[Iasc's Private Room] Understanding is the Key

[Iasc]

22 Bells, 2nd Autumn, 510 AV

The room was small but, only just, adequate. It had the basics of what he needed and nothing else. “Do I need anything else, though. Maybe a window or some more space.” Apart from his bed, his desk (table, actually) was where he spent most of his time, pouring over his books. The current book he was reading was a battered, old thing which had a list of the ten axioms of mathematics and a very brief explanation of them. It didn't even explain all of them, just the easy ones. It truly was a disaster of a book but beggars can't be choosers and, in this case, Iasc was a beggar.
He knew that these axiom's didn't need to be proven but he wanted to understand the logic behind them.

“1 - The whole is greater than the part.”
This one wasn't hard. Basic numerical skills will tell you that. He demonstrated it just to prove to himself that he did know it. After all, if something works theoretically it doesn't always work practically. He picked up four of his arrows. “I shall call this the whole.” He split them into one group of one and one group of three. “These are the parts. Both are less than the whole as one is less then three is less then four.”

A good start. He moved on to the next one without much celebration.

“2 - If equals are added to equals, then the wholes are equal.”
“This is another simple problem of addition.” He decided not to use arrows this time but to do it theoretically. He drew a straight line and put the letters A and C on either endpoints. “By axion four a straight line can be drawn between any two points.” He then put a point B about halfway between the endpoints. “AB + BC = AC.”

Not as easy as axion one but still no cause for celebration.

“3 - If equals are subtracted from equals, then the remainders are equal.”
He ignored this one as being a rearranging of axiom 2.

“4 - A straight line can be drawn between any two points.”
He had already shown this in axiom two. He circled it as if to prove to himself that he wasn't cheating.

“5 - A line segment can be extended indefinitely.”
He was still trying to get his head around this one. “Indefinitely.” The word bounced around his consciousness, always out of reach. He knew what the word meant but it seemed to be just beyond his comprehension. “How can something have no end. All things must end at some stage. Are lines an exception. If so, then there must be more exceptions.” This seemed to be more philosophy than maths to him and he resolved to tackle it at some point, but not now.
“Don't worry if you don't understand it now”, Sador, his uncle and mentor, had reassured him. “It will make itself clear to you when you are ready.”
He had no reason to distrust his uncle.

“7 - Things that equal the same thing also equal one another.”
He used the same logic as axiom two for this. He redrew the line from that axiom. He then extended both endpoints a small bit (as he did this he considered trying to extend it indefinitely but dismissed the idea as impossible.) He assigned the letters D and E to the new endpoints and made A the origin.
His original equation, AB + BC = AC, still stood. On top of this he now had a new equation, namely, AD + AE = AC. Therefore, by axiom seven, AB + BC = AD + AE.


He had looked at more then half of the axioms now. He had analyzed all of the algebraic ones. Algebra was all he really knew. From trigonometry he knew a little of right angles but not enough to proceed. He reluctantly closed his book and headed toward his bed.

[Iasc]

The sun was shining today, with not a cloud in the sky. Or at least he assumed that was what the day was bringing. Being in the bowels of the castle it was hard to tell. He felt very good anyway, whatever the weather.

There was a knock at the door. “Come in”, Iasc called, still lying in bed looking at the ceiling. He heard the door open and footsteps cross the distance between door and bed. His view of the ceiling was obscured by the gruff appearance of his uncle, his black beard hovering just above Iasc's face.

“Sador”, he cried. “How good to see you this morning. How are you?”
“I feel great”, he said, a hearty laugh emanating from his lungs.
“What brings you here?”
“We have work to do.” Sador was serious now. “I wish to explain the last remaining axioms to you. You shall begin your forays into trigonometry soon and we need you to have the basics.”

He moved towards the table where Iasc had done his work the night before.
“This is some good work. You seem to know enough to proceed.”
Iasc followed him to the table and sat on a chair. “Can I have some food first?”, he asked hopefully. His stomach beginning to rumble.
Sador produced a shiny red apple from his pocket and gave it to Iasc who took it greedily and started eating.

When breakfast was over, Sador took out another sheet and wrote the heading:

“6 - All right angles are congruent.”
“You know what right angles are.” Sador traced a right angle. “The word congruent means similar.” He drew another right angle, putting the number two beside it and number one beside the first one. “Alright so far?”
Iasc nodded that he understood so far.
"The definition of a right angle is that they both measure ninety degrees. Therefore, angle one is equal to angle two. Therefore, the are similar. Therefore, they are congruent.” Direct and to the point. The way Sador usually operated.

“That's it?”
“That's it”, Sador replied with a smile.
Iasc was very suprised. He had imagined something much harder. “It's the word congruent that put me off”, he claimed with a smile.


“Quite right. Now onto the next one.”

“8 - Things that coincide with one another equal one another.”
Sador drew a line and called the endpoints A and C, respectively. He then put a letter B in between the two points. This was not unlike what Iasc had done the previous night.
“In this figure, AC coincides with AB + BC. By axiom eight things which coincide with one another are equal to one another. Therefore, these things are equal." Sador scratched his head "This is not really an axiom that can be shown to be true, it's just a statement that is true.”
Iasc didn't like the explanation. He preferred to see things clearly make sense before his eyes.
“You may not understand it now but think about for a while and it will make sense to you.”
Iasc nodded in approval, although it was already starting to make sense.


“We will skip axiom nine as it deals with circle which you will not deal with for a while yet. Let's proceed straight the last axiom. This one is off putting even when you see the image but don't despair.”

“10 - If a pair of lines are drawn which intersect a third in such a way that the sum of their interior angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side upon sufficient extension.”
Sador drew an image:

He marked the angles A and B.
Iasc looked bewildered as he tried to make sense of the image.

“The axiom states that if A + B, the sum of the interior angles, is less than two right angles, that is 180 degrees, the two lines intersect.” He pointed to where the lines intersect.

Iasc stared at the drawing. “A + B is less than two right angles.” He traced the two lines with his fingers down to where they meet. He took his time thinking about this. Sador didn't say a word. He prefered his student to think himself. Better for the mind that way.

“If both angles are less than ninety degrees”, he mused aloud. “Then they will clearly intersect. If only one is less than ninety degrees, then that angle is sufficiently small that it must intersect with other line.”
He looked up at Sador hopefully, who was beaming. “Very good m'boy. Very good indeed. You have come a long way. I'm very proud of you.”

Iasc was proud of himself as well. "I am closer than ever to learning the language of mathematics", he said to himself, sitting back on his chair, dreaming of things to come.

[Leviathan]

Thread Award! Iasc Exp +3 Mathematics, +1 Observation Lore N/A Additional Comments Sorry it took me so long to get this graded. I'm very busy with life. If you have questions, comments, or concerns, feel free to PM me. Nice job and keep up the good writing.