Professor Fox nodded. "Excellent answers. Both 0 and 1 possess qualities that yield the identities of other numbers in various operations. As Ms. Arbon and Ms. Rey explained, zero added or negated from any other number will result in the original number. Any number multiplied or divided by one will again result in the original number." He wrote this on the chalk board.

"Unfortunately, Ms. Rey, it is only multiplication by zero that yields zero. Division, as both you and Ms. Arbon pointed out, real division by zero is not defined. Furthermore, exponentiation by zero yields 1, and addition and subtraction by zero both yield the original number." He faced the class again and added encouragingly, "Though it is easy enough to make that mistake. Now, can anyone hazard a guess, correctly or incorrectly about any magical or supernatural qualities associated with these numbers?

" errm proffessor is it because 3 is a very used number or something 3 has more power or summat like the 3 little pigs and 3 blind mice or 3 musketeers?" luna said puttering her hand up

Rosa raised her hand. 'well, zero is nothing. but if you had 2 of something, that would be two numbers away of the zero, but still you couldn't say it was two times as much. you could say zero is an impossible number. one that we only invented to make calculating easier, but that doesn't really exist.'

Um. Had she... made that mistake? Well, by convention, x / 0 equals... Nadia shook her head. Maths. She was better at them at youngth.

Raising her hand again, "Zero (0) as representative of void/emptiness/nothingness, for example. The concept of nothing is very important: specifically, nothing as embodied in the simple circular form of the zero. The zero was invented about 1500 years ago by a group of Indian astronomers, and it was viewed with great suspicion. Sometimes it was nothing at all. Sometimes it multiplied the value of other numbers by 10. Sounding ominously magical and mysterious, it was thought at the time it would it be the work of Satan.

"Masculine number one is said to represent initiative, independence, forcefulness, unity, creation, between other things. I would hazard a guess it deals with every magic that must be controlled invidually? Though it could also be an alienation power, for example, 'the wizard and his wand are the same and one'."

Terry raises her hand. She has more of a question to ask than an answer to the professor's question. "Professor, does 1 and 0 have something to do with eternity? Is that their magical property?"

Professor Fox nodded empathetically with Luna as she spoke, "Ah, well, that is certainly an interesting theory. However, we are discussing the numbers 0 and 1. Please keep your answers on topic."

"Zero is a real number. Though originally kept as only a place keeper by the Sumerians, zero is in fact a true number, and has its own specific properties. As for your answer Ms. Rey, you are quite right. Zero represents nothingness and emptiness, though I believe it was actually discovered earlier than 1500 years ago, by Indian scholars nonetheless." He wrote this last answer on the board for the students. "Zero is indeed representative of emptiness, and as Ms. Rey stated, that was a very important concept to muggles and wizards alike when it was made. While not specifically representative of eternity, zero and one both have certain magical elements similar to the permenance associated with 7, good answer Ms. Weasley."

"As for one, Ms. Rey, you are correct, it does have a large element associated with unity. We will ge to its applications later." He said, "Can anyone else hazard a guess concerning one along the same lines as Ms. Rey?"

Erica scrambled her brains for an answer to the professor's question. Unfortunately, she was unable to prepare herself for the lesson and found herself quite short-handed at answers. Hesitantly, she raised her hand and offered up a pure guess.

"Um...all I remember about the number one in Arithmancy is what Professor Bree taught me. I remember it had something to do with leadership qualities and that lots of people who were great leaders were born with that number as their life path number. So, perhaps, it has something to do with leading in magic...I don't know."

1 (one) is a number, numeral, and glyph. It is the natural number following 0 and preceding 2. It represents a single entity. One is sometimes referred to as unity, and unit is sometimes used as an adjective in this sense. (For example, a line segment of "unit length" is a line segment of length 1.)

Some ancient Greeks did not consider one as a number. They considered it to be a unit two being the first proper number as it represented a multiplicity.

Professor Fox nodded. "Thank you both Ms. Arbon and Ms. Weasley. We will just continue. To begin with, let us examine the number zero."

"Now, as has been mentioned, zero is representative of emptiness and nothingess. It is the only number of the 10 basic numbers that represents such. This however is not a negative aspect of the number. Zero also represents change, because without emptiness, there is no niche or no room for change. Because of its mathematical properties, the number represents how pwerful emptiness and change are. Additively, the number reveals the true nature of other numbers, multiplicatively it changes all other numbers into itself. Thus, it reminds us that all numbers and all things possess the potential for nothingness."

"As a side note, 0 is frequently used in all areas of magic. It is a powerful number to use when dechiphering ancient magical spells and curses. Are there any questions?" He said, watching the class.

Before it was 'anybody else' but her, hence why she hadn't spoken, but now Nadia raised her hand, "Professor, I have heard such thing that zero is used for ciphering and dechipering - can you explain how this happens?"

"And," she added, "you mentioned zero represents change as in what it should become of things. But, can we think of something that stays in 'zero'? Or would that be 'constant change'?" She finished, not entirely sure she had been clear about her question.

Professor Fox looked to Nadia and nodded, removing his glasses before answering. "Well, deciphering and encyrption is a far more complicated subject, and one which I cannot justly address within the confines of this classroom Ms. Rey." He frowned at her second question, thinking about it a moment, then said, "I'm not entirely certain what specifically you are asking Ms. Rey. Could you perhaps give me a more specific situation? If you are asking whether it is possible to have constant change, of course it is Ms. Rey. The saying, 'the only thing that is truly permenant is change' seems appropriate when discussing the number zero. If you are asking me whether it is possible to have something remain permenant indefinatley I can only answer this Ms. Rey: all things physical that exist must inevitably decay, to attempting to prevent such requries ever increasing amounts of energy, magical or otherwise. I hope I've managed to somewhat answer your question." He said, appearing somewhat dissapointed in his own answer. "Are there any other questions before we move on?"

