licked her lower lip. Pudiente made eye contact with her at that exact moment, both turning red in an instant. After a few moments, she stood in front of everyone.
― Doina ― I think the plebeians have won. Anything to add? ―
― The dearest students of Dalia. ― The blond one's name, I want to be his girlfriend. ―
Neo was about to respond, and I signaled him to stay quiet.
The girls' screams didn't stop, directed at Neo, Pudiente, and me, until a powerful presence filled the room. It was our ladies, who had arrived at some point during the discussion.
Selene and Dalia descended the central stairs, extremely angry. As they passed, the students sat down in fear. They confronted us and started scolding us.
― Selene ― Do you think this is normal…? ― I had to quiet her somehow, so I kissed her, leaning her backward. When I finished, I looked at Neo and winked repeating the action, Dalia turned as red as a tomato.
― Doina ― I want to do it too. ― She whispered, looking at Pudiente. He approached and tried, putting in all his effort, but the Marchioness had to intervene.
― Sorina ― BUT WHAT! ―