Disclaimer: Same as before.
To all readers: WOW you're still with me! I heart you corageous folk. Now on with the story!
Chapter One
Stella came to, feeling as though she had been run over by a train. She was sprawled out on her back, lying on some sort of soft surface. A sun shone down and warmed her face, while birds chirped overhead. Slowly, she opened her eyes, and at once was met by a bright robin's-egg-blue sky. A lush forest surrounded her, and the ground was covered by vivid, emerald green grass. A soft breeze ruffled the leaves on the trees, making them sound as if they were whispering amongst themselves in a language meant for only tree ears alone. The atmosphere was extremely peaceful, and for a few moments Stella tried to remember how she came by the place. A flashback of a roaring wind, a dark window, and a leather book went through her mind as her memory returned. Stella groaned and placed a hand on her head. Oh how it hurt! Oh how confused she was!
An outburst of jovial laughter broke through her muddled thoughts, and she sat up, startled. A man dressed in eighteenth century clothing glanced down at her from his position near a table laden with papers, pencils, and calculators. A whiteboard stood on his other side, scrawled with many different math symbols, which all appeared to be hastily drawn and very advanced math.
"I must admit, it is a very strange sight to see a girl appear out of air, thought I have seen stranger things." He spoke, with a deep, lightly accented voice. "Tell me how it is that you come by this remote location?"
"I-I'm not quite sure." Stella managed to say, still feeling quite shaken about her past ordeals. "I opened a book, and then a whirling wind filled toe room, and then I found myself here."
"Well, if you are here, that must mean that you are no longer where you started, correct?"
Stella nodded. "I am most definitely not in my Maths classroom."
"Well then! We can use inductive reasoning to find out what has occurred!" The man exclaimed, with a delighted smile.
Stella glanced at him, perplexed. "I beg your pardon? What is inductive reasoning? How will it help my predicament?"
"All will be explained! But first, I would like a formal introduction. My name is Goldbach. What is yours, young lady?"
"My name is Stella. Where exactly am I?"
"That can also be explained by logic!" The man gestured at the paper-strewn table. "And, I cannot be modest at all when I confess I am quite familiar with logic."
"How so?" Stella asked, intrigued by the strange man.
"Well, for many years now I have been working on a certain math problem. It is entitled Goldbach's conjecture. But let's save that for another time."
"Alright," Stella agreed.
"Now, I am going to teach you how to use inductive reasoning. Not only can you use this technique in finding out your location, but in many other everyday problems. Inductive reasoning is also very important in geometry."
"How can logic be important for math?"
"In order to prove that objects are really what they seem to be. It also has other uses, but we won't worry about those for now."
"Oh."
Goldbach gestured towards one of two stools near the whiteboard. "Would you like to take a seat?"
"Thank you," Stella said as she seated herself in one stool. Goldbach erased the scribblings on the whiteboard and produced a blue whiteboard marker from somewhere inside of his brown overcoat. Uncapping it, he turned to Stella.
"Part of the inductive reasoning is looking for patterns and making conjectures. A conjecture is an unproven statement that is based on observations. Do you have a conjecture for your present situation, Stella?"
"Well, I suppose I could say that I'm inside a book."
"Good. Now, a counterexample is an example that shows a conjecture is false. It is used to help prove conjectures."
"How do you prove that conjectures are true by proving them false?"
"Because you need to be able to prove a conjecture is true in all cases. If you find a counterexample, it means that it is not true in all cases."
"Oh, I see."
"Conjectures also don't have to be only right or wrong. Some are not known to be either. They are called unproven. Goldbach's conjecture, the math problem I am working on, is an unproven conjecture,"
"How interesting!"
"I know of a method you are probably familiar with using, but not in math. Do you know what a conditional statement is?
"Oh of course! We use those in science all the time. It is a sentence in two parts, and contains a hypothesis and conclusion. A conditional statement is always written in if-then form, where the 'if' represents the hypothesis, and the 'then' represents the conclusion. (How does the book do it?)"
"Excellent. I couldn't have said it better myself. Anyhow, a conditional sentence can be used in math." Goldbach turned to the whiteboard and drew the letter P. "I am writing the letter P up here to represent the 'if' part of the statement." He turned and wrote the letter Q in a space near the letter P. "The letter Q will represent the 'then' part of the statement." He drew an arrow from P, pointing to Q. "The arrow symbolifies that P implies Q. Now, if the arrow were to be reversed…" Gold berg erased the arrow and drew a new one, from Q to P "…then the statement would read Q implies P. This is also called the converse. Goldbach paused. "Are you getting all of this so far?"