"You have," Nadia said, nodding and while she was lowering her hand, taking notes of the subject in question. Of course, it had been quite the obvious answer. Peculiar, then, why hadn't she thought of it? Everything is in constant movement... was it Heraclito the one to say that? No, he believed in stable things, in a kind of paradox.

She was pretty much failing to keep with her own thoughts. In fact, something seemed to ring on her head as she wrote down more notes, then finally, after watching nobody would raise her hand again, Nadia did.

"What about non psychical things. Like magic, like -" she shrugged - "the universe which is suppossed to go on forever in time. Time itself. The concept of soul, the thoughts, memory. Those type of thing, can they be thought as nothingness? Do they have part of the nothingness, or are part of the nothingness?"

Or they are the nothingness?

Kalaya listened intently taking notes on anything she wanted to. Ms. Rey as the Professor called her (she wasn't sure of her actual name) had just hit upon an interesting topic, one of which she was sure however, could be confusing if you went into to much detail. At least for Kalaya.

Professor Fox kept from giving away a small smile. The students were finally beginning to show some inquisitiveness about the subject. He lifted up an open hand as if signalling Nadia to stop. "Yes, Ms. Rey, I know this is an interesting subject, but we must confine our curiosity on it to within the realm covered by the lecture." He walked around to the blackboard and wrote on it.

"As I have written on the board, and as we have discussed, the ability of zero to change all other numbers into specific identity numbers, means that as all spells can be represented by numbers, there exist no spells which cannot be undone." Before he had even properly finished, he added, "Of course, there is a single notable exception, which we will not address here." He paused a moment, to let it sink in, then continued. "Are there any further quesitons concerning zero before we move on to one?"

Seeing as how the students did not appear to have any more questions, Professor Fox continued. "Moving along. The number one is both an identity number, in that certain instances it reveals the identity of other numbers, and a basic number similar to two. Now, keeping in mind that two represents duality and dichotomous thinking, three represents more complicated and less dualistic thinking, one similarly represents monoist thinking. The idea that all things are integratd into a cohesive whole. The idea that all things magical and non-magical, phsyical and non-physical are part of a greater whole is the essence of the number one. For this reason, monoistic thinking often proves difficult for dualistic thinking people to understand. Thus, one represents identity, unity, and interconnectedness."

"As a side note, one is often associated with centaurs, due to their ability to naturally see the world and all things within it as parts of a cohesive whole. This also helps explain their disdain for the 'categorization' that humans often need to employ, as we discussed during the last lecture." He paused and watched the class, "Are there any questions about one, its meaning, or its applications before we continue to discuss the properties of zero and one as identity numbers?"

Nadia frowned and though already had a small idea what it was, raised her hand. "Just wanted to ask, professor, we have already mentioned before but, what exactly is an identity number?"

Professor Fox turned his attention to Nadia. "Ah, well Ms. Rey, it is a good question and I will address that topic next. Are there any questions concerning the properties of one?"

Peony raised her hand as well. "Professor," she called out, "does the 'identity' number-" she had been about to add 'thingamajiggy' but Peony sensed that the uptight professor with the nose for trouble like a fox for chickens wouldn't appreciate the colloquialism. "-have anything to do with the personal, heart, and future numbers that they use in Numerology?"

Professor Fox shook his head. "I'm afraid not. As I mentioned in the first class, Professor Bree prepared everyone well in the area of Greek Arithmancy, and I will not be covering similar areas of Arithmancy this year I'm afraid, at the behest of others." He said encouragingly. "Are there any other questions concerning one specifically before I explain the importance of the identity numbers 0 and 1?"

Terry finds it interesting that centaurs are so closely related to 0 and 1. That helps explain why they find humans so distasteful. We tend to see things too much in shades of gray. She remembers from math that an identity number is a number that leaves other numbers unchanged when combined with it. 0 would be an identity number in muggle addition, and 1 would be an indentity number in muggle multiplication. "I wonder how that can be useful in magic?" Terry thinks to herself.

Patrick looked up from taking notes. He had an ink smudge on his nose but he had to focus on these notes. He need this course for NEWT so his OWL result was valid. He turned his attention back to the notes.

Xavier wasn't about to ask a question, because it would probably involve him questioning why the heck they had to learn a whole new system.

He leans back in his chair wondering to himself why there always had to be two of everything?

Erica listened attentively as the professor went over the magical qualities of both numbers. The idea of centaurs having a different perspective in life compared to humans struck her as interesting and she raised her hand once again, momentarily surprised at herself in her interest in a seemingly dull and boring subject.

"Professor, would it be possible for you to give an example of a situation and how a centaur sees things differently as compared with us humans?"

Even thought Alex had been quiet the whole time, truth was she was listening intently for the Professor's comments. And at the speed that Nadia was asking the questions, there wasn't even time for other people to ask anything, really. There was just too many things to jot down.

When hearing the Professor's question, Alex shook her head and scanned the classroom to see if anyone else had questions.

Drusilla writes down notes quite machanically, as she is still shocked about her homework points. She cannot think of anything else to ask than which one(s) weren´t correct. But she don´t think that would be a proper question to be made in the middle of the class.