"Yes." Stella nodded.
"Wonderful! Let's go on! Where was I…?"
"We were talking about converses."
"Ah yes. Now, the inverse is formed by negating the original statement. So, where it was P implies Q, it is now the negative of P implies the negative of Q." Goldbach erased the arrow and changed it back to what it was originally, so that P implied Q. "The symbol for negation looks like a little squiggly line, so when I draw one in front of P and one in front of Q, we get the inverse!" Goldbach drew two symbols, one in front of each letter. "Math can incorporate these statements easily. Let's say P represents 'If 2+n5', while Q represents 'then n would equal 3'. What would the three statements be?"
"Well, the conditional statement would be: If 2+n5, then n equal would equal 3. The converse of the statement would be: if n equals 3, then 2+n5. The inverse would be: if 2+n(does not)5, then n does not equal 3. Am I right?"
"Perfect!" Goldbach beamed. "You'll make a good mathematician some day, Stella. Now, moving on. A contrapositive statement combines the inverse and the converse together into one statement."
"Do you mean it reverses the hypothesis and the conclusion, as well as negating them?"
"Exactly! When we say it, the contrapositive would be negative Q implies negative P. Using the statement before, what would the contrapositive be?
"If n does not equal 3, then 2+n(does not)5."
"Correct! You should also remember that those statements don't necessarily have to be all true. In fact, it isn't expected for both an inverse and a contrapositive to be correct. However, the Inverse and the converse are expected to be incorrect, and the original statement and the contrapositive are expected to be true. When two statements are both true or both false, they are called equivalent statements. Now, we must drift away from conditional statements and learn about something new: biconditional statements!"
"What are those?"
"A biconditional statement is a phrase written as 'If and only if.'"
"How is that different from a conditional statement?"
"A biconditional statement states that something must be true."
"I see. So if we used the same phrases from the earlier conditional statement and plug it in here, we would get: 2+n5 if and only if n equals three, yes?"
"Correct!"
"But wouldn't we also say that it is similar to a converse?"
"Yes! That is what a biconditional statement is: equivalent to both a conditional statement and its converse."
"Oh, I see!"
"Well, you seem to understand inductive reasoning very well, so now I am going to introduce you to deductive reasoning."
"That term sounds familiar."
"Is should: you do it every day!"
"I do?"
"Yes! Deductive reasoning uses facts, definitions, and accepted properties in a logical order to write a logical statement."
"How does that differ from inductive reasoning?"
"Inductive reasoning uses previous examples and patterns to form conjectures."
"Ah! I understand."
"Wonderful! Now, there are two laws included in deductive reasoning."
"Laws? You mean like rules?"
"Yes! They are called the Law of Detachment and the Law of Syllogism."
"What do they say?"
"The Law of Detachment says that if 'P implies Q' is a true conditional statement, then both P and Q are true. The Law of Syllogism states that if 'P implies Q' and 'Q implies R' are--"
"Wait a second. What is R?"
"R is a third term. For example, we could say: if the sun is shining, then it is a beautiful day. That is P implies Q. Then we could go on to say: if it is a beautiful day, then we will have a picnic. That is Q implies R."
"Ah! I see now! Please, go on."
"All right. The Law of Syllogism states that if 'P implies Q' and 'Q implies R' are both true conditional statements, then 'P implies R' is also true."
"So it would be like saying if the sun is shining, then we will have a picnic?"
"Exactly! Very good, Stella. You did a wonderful job."
"Thank you. Does that mean you are done teaching me?"
"I'm afraid that's all I can teach you."
"But I still don't know where I am!"
"In Geomno, of course!"
"What is Geomno?"
"Geomno is the land in which you are in!"
"But how did I get here?"
Goldbach shrugged. "I don't know. It's not everyday a girl suddenly appears out of thin air, right under my nose! You might try asking someone else. I have some friends down the road." He pointed towards a well-worn dirt path that Stella had overlooked earlier. It led out of the clearing and deep into the woods. "You could try asking them."
"Oh, alright then! Thank you very much, Mr. Goldbach!"
"You're welcome, Stella. I hope you find what you're looking for."
"I hope so too! Goodbye!"
"Goodbye. May we meet again." Goldbach waved at her before turning back to his whiteboard. Stella turned around and set off down the road.
